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-   -   Minimal set of the strings for primes with at least two digits (https://www.mersenneforum.org/showthread.php?t=24972)

sweety439 2022-08-13 20:22

Integers b>=2 sorted by [URL="https://oeis.org/A062955"]A062955[/URL](b):

2 (1), 3 (4), 4 (6), 6 (10), 5 (16), 8 (28), 7&10 (36), 12 (44), 9 (48), 14 (78), 11 (100), 18 (102), 15 (112), 16 (120), 13 (144), 20 (152), 24 (184), 22 (210), 30 (232), 21 (240), 17 (256), 26 (300), 19&28 (324), 36 (420), 27 (468), 25 (480), 23 (484), 42 (492), 32 (496), 34 (528), 40 (624), 33 (640), 38 (666), 48 (752), 29 (784), 35 (816), 44 (860), 31 (900), 39 (912), 60 (944), 54 (954), 50 (980), 46 (990), ...

Integers b>=2 sorted by number of minimal primes (starting with b+1) base b: (not sure if 26 and 28 are before 17 and 21)

2 (1), 3 (3), 4 (5), 6 (11), 5 (22), 7 (71), 8 (75), 10 (77), 12 (106), 9 (151), 18 (549), 14 (650), 11 (1068), 15 (1284), 16 (2346~2347), 30 (2619), 13 (3195~3197), 20 (3314), 24 (3409), 22 (8003), 17 (10405~10428), 21 (13373~13395), ...

Integers b>=2 sorted by length of largest minimal prime (starting with b+1) base b:

2 (2), 3&4 (3), 6 (5), 7 (17), 10 (31), 12 (42), 5 (96), 15 (157), 8 (221), 9 (1161), 18&20 (6271), 24 (8134), 14 (19699), 22 (22003), 30 (34206), 11 (62669), ...

These three sequences are conjectured to be similar, the integers b = 7 and b = 15 for the third sequence is relatively small since they (as well as b = 3) are high-weight bases (like [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] bases R7, R15, S7, S15, they are high-weight bases), i.e. they are very "primeful", while b = 5 and b = 8 and b = 11 and b = 14 are relatively large, since they are low-weight bases (like [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL], bases == 2 mod 3 are low-weight bases), although this does not hold for b = 20, which is also == 2 mod 3

sweety439 2022-08-13 20:35

[QUOTE=sweety439;609695]Number of totally digits of minimal primes (start with b+1) in base b

Sum of all minimal primes (start with b+1) in base b

Product of all minimal primes (start with b+1) in base b

Base 2:

1 primes, totally 2 digits, [URL="http://factordb.com/index.php?id=3"]sum[/URL], [URL="http://factordb.com/index.php?id=3"]product[/URL]

Base 3:

3 primes, totally 7 digits, [URL="http://factordb.com/index.php?id=25"]sum[/URL], [URL="http://factordb.com/index.php?id=455"]product[/URL]

Base 4:

5 primes, totally 11 digits, [URL="http://factordb.com/index.php?id=77"]sum[/URL], [URL="http://factordb.com/index.php?id=205205"]product[/URL]

Base 5:

22 primes, totally 169 digits, [URL="http://factordb.com/index.php?id=1100000003799642708"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457822814"]product[/URL]

Base 6:

11 primes, totally 29 digits, [URL="http://factordb.com/index.php?id=7401"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457821560"]product[/URL]

Base 7:

71 primes, totally 288 digits, [URL="http://factordb.com/index.php?id=116315467894207"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457825324"]product[/URL]

Base 8:

75 primes, totally 523 digits, [URL="http://factordb.com/index.php?id=1100000003799644593"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002371473795"]product[/URL]

Base 9:

151 primes, totally 3004 digits, [URL="http://factordb.com/index.php?id=1100000003799645271"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003450366253"]product[/URL]

Base 10:

77 primes, totally 310 digits, [URL="http://factordb.com/index.php?id=1100000003799645582"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002370859491"]product[/URL]

Base 11:

1068 primes, totally 75414 digits, [URL="http://factordb.com/index.php?id=1100000003799646641"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003583737715"]product[/URL]

