A341449 Heinz numbers of integer partitions into odd parts > 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

Author

Gus Wiseman, Feb 15 2021

Keywords

• nonn

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.

Examples

			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)
		

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.

Cf. A026804 (A340932), A160786 (A340931), A340386, A340604, A340607, A340785, A340854/A340855, A341446.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&OddQ[Times@@primeMS[#]]&]