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A341449
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Heinz numbers of integer partitions into odd parts > 1.
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1
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1, 5, 11, 17, 23, 25, 31, 41, 47, 55, 59, 67, 73, 83, 85, 97, 103, 109, 115, 121, 125, 127, 137, 149, 155, 157, 167, 179, 187, 191, 197, 205, 211, 227, 233, 235, 241, 253, 257, 269, 275, 277, 283, 289, 295, 307, 313, 331, 335, 341, 347, 353, 365, 367, 379, 389
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OFFSET
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1,2
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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LINKS
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EXAMPLE
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The sequence of partitions together with their Heinz numbers begins:
1: () 97: (25) 197: (45) 307: (63)
5: (3) 103: (27) 205: (13,3) 313: (65)
11: (5) 109: (29) 211: (47) 331: (67)
17: (7) 115: (9,3) 227: (49) 335: (19,3)
23: (9) 121: (5,5) 233: (51) 341: (11,5)
25: (3,3) 125: (3,3,3) 235: (15,3) 347: (69)
31: (11) 127: (31) 241: (53) 353: (71)
41: (13) 137: (33) 253: (9,5) 365: (21,3)
47: (15) 149: (35) 257: (55) 367: (73)
55: (5,3) 155: (11,3) 269: (57) 379: (75)
59: (17) 157: (37) 275: (5,3,3) 389: (77)
67: (19) 167: (39) 277: (59) 391: (9,7)
73: (21) 179: (41) 283: (61) 401: (79)
83: (23) 187: (7,5) 289: (7,7) 415: (23,3)
85: (7,3) 191: (43) 295: (17,3) 419: (81)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[#]&&OddQ[Times@@primeMS[#]]&]
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CROSSREFS
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Note: A-numbers of ranking sequences are in parentheses below.
These partitions are counted by A087897.
The version for factorizations is A340101.
A112798 lists the prime indices of each positive integer.
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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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