A354991 Number of divisors d of n for which A344005(d) = A344005(n), where A344005(n) is the smallest positive integer m such that n divides m*(m+1).

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2

Offset: 1

Author

Antti Karttunen, Jun 17 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000005, A344005, A354990, A354992, A354994 (positions of 1's).

Cf. also A344590, A345935.

Programs

Mathematica

s[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[n_] := Module[{sn = s[n]}, 1 + DivisorSum[n, 1 &, # < n && s[#] == sn &]]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)

PARI

A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
A354991(n) = { my(x=A344005(n)); sumdiv(n, d, A344005(d)==x); };

Formula

a(n) = Sum_{d|n} [A344005(d) = A344005(n)], where [ ] is the Iverson bracket.

a(n) = A000005(n) - A354992(n).

a(n) <= A354990(n).