A088318 The number of possible values of the squarefree kernel (A007947) shared by at least two solutions x to A056239(x) = n.

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 24, 32, 40, 51, 66, 83, 103, 128, 158, 194, 237, 288, 348, 419, 500, 601, 718, 846, 1003, 1186, 1392, 1638, 1915, 2232, 2605, 3027, 3518, 4066, 4704, 5419, 6241, 7178, 8236, 9427, 10792, 12308, 14062, 15990, 18203, 20659

Offset: 1

Author

Naohiro Nomoto, Nov 06 2003

Keywords

• nonn

Comments

Previous name: Number of different values of A007947(m_k) when A007947(m_j) = A007947(m_k) and A056239(m_j) = A056239(m_k) = n, where k>1 ( j runs from 1 to k-1 ).

Without the restriction of being shared by at least two solutions, the number of possible values of the squarefree kernel of the solutions x to A056239(x) = n is A088314(n). - Amiram Eldar, Jun 18 2025

Examples

			a(7) = 2 because there are two different values, 10 and 6:
m_1 = 50, m_2 = 80, 10 = A007947(50) = A007947(80) and A056239(50) = A056239(80) = 7.
m_1 = 54, m_2 = 72, m_3 = 96, 6 = A007947(54) = A007947(72) = A007947(96) and A056239(54) = A056239(72) = A056239(96) = 7.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..99

Crossrefs

Cf. A007947, A056239, A088314, A088850.

Programs

Mathematica

a[n_] := Count[Tally[DeleteDuplicates /@ IntegerPartitions[n]][[;; , 2]], ?(# > 1 &)]; Array[a, 50] (* _Amiram Eldar, Jun 18 2025 *)

PARI

a(n) = {my(v = List(), c = 0); forpart(p = n, listput(v, vecprod(apply(prime, Set(p))))); v = matreduce(Vec(v))[,2]; for(i = 1, #v, if(v[i] > 1, c++)); c;} \\ Amiram Eldar, Jun 18 2025

Extensions

Name changed, a(29) corrected and (30)-a(53) added by Amiram Eldar, Jun 18 2025