A340931 - OEIS

%I #5 Feb 07 2021 19:43:35

%S 2,5,8,11,17,18,20,23,31,32,41,42,44,45,47,50,59,67,68,72,73,78,80,83,

%T 92,97,98,99,103,105,109,110,114,124,125,127,128,137,149,153,157,162,

%U 164,167,168,170,174,176,179,180,182,188,191,195,197,200,207,211

%N Heinz numbers of integer partitions of odd numbers into an odd number of parts.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is a bijective correspondence between positive integers and integer partitions.

%F Intersection of A026424 and A300063.

%e The sequence of terms together with the corresponding partitions begins:

%e       2: (1)             50: (3,3,1)        109: (29)

%e       5: (3)             59: (17)           110: (5,3,1)

%e       8: (1,1,1)         67: (19)           114: (8,2,1)

%e      11: (5)             68: (7,1,1)        124: (11,1,1)

%e      17: (7)             72: (2,2,1,1,1)    125: (3,3,3)

%e      18: (2,2,1)         73: (21)           127: (31)

%e      20: (3,1,1)         78: (6,2,1)        128: (1,1,1,1,1,1,1)

%e      23: (9)             80: (3,1,1,1,1)    137: (33)

%e      31: (11)            83: (23)           149: (35)

%e      32: (1,1,1,1,1)     92: (9,1,1)        153: (7,2,2)

%e      41: (13)            97: (25)           157: (37)

%e      42: (4,2,1)         98: (4,4,1)        162: (2,2,2,2,1)

%e      44: (5,1,1)         99: (5,2,2)        164: (13,1,1)

%e      45: (3,2,2)        103: (27)           167: (39)

%e      47: (15)           105: (4,3,2)        168: (4,2,1,1,1)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],OddQ[PrimeOmega[#]]&&OddQ[Total[primeMS[#]]]&]

%Y Note: A-numbers of Heinz-number sequences are in parentheses below.

%Y These partitions are counted by A160786.

%Y The even version is A236913 (A340784).

%Y The case of where the prime indices are also odd is A300272.

%Y A000009 counts partitions into odd parts (A066208).

%Y A001222 counts prime factors.

%Y A027193 counts odd-length partitions (A026424).

%Y A047993 counts balanced partitions (A106529).

%Y A056239 adds up prime indices.

%Y A058695 counts partitions of odd numbers (A300063).

%Y A072233 counts partitions by sum and length.

%Y A112798 lists the prime indices of each positive integer.

%Y Cf. A000041, A316413, A326845, A340385, A340386, A340387, A340604, A340607, A340854, A340855.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 05 2021