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A347455
Heinz numbers of integer partitions with non-integer alternating product.
4
15, 30, 33, 35, 51, 55, 60, 66, 69, 70, 77, 85, 91, 93, 95, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 154, 155, 161, 165, 170, 177, 182, 186, 187, 190, 201, 203, 204, 205, 209, 210, 215, 217, 219, 220, 221, 231, 238, 240, 246, 247, 249
OFFSET
1,1
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.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has non-integer reverse-alternating product.
EXAMPLE
The terms and their reversed prime indices begin:
15: (3,2) 102: (7,2,1) 161: (9,4)
30: (3,2,1) 105: (4,3,2) 165: (5,3,2)
33: (5,2) 110: (5,3,1) 170: (7,3,1)
35: (4,3) 119: (7,4) 177: (17,2)
51: (7,2) 120: (3,2,1,1,1) 182: (6,4,1)
55: (5,3) 123: (13,2) 186: (11,2,1)
60: (3,2,1,1) 132: (5,2,1,1) 187: (7,5)
66: (5,2,1) 135: (3,2,2,2) 190: (8,3,1)
69: (9,2) 138: (9,2,1) 201: (19,2)
70: (4,3,1) 140: (4,3,1,1) 203: (10,4)
77: (5,4) 141: (15,2) 204: (7,2,1,1)
85: (7,3) 143: (6,5) 205: (13,3)
91: (6,4) 145: (10,3) 209: (8,5)
93: (11,2) 154: (5,4,1) 210: (4,3,2,1)
95: (8,3) 155: (11,3) 215: (14,3)
For example, (4,3,2,1) has alternating product 4/3*2/1 = 8/3, so the Heinz number 210 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Select[Range[100], !IntegerQ[altprod[Reverse[primeMS[#]]]]&]
CROSSREFS
The reciprocal version is A028983, complement A028982.
Factorizations not of this type are counted by A347437.
Partitions not of this type are counted by A347446.
The complement of the reverse reciprocal version is A347451.
The complement in the odd-length case is A347453.
The complement of the reverse version is A347454.
The complement is A347457.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A347461 counts possible alternating products of partitions, reverse A347462.
Sequence in context: A343343 A318992 A343337 * A324970 A033898 A051967
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 04 2021
STATUS
approved