A141197 a(n) = the number of divisors of n that are each one less than a power of a prime.

1, 2, 2, 3, 1, 4, 2, 4, 2, 3, 1, 6, 1, 3, 3, 5, 1, 5, 1, 4, 3, 3, 1, 8, 1, 3, 2, 5, 1, 7, 2, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 6, 1, 4, 3, 3, 1, 10, 2, 3, 2, 5, 1, 5, 1, 6, 2, 3, 1, 10, 1, 3, 4, 5, 1, 6, 1, 3, 2, 5, 1, 11, 1, 2, 3, 3, 2, 6, 1, 8, 2, 3, 1, 9, 1, 2, 2, 6, 1, 8, 2, 4, 3, 2, 1, 11, 1, 3, 2, 5, 1, 5, 1

Offset: 1

Author

Leroy Quet, Jun 12 2008

Keywords

• nonn

Comments

A067513(n) <= a(n) <= A000005(n). [From Reinhard Zumkeller, Oct 06 2008]

a(A185208(n)) = 1. - Reinhard Zumkeller, Nov 01 2012

Examples

			The divisors of 9 are 1,3,9. 1 is one less than 2, a power of a prime. 3 is one less than 4, a power of a prime. And 9 is one less than 10, not a power of a prime. There are therefore 2 such divisors that are each one less than a power of a prime. So a(9)=2.
		

Links

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

Crossrefs

Cf. A141198.

Cf. A049073.

Programs

Haskell

a141197 = sum . map (a010055 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Nov 01 2012

Mathematica

a[n_] := Select[Divisors[n], PrimeNu[# + 1] == 1 &] // Length; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Aug 17 2013 *)
Table[DivisorSum[n, 1 &, PrimePowerQ[# + 1] &], {n, 103}] (* Michael De Vlieger, Aug 29 2017 *)

Formula

a(n) = sum (A010055(A027750(n,k)): k=1..A000005(n)). - Reinhard Zumkeller, Nov 01 2012

Extensions

Added more terms. - Steven Bi (chenhsi(AT)stanford.edu), Dec 22 2008

Added more terms (Terms 27 - 50). Steven Bi (chenhsi(AT)stanford.edu), Jan 09 2009

Corrected and extended by Ray Chandler, Jun 25 2009