A359905 Numbers whose prime indices and prime signature both have integer mean.

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 91, 94, 97, 100, 101, 103, 105, 107, 109, 110, 111, 113, 115, 118, 121

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.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Links

Table of n, a(n) for n=1..64.

Formula

Intersection of A316413 and A067340.

Example

The terms together with their prime indices begin:
     2: {1}          19: {8}            39: {2,6}
     3: {2}          21: {2,4}          41: {13}
     4: {1,1}        22: {1,5}          43: {14}
     5: {3}          23: {9}            46: {1,9}
     7: {4}          25: {3,3}          47: {15}
     8: {1,1,1}      27: {2,2,2}        49: {4,4}
     9: {2,2}        29: {10}           53: {16}
    10: {1,3}        30: {1,2,3}        55: {3,5}
    11: {5}          31: {11}           57: {2,8}
    13: {6}          32: {1,1,1,1,1}    59: {17}
    16: {1,1,1,1}    34: {1,7}          61: {18}
    17: {7}          37: {12}           62: {1,11}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
prisig[n_]:=If[n==1, {}, Last/@FactorInteger[n]];
Select[Range[100], IntegerQ[Mean[prix[#]]]&&IntegerQ[Mean[prisig[#]]]&]

Crossrefs

A058398 counts partitions by mean, see also A008284, A327482.

A067340 lists numbers whose prime signature has integer mean.

A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.

A124010 lists prime signature, mean A088529/A088530.

A316413 lists numbers whose prime indices have integer mean.

A326622 counts factorizations with integer mean, strict A328966.

Cf. A327473, A348551, A359904, A359908, A360005, A360068, A360069.

Sequence in context: A305504 A316413 A360009 * A316465 A004764 A128649

Adjacent sequences: A359902 A359903 A359904 * A359906 A359907 A359908

Keyword

nonn

Author

Gus Wiseman, Jan 25 2023

Status

approved