A346536 a(n) is the sum of the number of ways the numbers between the n-th and (n+1)st composite squarefree numbers can be properly factored.

6, 6, 0, 15, 0, 10, 8, 8, 0, 0, 10, 0, 8, 9, 19, 12, 7, 0, 13, 15, 0, 5, 0, 18, 8, 0, 18, 12, 0, 0, 19, 4, 0, 0, 38, 8, 0, 18, 0, 13, 0, 8, 0, 23, 0, 34, 0, 12, 0, 15, 12, 0, 0, 29, 0, 32, 0, 12, 0, 19, 17, 0, 24, 9, 16, 0, 28, 0, 7, 0, 0, 11, 32, 0, 38, 0, 0, 11

Offset

1,1

Comments

This is the sum of the number of ways numbers which are between two adjacent composite squarefree numbers (A120944), can be factored with all factors greater than 1 (A001055).

Links

Table of n, a(n) for n=1..78.

Daniel Yaqubi and Madjid Mirzavaziri, Some results on ordered and unordered factorization of a positive integer, arXiv:1412.0392 [math.CO], 2014-2018.

Example

A120944(2)=10 and A120944(3)=14 are composite squarefree numbers. So a(2) = A001055(11) + A001055(12) + A001055(13) = 1 + 4 + 1 = 6.

Prog

(PARI) fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s; }
f(n) = fcnt(n, n); \\ A001055
lista(nn) = {my(sum = 0, list = List()); for(x=7, nn, if (!isprime(x) && issquarefree(x), listput(list, sum); sum = 0, sum += f(x)); ); Vec(list); } \\ Michel Marcus, Jul 24 2021

Crossrefs

Cf. A001055, A120944.

Sequence in context: A153629 A154155 A021942 * A317577 A361739 A245173

Adjacent sequences: A346533 A346534 A346535 * A346537 A346538 A346539

Keyword

nonn

Author

Ahmadreza Afi, Jul 22 2021

Status

approved