20220115, 06:44  #1 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
A new idea for OEIS "triangle read by rows" sequence
Triangle read by rows: a(m,n) = the gcd(m,n)th number k such that sigma(k)/k = m/n (i.e. the abundancy of k is m/n, note that the abundancy is always >=1 and rational number), or 0 if no such k exists, for 1<=n<=m
All positive integers appear in this triangle exactly once (only 0 appears infinitely many times), however, computing this triangle is very hard, related topics: friendly number multiply perfect number hemiperfect number This triangle begins with: 1 (1 is the first (and the only) number k such that sigma(k)/k = 1/1 = 1) 6, 0 (6 is the first number k such that sigma(k)/k = 2/1 = 2, and there is no second number k such that sigma(k)/k = 2/2 = 1, 1 is the only one such number) 120, 2, 0 (120 is the first number k such that sigma(k)/k = 3/1 = 3, 2 is the first number k such that sigma(k)/k = 3/2, and there is no third number k such that sigma(k)/k = 3/3 = 1, 1 is the only one such number) 30240, 28, 3, 0 (30240 is the first number k such that sigma(k)/k = 4/1 = 4, 28 is the second number k such that sigma(k)/k = 4/2 = 2, 3 is the first number k such that sigma(k)/k = 4/3, and there is no fourth number k such that sigma(k)/k = 4/4 = 1, 1 is the only one such number) 14182439040, 24, (unknown, there is no k <= 2^24 such that sigma(k)/k = 5/3), (unknown, there is no k <= 2^24 such that sigma(k)/k = 5/4), 0 154345556085770649600, 672 (the second k such that sigma(k)/k = 6/2 = 3), 496, 0 (2 is solitary number, thus there is no second number k such that sigma(k)/k = 3/2), 5, 0 141310897947438348259849402738485523264343544818565120000, 4320, 12, 4, (unknown), (unknown), 0 8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000, 32760, 84, 8128, 15, 0 (3 is solitary number, thus there is no second number k such that sigma(k)/k = 4/3), 7, 0 ... keywords: nonn, tabl, more, hard Last fiddled with by sweety439 on 20220115 at 06:49 
20220115, 06:46  #2 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
Problem: Prove or disprove such terms are 0:
a(5,3): Are there any k such that sigma(k)/k = 5/3? a(5,4): Are there any k such that sigma(k)/k = 5/4? a(7,5): Are there any k such that sigma(k)/k = 7/5? a(7,6): Are there any k such that sigma(k)/k = 7/6? I conjectured that no such k exists, but this can be hard to prove or disprove, like that it is hard to prove or disprove that 10 is solitary. Last fiddled with by sweety439 on 20220115 at 06:47 
20220115, 06:55  #3 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
bfile (or afile):
(rows m = 1 to m = 12 of the triangle (indices i = 1 to i = 78), flattened, row m = 1 has index i = 1, row m = 2 has index i = {2,3}, row m = 3 has index i = {4,5,6}, row m = 4 has index i = {7,8,9,10}, ...) Code:
1 1 2 6 3 0 4 120 5 2 6 0 7 30240 8 28 9 3 10 0 11 14182439040 12 24 13 0 (conjectured, bfile cannot contain conjectured terms) 14 0 (conjectured, bfile cannot contain conjectured terms) 15 0 16 154345556085770649600 17 672 18 496 19 0 20 5 21 0 22 141310897947438348259849402738485523264343544818565120000 23 4320 24 12 25 4 26 0 (conjectured, bfile cannot contain conjectured terms) 27 0 (conjectured, bfile cannot contain conjectured terms) 28 0 29 8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000 30 32760 31 84 32 8128 33 15 34 0 35 7 36 0 37 56130808183737158999998793684026231356147190822348283579122819870557664808030968216100782148452765644947099984854756332066651809002612793115408005967022213284272150201873375214629478176342119709234895003815657961417701371450048608475283004587476685222825422086715415685343739904000000000 38 8910720 39 523776 40 40 41 10 42 0 43 0 (conjectured, bfile cannot contain conjectured terms) 44 0 (conjectured, bfile cannot contain conjectured terms) 45 0 46 448565429898310924320164584477824539743733611787032214093531166213832480352545322596103983279901594851618303191976881540265700979715663129717227174064951388089181836554261404658300202164569689002163541973673164520563541918204730545126953234301917651168672330930792980648798941714800067628476201295868684534260358385663132152404556573415114464345129374402241250321888006130154014722016581041848460470899369707101530131054233224151410021638247842255328134212216267585925319281368941986654342379111273890042490862210239469135812761018153626037271326011687320220816667938885176742884376178910446080293797341901619200000000000000000000000000000 47 31998395520 48 1080 49 91963648 50 33550336 51 0 (conjectured, bfile cannot contain conjectured terms) 52 0 (conjectured, bfile cannot contain conjectured terms) 53 0 (conjectured, bfile cannot contain conjectured terms) 54 0 (conjectured, bfile cannot contain conjectured terms) 55 0 56 