Wikipedia:Reference desk/Archives/Mathematics/2024 February 28

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February 28

Least base b such that there is a k such that k*b^m+1 (or k*b^m-1) has a covering set of modulus 2*n

For given positive integer n, find the least base b such that there is a k such that k*b^m+1 (or k*b^m-1) has a covering set of modulus n

For even n, b = 2 if and only if (sequence A257647 in the OEIS)(n/2) and (sequence A258154 in the OEIS)(n/2) and (sequence A289110 in the OEIS)(n/2) are nonzero.

For prime n, there is a sequence (sequence A146563 in the OEIS), but unfortunately there is no OEIS sequence for all n (instead of only the prime n)

For n = 1 through n = 16, I found the sequence (the prime n terms are given by (sequence A146563 in the OEIS):

3, 14, 74, 8, 339, 9, 2601, 8, 9, 25, 32400, 5, 212574, 51, 9, 5

the 1-cover of base 3 is (2)

the 2-cover of base 14 is (3, 5)

the 3-cover of base 74 is (7, 13, 61)

the 4-cover of base 8 is (3, 5, 13)

the 6-cover of base 9 is (5, 7, 13, 73)

the 8-cover of base 8 can be (3, 5, 17, 241) or (3, 13, 17, 241)

the 9-cover of base 9 is (7, 13, 19, 37, 757)

the 10-cover of base 25 is (11, 13, 41, 71, 521, 9161)

the 12-cover of base 5 is (3, 7, 13, 31, 601)

the 14-cover of base 51 is (13, 29, 43, 71, 421, 463, 11411, 1572887)

the 15-cover of base 9 is (7, 11, 13, 31, 61, 271, 4561)

the 16-cover of base 5 is (3, 13, 17, 313, 11489)

Could you extend this sequence to n = 25 or n = 50? 218.187.66.141 (talk) 17:01, 28 February 2024 (UTC)

Smallest triangular number with prime signature the same as A025487(n)

Smallest triangular number with prime signature the same as (sequence A025487 in the OEIS)(n), or 0 if no such number exists. (there is a similar sequence (sequence A081978 in the OEIS) in OEIS)

For n = 1 through n = 26, this sequence is: (I have not confirmed the n = 15 term is 0, but it seems that it is 0, i.e. it seems that there is no triangular number of the form p^3*q^2 with p, q both primes)

1, 3, 0, 6, 0, 28, 0, 136, 66, 0, 36, 496, 276, 0, 0, 118341, 120, 0, 1631432881, 300, 8128, 210, 0, 528, 0, 29403

Is it possible to extend this sequence to n = 100? 218.187.66.141 (talk) 17:12, 28 February 2024 (UTC)