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A341449
Heinz numbers of integer partitions into odd parts > 1.
1
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
OFFSET
1,2
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[#]&&OddQ[Times@@primeMS[#]]&]
CROSSREFS
Note: A-numbers of ranking sequences are in parentheses below.
Partitions with no ones are A002865 (A005408).
The case of even parts is A035363 (A066207).
These partitions are counted by A087897.
The version for factorizations is A340101.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A056239 adds up prime indices.
A078408 counts partitions with odd parts, length, and sum (A300272).
A112798 lists the prime indices of each positive integer.
A257991/A257992 count odd/even prime indices.
Sequence in context: A157847 A348934 A259548 * A314228 A314229 A314230
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 15 2021
STATUS
approved