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A372687
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Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.
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7
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0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157
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OFFSET
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0,7
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COMMENTS
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A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).
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LINKS
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EXAMPLE
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The a(2) = 1 through a(17) = 8 prime numbers:
2 3 5 . 17 11 19 . 257 131 73 137 97 521 4099 1031
7 13 67 41 71 263 2053 523
37 23 43 139 1033 269
29 83 193 163
53 47 149
31 101
89
79
The a(2) = 1 through a(11) = 3 strict partitions:
(2) (2,1) (3,1) . (5,1) (4,2,1) (4,3,1) . (9,1) (6,4,1)
(3,2,1) (5,2,1) (6,3,1) (8,2,1)
(7,2,1) (5,3,2,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]], {n, 0, 30}]
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CROSSREFS
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For all positive integers (not just prime) we get A000009.
Number of prime numbers p with A029931(p) = n.
Number of times n appears in A372429.
Number of rows of A372471 with sum n.
These (strict) partitions have Heinz numbers A372851.
A014499 lists binary indices of prime numbers.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.
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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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