25185041348399291877483713498452839916966096879183629830423600245683852142136706259560800348986386756884928746427450335660601307023373038634588577582918892533431450402183724117455334595958960646443710559359553217238301822034060822235497467583505902376632937749821904538948595692372746818119489645548364913224810802818959192770727375458075135612292574997001997278799175655795461301046784886677005760822388567001630151101782815381184988668092886418826403557592465895134406328103108387886302588280426808773105723398511413080140942576327322081815502240446525869836049273257585109346805290065013209442964638084455345164451071503934096586345646621435252985421032613820965126563181089299972811779771827292133331127950543939413593680479692537947729759573681603882467419425304239282710389518990539657568601363717555733252090589049779121854544304430943422632475378859195505972332643589920186230935827402032392512358317904205719007276528974346996804088475093174195795846288979499984245887890932762691427735635608946229415529311685622909161472374878023577290544985026475478842990208730092690868098537241205864463576592480291550144216695662591474662609220161635272633460803908095561758934546539203777010617081602775055346295159952823126174561031213792909432811583524975624624923782045771007518368762761898257156254594714798330150065789591832417356833304083655085709749622056190377608624429624702581833657422942180419191308548660976118472588542171257328553869306196831264880724956685291206292203673425203402099172683799298900379846778295601867178657268395893138340588006559551079505099888372453843841958093814447626891996590684617850572626445740708184605705926892483089270269513227148864728809213020684836052682352877501319442767645251307037335411271471315001229908824788094350629237006350011614937855042347271887606679208087512719593728227162814693769216000000000000000000000000000000000000000000000000000000000000000000 57 17116004505600 58 35640 59 47616 60 0 (conjectured, bfile cannot contain conjectured terms) 61 0 (conjectured, bfile cannot contain conjectured terms) 62 0 (conjectured, bfile cannot contain conjectured terms) 63 0 (conjectured, bfile cannot contain conjectured terms) 64 0 (conjectured, bfile cannot contain conjectured terms) 65 0 (conjectured, bfile cannot contain conjectured terms) 66 0 67 (first 12perfect number, should be exist) 68 9186050031556349952000 69 2178540 70 459818240 71 30 72 8589869056 73 14 74 0 75 0 76 0 77 11 78 0 Last fiddled with by sweety439 on 20220116 at 07:43 
20220528, 06:03  #4 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
111001100010_{2} Posts 
Ideas of OEIS sequences
* Smallest base b such that n is a unique period (see unique prime) (cf. A085398):
a(n) for n = 1 to 100: 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 2, 7, 2, 5, 7, 19, 3, 2, 2, 3, 3, 2, 9, 46, 47, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2, 11, 4, 2, 6, 44, 4, 12, 2, 63, 20 * Smallest totient k > 1 such that n*k is not a totient, or 0 if no such k exists: (A301587: numbers n such that a(n) = 0) (A350085(n) = a(A007617(n))) a(n) for n = 1 to 29: 0, 0, 30, 0, 10, 0, 2, 0, 10, 110, 22, 0, 2, 22, 6, 0, 2, 0, 2, 0, 54, 22, 10, 0, 2, 22, 22, <=28^2*29^110, 6 * Infinitytouchable numbers (assuming the strong version of Goldbach conjecture is true) includes all numbers < 208 not in A005114, the first numbers which is neither in this sequence nor in A005114 are 208, 250, ...) * Smallest starting value of exactly n1 numbers with exactly n divisors, or 0 if no such number exists: (cf. A072507, A292580) a(n) for n = 2 to 11: 5, 0, 33, 0, 10093613546512321, 0, 171893, 0, 0, 0 5 should be replaced by 2 if we use "at least n1 numbers" instead of "exactly n1 numbers" a(12) <= 677667095479412562100444 (a(12) <= 247239052981730986799644 if we use "at least n1 numbers" instead of "exactly n1 numbers) If k = floor(log_2(n1)), there must be at least one term exactly divisible by 2^j for any j < k; hence the number of divisors must be divisible by j+1, or more generally by lcm_{i<=k} i. The only values of n divisible by this lcm are 1, 2, 3, 4, 6, 8, 12, 24, 60, 120 (e.g. for n = 30, there must be an element divisible by 8 but not by 16, so its number of divisors is divisible by 4, and for n = 36, there must be an element divisible by 16 but not by 32, so its number of divisors is divisible by 5), for n = 60, there must by two numbers 8k and 8(k+2) with k odd; then k and k+2 must each have 15 divisors, making them squares, thus a(n) = 0 for all n except 2, 3, 4, 6, 8, 12, 24, 120 (a(1) is not defined in this sequence), whether a(24) and a(120) exist is an open question, they exist if Schinzel's hypothesis H is true. Last fiddled with by sweety439 on 20220602 at 09:15 
20220528, 06:20  #5 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
store my text file here

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