This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347455 #4 Oct 27 2021 22:22:33
%S A347455 15,30,33,35,51,55,60,66,69,70,77,85,91,93,95,102,105,110,119,120,123,
%T A347455 132,135,138,140,141,143,145,154,155,161,165,170,177,182,186,187,190,
%U A347455 201,203,204,205,209,210,215,217,219,220,221,231,238,240,246,247,249
%N A347455 Heinz numbers of integer partitions with non-integer alternating product.
%C A347455 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.
%C A347455 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347455 Also numbers whose multiset of prime indices has non-integer reverse-alternating product.
%e A347455 The terms and their reversed prime indices begin:
%e A347455 15: (3,2) 102: (7,2,1) 161: (9,4)
%e A347455 30: (3,2,1) 105: (4,3,2) 165: (5,3,2)
%e A347455 33: (5,2) 110: (5,3,1) 170: (7,3,1)
%e A347455 35: (4,3) 119: (7,4) 177: (17,2)
%e A347455 51: (7,2) 120: (3,2,1,1,1) 182: (6,4,1)
%e A347455 55: (5,3) 123: (13,2) 186: (11,2,1)
%e A347455 60: (3,2,1,1) 132: (5,2,1,1) 187: (7,5)
%e A347455 66: (5,2,1) 135: (3,2,2,2) 190: (8,3,1)
%e A347455 69: (9,2) 138: (9,2,1) 201: (19,2)
%e A347455 70: (4,3,1) 140: (4,3,1,1) 203: (10,4)
%e A347455 77: (5,4) 141: (15,2) 204: (7,2,1,1)
%e A347455 85: (7,3) 143: (6,5) 205: (13,3)
%e A347455 91: (6,4) 145: (10,3) 209: (8,5)
%e A347455 93: (11,2) 154: (5,4,1) 210: (4,3,2,1)
%e A347455 95: (8,3) 155: (11,3) 215: (14,3)
%e A347455 For example, (4,3,2,1) has alternating product 4/3*2/1 = 8/3, so the Heinz number 210 is in the sequence.
%t A347455 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347455 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347455 Select[Range[100],!IntegerQ[altprod[Reverse[primeMS[#]]]]&]
%Y A347455 The reciprocal version is A028983, complement A028982.
%Y A347455 Factorizations not of this type are counted by A347437.
%Y A347455 Partitions not of this type are counted by A347446.
%Y A347455 The complement of the reverse reciprocal version is A347451.
%Y A347455 The complement in the odd-length case is A347453.
%Y A347455 The complement of the reverse version is A347454.
%Y A347455 The complement is A347457.
%Y A347455 A056239 adds up prime indices, row sums of A112798.
%Y A347455 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347455 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A347455 A347461 counts possible alternating products of partitions, reverse A347462.
%Y A347455 Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450, A347465.
%K A347455 nonn
%O A347455 1,1
%A A347455 _Gus Wiseman_, Oct 04 2021