A285313 Number of pairs (a,b) such that a*b = n and d(a) = d(b) with d = A000005 and a <= b.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1

Offset: 1

Author

Michel Marcus, Apr 17 2017

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Project Euler, Problem 598: Split Divisibilities

Crossrefs

Cf. A000005, A277621 (for n!).

Programs

Mathematica

a[n_]:=Sum[Boole[d<=(n/d) && DivisorSigma[0, d] == DivisorSigma[0, n/d]], {d, Divisors[n]}]; Table[a[n], {n, 100}] (* Indranil Ghosh, Apr 18 2017 *)

PARI

a(n) = sumdiv(n, d, (d <= n/d) && (numdiv(d) == numdiv(n/d)));

Python

from sympy import divisors, divisor_count
def a(n): return sum([d<=(n/d) and divisor_count(d)==divisor_count(n/d) for d in divisors(n)]) # Indranil Ghosh, Apr 18 2017

Formula

a(p) = 0; for prime p and for an odd power of a prime.

a(p^2k) = 1, for an even power of a prime.