A195150 Number of divisors d of n such that d-1 does not divide n.

0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 3, 4, 1, 4, 1, 4, 3, 2, 3, 5, 1, 2, 3, 5, 1, 4, 1, 4, 5, 2, 1, 6, 2, 4, 3, 4, 1, 5, 3, 5, 3, 2, 1, 6, 1, 2, 5, 5, 3, 5, 1, 4, 3, 6, 1, 7, 1, 2, 5, 4, 3, 5, 1, 7, 4, 2, 1, 7, 3, 2

Offset

1,8

Comments

Define "subdivisor" of n to be the positive integer b such that b = d - 1, if d divides n and b does not divide n. For the list of subdivisors of n see A195153.

First occurrence of k is given in A173569. - Robert G. Wilson v, Sep 23 2011

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = A137921(n) - 1. - Robert G. Wilson v, Sep 23 2001

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

Example

a(24) = 4 since the divisors of 24 are 1,2,3,4,6,8,12,24, so the subdivisors of 24 are 5,7,11,23 because 6-1 = 5, 8-1 = 7, 12-1 = 11 and 24-1 = 23. Note that the positive integers 1,2,3 are not subdivisors of 24 because they are divisors of 24.

Mathematica

f[n_] := Module[{d = Divisors[n]}, Length[Select[Rest[d-1], Mod[n, #] > 0 &]]]; Table[f[n], {n, 100}] (* T. D. Noe, Sep 22 2011 *)

Prog

(Haskell)
a195150 n = length [d | d <- [3..n], mod n d == 0, mod n (d-1) /= 0]
-- Reinhard Zumkeller, Sep 23 2011

Crossrefs

Cf. A000005, A001620, A027750, A066272, A195153, A137921, A173569.

Sequence in context: A280314 A080354 A238529 * A022307 A029413 A237523

Adjacent sequences: A195147 A195148 A195149 * A195151 A195152 A195153

Keyword

nonn,easy

Author

Omar E. Pol, Sep 19 2011

Status

approved