|
%I #36 Feb 14 2020 16:30:46
%S 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,
%T 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,
%U 35,1,11,56,1,48,64,1,28,44,70,1,14,1,12,42,36,42
%N 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.
%C 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).
%C 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
%C 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
%D B. E. Sagan, The Symmetric Group, Springer, 2001, New York.
%H Alois P. Heinz, <a href="/A153452/b153452.txt">Table of n, a(n) for n = 1..10000</a>
%e For n=10; (m=5; 2*5 = 10), (m=6; (6/3)*nextprime(3) = 10), hence a(10) = a(5) + a(6) = 3.
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=1, 1,
%p add(a(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Aug 09 2012
%t 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_ *)
%Y Cf. A285175.
%K easy,nonn
%O 1,6
%A _Naohiro Nomoto_, Dec 26 2008
%E New name from _Emeric Deutsch_, May 14 2015
|
