A341447 Heinz numbers of integer partitions whose only even part is the smallest.

3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317

Offset: 1

Author

Gus Wiseman, Feb 13 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 even prime index (counting multiplicity) is the smallest.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      3: (2)         77: (5,4)     165: (5,3,2)
      7: (4)         79: (22)      173: (40)
     13: (6)         89: (24)      177: (17,2)
     15: (3,2)       93: (11,2)    181: (42)
     19: (8)        101: (26)      193: (44)
     29: (10)       107: (28)      199: (46)
     33: (5,2)      113: (30)      201: (19,2)
     37: (12)       119: (7,4)     217: (11,4)
     43: (14)       123: (13,2)    219: (21,2)
     51: (7,2)      131: (32)      221: (7,6)
     53: (16)       139: (34)      223: (48)
     61: (18)       141: (15,2)    229: (50)
     69: (9,2)      151: (36)      239: (52)
     71: (20)       161: (9,4)     249: (23,2)
     75: (3,3,2)    163: (38)      251: (54)
		

Crossrefs

These partitions are counted by A087897, shifted left once.

Terms of A340933 can be factored into elements of this sequence.

The odd version is A341446.

A000009 counts partitions into odd parts, ranked by A066208.

A001222 counts prime factors.

A005843 lists even numbers.

A026804 counts partitions whose least part is odd, ranked by A340932.

A026805 counts partitions whose least part is even, ranked by A340933.

A027187 counts partitions with even length/max, ranked by A028260/A244990.

A031215 lists even-indexed primes.

A055396 selects least prime index.

A056239 adds up prime indices.

A058696 counts partitions of even numbers, ranked by A300061.

A061395 selects greatest prime index.

A066207 lists numbers with all even prime indices.

A112798 lists the prime indices of each positive integer.

Cf. A026804, A035363, A036554, A160786, A244991, A257991, A257992, A300272, A300063, A340854/A340855.

Programs

Mathematica

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