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.
These partitions are counted by A087897.
The version for factorizations is A340101.
A001222 counts prime factors.
A056239 adds up prime indices.
A112798 lists the prime indices of each positive integer.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 15 2021
STATUS
approved