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A153452
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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.
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58
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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
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OFFSET
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1,6
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COMMENTS
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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
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REFERENCES
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B. E. Sagan, The Symmetric Group, Springer, 2001, New York.
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LINKS
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EXAMPLE
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For n=10; (m=5; 2*5 = 10), (m=6; (6/3)*nextprime(3) = 10), hence a(10) = a(5) + a(6) = 3.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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