login
Heinz numbers of integer partitions whose only even part is the smallest.
3

%I #13 Feb 14 2021 13:14:23

%S 3,7,13,15,19,29,33,37,43,51,53,61,69,71,75,77,79,89,93,101,107,113,

%T 119,123,131,139,141,151,161,163,165,173,177,181,193,199,201,217,219,

%U 221,223,229,239,249,251,255,263,271,281,287,291,293,299,309,311,317

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

%C 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.

%e The sequence of partitions together with their Heinz numbers begins:

%e 3: (2) 77: (5,4) 165: (5,3,2)

%e 7: (4) 79: (22) 173: (40)

%e 13: (6) 89: (24) 177: (17,2)

%e 15: (3,2) 93: (11,2) 181: (42)

%e 19: (8) 101: (26) 193: (44)

%e 29: (10) 107: (28) 199: (46)

%e 33: (5,2) 113: (30) 201: (19,2)

%e 37: (12) 119: (7,4) 217: (11,4)

%e 43: (14) 123: (13,2) 219: (21,2)

%e 51: (7,2) 131: (32) 221: (7,6)

%e 53: (16) 139: (34) 223: (48)

%e 61: (18) 141: (15,2) 229: (50)

%e 69: (9,2) 151: (36) 239: (52)

%e 71: (20) 161: (9,4) 249: (23,2)

%e 75: (3,3,2) 163: (38) 251: (54)

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

%t Select[Range[2,100],EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]

%Y These partitions are counted by A087897, shifted left once.

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

%Y The odd version is A341446.

%Y A000009 counts partitions into odd parts, ranked by A066208.

%Y A001222 counts prime factors.

%Y A005843 lists even numbers.

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

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

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

%Y A031215 lists even-indexed primes.

%Y A055396 selects least prime index.

%Y A056239 adds up prime indices.

%Y A058696 counts partitions of even numbers, ranked by A300061.

%Y A061395 selects greatest prime index.

%Y A066207 lists numbers with all even prime indices.

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 13 2021