A085497 -id:A085497 - cp's OEIS frontend

A085498 Primes p having at least one partition into distinct divisors of p + 1.

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 103, 107, 127, 131, 139, 149, 167, 179, 191, 197, 199, 223, 227, 233, 239, 251, 263, 269, 271, 293, 307, 311, 347, 349, 359, 367, 379, 383, 389, 419, 431, 439, 449, 461, 463, 467, 479, 499, 503

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2000

Subsequence of A085493.

Cf. A085496, A085497, A085499.

seqQ[p_] := Module[{d = Most[Divisors[p+1]]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, p}], p] > 0]; Select[Range[500], PrimeQ[#] && seqQ[#] &] (* Amiram Eldar, Jan 13 2020 *)
Select[Prime[Range[100]],MemberQ[Total/@Subsets[Divisors[#+1]],#]&] (* Harvey P. Dale, Oct 04 2020 *)

A085496(a(n)) > 0.

A085492 Numbers n having no partition into distinct divisors of n+1.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 13, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 37, 38, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96, 97

Offset: 1

Reinhard Zumkeller, Jul 03 2003

nonn

A085491(a(n)) = 0; complement of A085493;

A006093 is a subsequence (prime minus 1).

Cf. A085497.

A320515 Squarefree k > 1 with sigma(sigma(k)) < 2*k + 1.

Original entry on oeis.org

2, 13, 37, 43, 61, 67, 73, 97, 109, 151, 157, 163, 181, 193, 211, 229, 241, 277, 283, 313, 331, 337, 373, 397, 409, 421, 433, 457, 487, 523, 541, 547, 577, 601, 613, 631, 661, 673, 691, 709, 733, 751, 757, 787, 823, 829, 853, 877, 883, 907, 937, 997

Offset: 1

Peter Luschny, Oct 14 2018

nonn

Conjecturally a subsequence of A085497.

This conjecture is false, the first counterexample is a(113) = 2257 = 37 * 61 which is the least composite term in this sequence. - Amiram Eldar, Jun 17 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A085497, A000668.

isA320515 := n -> (n > 1) and issqrfree(n) and (sigma(sigma(n)) < 2*n+1):
select(isA320515, [$1..1000]);

Rest[Select[Range[1000], SquareFreeQ[#] && DivisorSigma[1, DivisorSigma[1, #]] < 2*# + 1 &]] (* Vaclav Kotesovec, Oct 14 2018 *)

isok(n) = (n>1) && issquarefree(n) && (sigma(sigma(n)) < 2*n + 1); \\ Michel Marcus, Oct 14 2018

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A085498 Primes p having at least one partition into distinct divisors of p + 1.

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A085492 Numbers n having no partition into distinct divisors of n+1.

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A320515 Squarefree k > 1 with sigma(sigma(k)) < 2*k + 1.

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