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A353398
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Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.
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9
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1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 6, 5, 4, 4, 6, 6, 8, 8, 13, 16, 13, 16, 18, 16, 20, 21, 27, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156
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OFFSET
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0,8
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COMMENTS
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We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
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LINKS
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EXAMPLE
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The a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
3311 711 61111 521111 5511 B11 A1111
321111 3221111 9111 721111 731111
531111 811111 33221111
3321111 5221111 422111111
22221111 43111111
42111111
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MATHEMATICA
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red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n], Times@@red/@#==Times@@Length/@Split[#]&]], {n, 0, 30}]
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CROSSREFS
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The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353399.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702.
A339095 counts partitions by product (or factorizations by sum).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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