Base 12:

106 primes, totally 433 digits, [URL="http://factordb.com/index.php?id=1100000003799647067"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457818232"]product[/URL]

Base 14:

650 primes, totally 25404 digits, [URL="http://factordb.com/index.php?id=1100000003799647609"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003575953976"]product[/URL]

Base 15:

1284 primes, totally 8286 digits, [URL="http://factordb.com/index.php?id=1100000003799647942"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003588261354"]product[/URL]

Base 18:

Conjecture: the sum of all minimal primes (start with b+1) base b is always in [URL="https://oeis.org/A063538"]https://oeis.org/A063538[/URL], i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers.[/QUOTE]

Some interesting sequences: (since I have a conjecture that the sum of all minimal primes (start with b+1) base b is always in [URL="https://oeis.org/A063538"]https://oeis.org/A063538[/URL], i.e. it must have a prime factor >= its square root, I have run the "greatest prime factor ^2-1" sequences for them, while it is meaningless for running Aliquot sequences and home prime sequences for them)

Base 5:

[URL="http://factordb.com/sequences.php?se=1&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=5&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=15&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=2524354896707237777317531408904915934954260592348873615264892600018&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with sum[/URL]

Base 7:

[URL="http://factordb.com/sequences.php?se=1&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=7&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=17&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=116315467894207&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with sum[/URL]

Base 8:

[URL="http://factordb.com/sequences.php?se=1&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=8&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=18&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064067632208136437&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence start with sum[/URL]

Base 10:

[URL="http://factordb.com/sequences.php?se=1&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=10&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=20&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=5000000000000000000555857895791&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence start with sum[/URL]

Base 12: (no inverse home prime sequence available)

[URL="http://factordb.com/sequences.php?se=1&aq=295379858346459160321805517996534469599028406080898251334958136688449432694413591686670702771988538717608698226463191652828824496701353686057825770706609766177376504079608957349577470384478406844265668777103440858249254803548508509588031749392867767312098917190750874886680388900285905326905017101113635039084401476984289722816777990643065656223617398811312978513004322459271352884623281096212482243698557359065144801241770738235059141453&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=29&aq=295379858346459160321805517996534469599028406080898251334958136688449432694413591686670702771988538717608698226463191652828824496701353686057825770706609766177376504079608957349577470384478406844265668777103440858249254803548508509588031749392867767312098917190750874886680388900285905326905017101113635039084401476984289722816777990643065656223617398811312978513004322459271352884623281096212482243698557359065144801241770738235059141453&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=295379858346459160321805517996534469599028406080898251334958136688449432694413591686670702771988538717608698226463191652828824496701353686057825770706609766177376504079608957349577470384478406844265668777103440858249254803548508509588031749392867767312098917190750874886680388900285905326905017101113635039084401476984289722816777990643065656223617398811312978513004322459271352884623281096212482243698557359065144801241770738235059141453&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=705490352625379091234039122063021997484453506&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with sum[/URL]

sweety439 2022-08-13 20:40

For the interest of "greatest prime factor ^2-1" sequences:

* [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality proving[/URL]
* [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality proving[/URL]
* [URL="https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm"]P-1 integer factorization method[/URL]
* [URL="https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm"]P+1 integer factorization method[/URL]

[URL="https://oeis.org/A087713"]https://oeis.org/A087713[/URL] (greatest prime factor of p^2-1)
[URL="https://oeis.org/A024710"]https://oeis.org/A024710[/URL] (the same sequence (start with p=11) of the A087713, which is the greatest prime factor of A024702)
[URL="https://oeis.org/A024702"]https://oeis.org/A024702[/URL] ((p^2-1)/24)
[URL="https://oeis.org/A084920"]https://oeis.org/A084920[/URL] (p^2-1)
[URL="https://oeis.org/A001248"]https://oeis.org/A001248[/URL] (p^2)
[URL="https://oeis.org/A001318"]https://oeis.org/A001318[/URL] (generalized pentagonal numbers, (n^2-1)/24)


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