A347453 - OEIS

%I #7 Sep 27 2021 07:56:25

%S 2,3,5,7,8,11,12,13,17,18,19,20,23,27,28,29,31,32,37,41,42,43,44,45,

%T 47,48,50,52,53,59,61,63,67,68,71,72,73,75,76,78,79,80,83,89,92,97,98,

%U 99,101,103,107,108,109,112,113,114,116,117,124,125,127,128,130

%N Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

%C 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 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

%C Also numbers whose multiset of prime indices has odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.

%e The terms and their prime indices begin:

%e       2: {1}         29: {10}            61: {18}

%e       3: {2}         31: {11}            63: {2,2,4}

%e       5: {3}         32: {1,1,1,1,1}     67: {19}

%e       7: {4}         37: {12}            68: {1,1,7}

%e       8: {1,1,1}     41: {13}            71: {20}

%e      11: {5}         42: {1,2,4}         72: {1,1,1,2,2}

%e      12: {1,1,2}     43: {14}            73: {21}

%e      13: {6}         44: {1,1,5}         75: {2,3,3}

%e      17: {7}         45: {2,2,3}         76: {1,1,8}

%e      18: {1,2,2}     47: {15}            78: {1,2,6}

%e      19: {8}         48: {1,1,1,1,2}     79: {22}

%e      20: {1,1,3}     50: {1,3,3}         80: {1,1,1,1,3}

%e      23: {9}         52: {1,1,6}         83: {23}

%e      27: {2,2,2}     53: {16}            89: {24}

%e      28: {1,1,4}     59: {17}            92: {1,1,9}

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

%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];

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

%Y The reciprocal version is A000290.

%Y Allowing any alternating product <= 1 gives A001105.

%Y Allowing any alternating product gives A026424.

%Y Factorizations of this type are counted by A347441.

%Y These partitions are counted by A347444.

%Y Allowing any length gives A347454.

%Y Allowing any alternating product > 1 gives A347465.

%Y A027193 counts odd-length partitions.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A316524 gives the alternating sum of prime indices (reverse: A344616).

%Y A335433 lists numbers whose prime indices are separable, complement A335448.

%Y A344606 counts alternating permutations of prime indices.

%Y A347446 counts partitions with integer alternating product.

%Y A347457 ranks partitions with integer alt product, complement A347455.

%Y A347461 counts possible alternating products of partitions.

%Y A347462 counts possible reverse-alternating products of partitions.

%Y Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 24 2021