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A359905
Numbers whose prime indices and prime signature both have integer mean.
12
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.
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.
Sequence in context: A305504 A316413 A360009 * A316465 A004764 A128649
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 25 2023
STATUS
approved