OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers whose multiset of prime indices has alternating sum <= 1.
EXAMPLE
The initial terms and their prime indices:
1: {} 26: {1,6} 56: {1,1,1,4}
2: {1} 32: {1,1,1,1,1} 57: {2,8}
4: {1,1} 33: {2,5} 58: {1,10}
6: {1,2} 34: {1,7} 60: {1,1,2,3}
8: {1,1,1} 35: {3,4} 62: {1,11}
9: {2,2} 36: {1,1,2,2} 64: {1,1,1,1,1,1}
10: {1,3} 38: {1,8} 65: {3,6}
14: {1,4} 39: {2,6} 69: {2,9}
15: {2,3} 40: {1,1,1,3} 72: {1,1,1,2,2}
16: {1,1,1,1} 46: {1,9} 74: {1,12}
18: {1,2,2} 49: {4,4} 77: {4,5}
21: {2,4} 50: {1,3,3} 81: {2,2,2,2}
22: {1,5} 51: {2,7} 82: {1,13}
24: {1,1,1,2} 54: {1,2,2,2} 84: {1,1,2,4}
25: {3,3} 55: {3,5} 85: {3,7}
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], altprod[primeMS[#]]<=1&]
CROSSREFS
The additive version (alternating sum <= 0) is A028260.
Allowing any alternating product < 1 gives A119899.
Partitions of this type are counted by A347443.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A344606 counts alternating permutations of prime indices.
A347457 lists Heinz numbers of partitions with integer alternating product.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 24 2021
STATUS
approved