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A347465
Numbers whose multiset of prime indices has alternating product > 1.
10
3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127
OFFSET
1,1
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)).
All terms have odd bigomega (A001222).
Also Heinz numbers integer partitions with reverse-alternating product > 1.
EXAMPLE
The terms and their prime indices begin:
3: {2} 37: {12} 68: {1,1,7}
5: {3} 41: {13} 70: {1,3,4}
7: {4} 42: {1,2,4} 71: {20}
11: {5} 43: {14} 73: {21}
12: {1,1,2} 44: {1,1,5} 75: {2,3,3}
13: {6} 45: {2,2,3} 76: {1,1,8}
17: {7} 47: {15} 78: {1,2,6}
19: {8} 48: {1,1,1,1,2} 79: {22}
20: {1,1,3} 52: {1,1,6} 80: {1,1,1,1,3}
23: {9} 53: {16} 83: {23}
27: {2,2,2} 59: {17} 89: {24}
28: {1,1,4} 61: {18} 92: {1,1,9}
29: {10} 63: {2,2,4} 97: {25}
30: {1,2,3} 66: {1,2,5} 99: {2,2,5}
31: {11} 67: {19} 101: {26}
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 squarefree case is A030059 without 2.
The reverse version is A028983, counted by A119620.
The opposite version (< 1 instead of > 1) is A119899.
Factorizations of this type are counted by A339890, reverse A347705.
The weak version (>= 1 instead of > 1) is A344609.
Partitions of this type are counted by A347449, reverse A347448.
The complement is A347450, counted by A339846 or A347443.
Allowing any integer reverse-alternating product gives A347454.
Allowing any integer alternating product gives A347457.
A335433 ranks inseparable partitions, complement A335448.
A347446 counts partitions with integer alternating product, reverse A347445.
Sequence in context: A109908 A372028 A152212 * A249370 A282810 A102941
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 27 2021
STATUS
approved