A153452 a(1) = 1; if n > 1, then a(n) = Sum_{prime q |n} a(n*q' /q), where q' = prevprime(q) for q>2 and 2' = 1.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 3, 1, 4, 5, 1, 1, 5, 1, 6, 9, 5, 1, 4, 5, 6, 5, 10, 1, 16, 1, 1, 14, 7, 14, 9, 1, 8, 20, 10, 1, 35, 1, 15, 21, 9, 1, 5, 14, 21, 27, 21, 1, 14, 28, 20, 35, 10, 1, 35, 1, 11, 56, 1, 48, 64, 1, 28, 44, 70, 1, 14, 1, 12, 42, 36, 42

Offset: 1

Author

Naohiro Nomoto, Dec 26 2008

Keywords

• easy
• nonn

Comments

Original name was: a(1)=1, for n>1, if 2*m = n or (m/p)*nextprime(p) = n, where p is a prime factor of m ( m runs from 1 to n-1 ), then a(n) = Sum_{m} a(m).

The number of standard tableaux of the integer partition with Heinz number n (for the definition of the Heinz number of a partition see the next comment). The proof follows from Lemma 2.8.2 of the Sagan reference. Examples: (i) a(6)=2; indeed 6 = 2*3 is the Heinz number of the partition [1,2] and, obviously, the Ferrers board admits 2 standard tableaux; (ii) a(60)=35; indeed, 60 = 2*2*3*5 is the Heinz number of the partition [1,1,2,3] and the hook-lengths of its Ferrer board are 6,3,1,4,1,2,1; then, the number of standard tableaux is 7!/(6*3*4*2) = 35. - Emeric Deutsch, May 24 2015

The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition; for example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436). - Emeric Deutsch, May 24 2015

Examples

			For n=10; (m=5; 2*5 = 10), (m=6; (6/3)*nextprime(3) = 10), hence a(10) = a(5) + a(6) = 3.
		

References

• B. E. Sagan, The Symmetric Group, Springer, 2001, New York.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A285175.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
      add(a(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 09 2012

Mathematica

a[n_] := a[n] = If[n == 1, 1, Sum[a[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 04 2016, after Alois P. Heinz *)

Extensions

New name from Emeric Deutsch, May 14 2015