A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

2, 5, 6, 11, 14, 17, 18, 23, 26, 31, 35, 38, 41, 42, 47, 54, 58, 59, 65, 67, 73, 74, 78, 83, 86, 95, 97, 98, 103, 106, 109, 114, 122, 126, 127, 137, 142, 143, 145, 149, 157, 158, 162, 167, 174, 178, 179, 182, 185, 191, 197, 202, 209, 211, 214, 215, 222, 226

Offset: 1

Author

Gus Wiseman, Feb 12 2021

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only odd prime index (counting multiplicity) is the smallest.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)         54: (2,2,2,1)    109: (29)
      5: (3)         58: (10,1)       114: (8,2,1)
      6: (2,1)       59: (17)         122: (18,1)
     11: (5)         65: (6,3)        126: (4,2,2,1)
     14: (4,1)       67: (19)         127: (31)
     17: (7)         73: (21)         137: (33)
     18: (2,2,1)     74: (12,1)       142: (20,1)
     23: (9)         78: (6,2,1)      143: (6,5)
     26: (6,1)       83: (23)         145: (10,3)
     31: (11)        86: (14,1)       149: (35)
     35: (4,3)       95: (8,3)        157: (37)
     38: (8,1)       97: (25)         158: (22,1)
     41: (13)        98: (4,4,1)      162: (2,2,2,2,1)
     42: (4,2,1)    103: (27)         167: (39)
     47: (15)       106: (16,1)       174: (10,2,1)
		

Crossrefs

These partitions are counted by A035363 (shifted left once).

Terms of A340932 can be factored into elements of this sequence.

The even version is A341447.

A001222 counts prime factors.

A005408 lists odd numbers.

A026804 counts partitions whose smallest part is odd.

A027193 counts odd-length partitions, ranked by A026424.

A031368 lists odd-indexed primes.

A032742 selects largest proper divisor.

A055396 selects smallest prime index.

A056239 adds up prime indices.

A058695 counts partitions of odd numbers, ranked by A300063.

A061395 selects largest prime index.

A066207 lists numbers with all even prime indices.

A066208 lists numbers with all odd prime indices.

A112798 lists the prime indices of each positive integer.

A244991 lists numbers whose greatest prime index is odd.

A340932 lists numbers whose smallest prime index is odd.

Cf. A006141, A160786, A257991, A257992, A300272, A340604, A340607, A340854/A340855, A340933.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[First[primeMS[#]]]&&And@@EvenQ[Rest[primeMS[#]]]&]

Formula

Also numbers n > 1 such that A055396(n) is odd and A032742(n) belongs to A066207.