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Forum: sweety439 2022-01-22, 18:56
Replies: 402
Views: 29,973
Posted By sweety439
Numbers n such that Phi(n,a,b) =...

Numbers n such that Phi(n,a,b) = b^eulerphi(n)*Phi(n,a/b) is prime, "*" for Phi(n,a,b)/gcd(Phi(n,a,b),n) is prime

reference: https://oeis.org/A250201
Forum: sweety439 2022-01-22, 10:47
Replies: 18
Views: 996
Posted By sweety439
Related conjectures: 11 is the only...

Related conjectures:

11 is the only non-generalized pentagonal number which cannot be written as sum of a prime and a positive generalized pentagonal number.

besides 1 and 3, 79 is the only odd...
Forum: sweety439 2022-01-21, 21:51
Replies: 18
Views: 996
Posted By sweety439
The numbers in A347567: {1, 3, 4, 6, 10, 17,...

The numbers in A347567:

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510,...
Forum: sweety439 2022-01-21, 21:32
Replies: 18
Views: 996
Posted By sweety439
For A347567, 2*square+p, where p is either odd...

For A347567, 2*square+p, where p is either odd prime or twice an odd prime, the set is:

{1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878,...
Forum: sweety439 2022-01-21, 15:52
Replies: 282
Views: 29,587
Posted By sweety439
the last (i.e. Proving that all primes > 10...

the last (i.e. Proving that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409,...
Forum: sweety439 2022-01-21, 15:44
Replies: 282
Views: 29,587
Posted By sweety439
zipped file for the minimal primes (start with...

zipped file for the minimal primes (start with b+1) in bases 17<=b<=36
Forum: sweety439 2022-01-20, 22:18
Replies: 18
Views: 996
Posted By sweety439
The conjecture 2 in post #1...

The conjecture 2 in post #1 (https://mersenneforum.org/showpost.php?p=592782&postcount=1) (i.e. A347568 (https://oeis.org/A347568) is full with 8 terms: {1, 3, 4, 10, 14, 122, 422, 432}):

Zhi-Wei...
Forum: sweety439 2022-01-20, 10:26
Replies: 402
Views: 29,973
Posted By sweety439
https://sites.google.com/view/williams-primes ...

https://sites.google.com/view/williams-primes

https://sites.google.com/view/nexus-prime
Forum: Prime Wiki 2022-01-20, 10:21
Replies: 195
Sticky: Prime-Wiki
Views: 70,633
Posted By sweety439
Now I search the Williams primes...

Now I search the Williams primes (https://www.rieselprime.de/ziki/Williams_prime) (of all 8 kinds) to n=5000 (now done to base 45) and create a website (https://sites.google.com/view/williams-primes)...
Forum: sweety439 2022-01-18, 10:11
Replies: 1,256
Views: 88,028
Posted By sweety439
Sierpinski conjectures (k*b^n+1)/gcd(k-1,b+1):...

Sierpinski conjectures (k*b^n+1)/gcd(k-1,b+1): [b,k]

(GFN and half GFN not counted)

[2,21181]

b=2, k<=21180

[3,1187]
Forum: sweety439 2022-01-18, 10:01
Replies: 1,256
Views: 88,028
Posted By sweety439
Riesel conjectures (k*b^n-1)/gcd(k-1,b-1): [b,k] ...

Riesel conjectures (k*b^n-1)/gcd(k-1,b-1): [b,k]

[2,23669]

b=2, k<=23668

[3,1613]

b<=4, k<=1612
Forum: Aliquot Sequences 2022-01-18, 09:53
Replies: 1,027
Views: 124,855
Posted By sweety439
Is this a new record for the largest number in a...

Is this a new record for the largest number in a terminate aliquot sequence? (the largest number in this sequence is index 2789 (http://factordb.com/index.php?id=1100000001517282545))
Forum: sweety439 2022-01-17, 06:17
Replies: 282
Views: 29,587
Posted By sweety439
find the set of the minimal primes (start with...

find the set of the minimal primes (start with b+1) base b for various bases b (2<=b<=36) is the target of the project in this thread.

minimal prime (start with b+1) base b is always minimal prime...
Forum: sweety439 2022-01-16, 17:13
Replies: 0
Views: 18
Posted By sweety439
Off-topic

Like Cunningham table (https://homes.cerias.purdue.edu/~ssw/cun/index.html) to various bases, is there any interest to factor the numbers similar to Fibonacci numbers A000045...
Forum: sweety439 2022-01-16, 17:05
Replies: 18
Views: 996
Posted By sweety439
There are complement of A065377...

There are complement of A065377 (https://oeis.org/A065377) and A065397 (https://oeis.org/A065397) of the primes A000040 (https://oeis.org/A000040) in OEIS: A065376 (https://oeis.org/A065376) and...
Forum: sweety439 2022-01-16, 16:55
Replies: 18
Views: 996
Posted By sweety439
Compare with the numbers with only one way to...

Compare with the numbers with only one way to write as:

"twice a positive square number" + "odd prime or twice an odd prime"

or

"twice a positive triangular number" + "odd prime or twice an...
Forum: sweety439 2022-01-16, 11:48
Replies: 18
Views: 996
Posted By sweety439
Related research: perfect power...

Related research: perfect power (https://oeis.org/A001597) (0 and 1 are not counted as perfect power) + prime (the even prime 2 is allowed)

OEIS sequences for this: A119748...
Forum: sweety439 2022-01-15, 07:11
Replies: 282
Views: 29,587
Posted By sweety439
Conjecture: There is no base b such that the...

Conjecture: There is no base b such that the largest minimal prime (start with b+1) and the second-largest minimal prime (start with b+1) have the same number of digits in base b, note that in the...
Forum: sweety439 2022-01-15, 06:55
Replies: 2
Views: 33
Posted By sweety439
b-file (or a-file): (rows m = 1 to m = 12 of...

b-file (or a-file):

(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...
Forum: sweety439 2022-01-15, 06:46
Replies: 2
Views: 33
Posted By sweety439
Problem: Prove or disprove such terms are 0: ...

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...
Forum: sweety439 2022-01-15, 06:44
Replies: 2
Views: 33
Posted By sweety439
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 (https://oeis.org/A000203)(k)/k = m/n (i.e. the abundancy (https://mathworld.wolfram.com/Abundancy.html) of k is m/n, note...
Forum: sweety439 2022-01-14, 02:44
Replies: 18
Views: 996
Posted By sweety439
Now OEIS has sequences about these two...

Now OEIS has sequences about these two conjectures!!! A347567 (https://oeis.org/A347567) A347568 (https://oeis.org/A347568)
Forum: sweety439 2022-01-13, 11:42
Replies: 282
Views: 29,587
Posted By sweety439
all bases b<=36 are done. the original text...

all bases b<=36 are done.

the original text file for base 35 is too large (1743 KB) to upload, thus zipped it
Forum: Aliquot Sequences 2022-01-13, 08:04
Replies: 13
Views: 10,661
Posted By sweety439
No, see the graph in page 8 of...

No, see the graph in page 8 of https://arxiv.org/pdf/1610.07471.pdf, 137438691328 is also such number (untouchable perfect number, hermit number) like 28
Forum: Aliquot Sequences 2022-01-13, 08:01
Replies: 13
Views: 10,661
Posted By sweety439
I found that there are sequences about this in...

I found that there are sequences about this in OEIS:

A283152 (https://oeis.org/A283152) 2-untouchable numbers: {208, 250, 362, 396, 412, 428, 438, 452, 478, 486, 494, 508, 672, 712, 716, 772, 844,...
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