A359903 - OEIS

A359903

Numbers whose prime indices and prime signature have the same mean.

1, 2, 9, 88, 100, 125, 624, 756, 792, 810, 880, 900, 1312, 2401, 4617, 4624, 6240, 7392, 7560, 7920, 8400, 9261, 9604, 9801, 10648, 12416, 23424, 33984, 37760, 45792, 47488, 60912, 66176, 71552, 73920, 75200, 78720, 83592, 89216, 89984, 91264, 91648, 99456

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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.

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..43.

EXAMPLE

The terms together with their prime indices begin:

1: {}

2: {1}

9: {2,2}

88: {1,1,1,5}

100: {1,1,3,3}

125: {3,3,3}

624: {1,1,1,1,2,6}

756: {1,1,2,2,2,4}

792: {1,1,1,2,2,5}

810: {1,2,2,2,2,3}

880: {1,1,1,1,3,5}

900: {1,1,2,2,3,3}

1312: {1,1,1,1,1,13}

2401: {4,4,4,4}

4617: {2,2,2,2,2,8}

4624: {1,1,1,1,7,7}

6240: {1,1,1,1,1,2,3,6}

7392: {1,1,1,1,1,2,4,5}

7560: {1,1,1,2,2,2,3,4}

7920: {1,1,1,1,2,2,3,5}

Example: 810 has prime indices {1,2,2,2,2,3} and prime exponents (1,4,1), both of which have mean 2, so 810 is in the sequence.

Example: 78720 has prime indices {1,1,1,1,1,1,1,2,3,13} and prime exponents (7,1,1,1), both of which have mean 5/2, so 78720 is in the sequence.

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[1000], Mean[prix[#]]==Mean[prisig[#]]&]

CROSSREFS

Prime indices are A112798, sum A056239, mean A326567/A326568.

Prime signature is A124010, sum A001222, mean A088529/A088530.

For prime factors instead of indices we have A359904.

Partitions with these Heinz numbers are counted by A360068.

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

A067340 lists numbers whose prime signature has integer mean.

A316413 lists numbers whose prime indices have integer mean.

A360005 gives median of prime indices (times two).

Cf. A240219, A316313, A326622, A327473, A327476, A348551, A359905, A359908, A360008, A360069.

Sequence in context: A037172 A106163 A368186 * A360433 A278332 A135747

Adjacent sequences: A359900 A359901 A359902 * A359904 A359905 A359906

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 24 2023

STATUS

approved