A347465 Numbers whose multiset of prime indices has alternating product > 1.

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.

Links

Table of n, a(n) for n=1..61.

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.

Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.

Sequence in context: A109908 A372028 A152212 * A249370 A282810 A102941

Adjacent sequences: A347462 A347463 A347464 * A347466 A347467 A347468

Keyword

nonn

Author

Gus Wiseman, Sep 27 2021

Status

approved