A176552 -id:A176552 - cp's OEIS frontend

A175397 Numbers such that both arithmetic means of distinct and all prime factors are not integers.

1, 6, 10, 12, 14, 18, 22, 24, 26, 28, 30, 34, 36, 38, 40, 46, 48, 52, 54, 56, 58, 62, 66, 70, 72, 74, 76, 80, 82, 86, 88, 90, 94, 96, 98, 100, 102, 104, 106, 108, 118, 120, 122, 124, 130, 132, 134, 136, 138, 142, 144, 146, 148, 150, 152, 154, 158, 160, 162, 165, 166, 172, 174, 176, 178, 182, 184

Offset: 1

Jaroslav Krizek, May 01 2010

nonn

Contains all even semiprimes. - Robert Israel, Nov 10 2024

			For a(13) = 36: 36 = 2^2*3^3; both (2+2+3+3)/4 and (2+3)/2 are not integers.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A176552, A175352 and A176587. Complement of A175418. Cf. A174894.

filter:= proc(n) local F,t,m;
  F:= ifactors(n)[2]; m:= nops(F);
  not (add(t[1],t=F)/m)::integer and not (add(t[1]*t[2],t=F)/add(t[2],t=F))::integer
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Nov 10 2024

a(27) corrected, and more terms from Robert Israel, Nov 10 2024

A174894 Numbers such that the arithmetic mean of their distinct prime factors and the arithmetic mean of all of their prime factors are both integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 101

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

Subsequence of A078174 and A078175.

Complement of A176552. [From Jaroslav Krizek, Apr 21 2010]

			For a(11) = 16: 16 = 2^4; both (2+2+2+2)/4 and 2/1 are integers.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

mdmaQ[n_]:=With[{fi=FactorInteger[n]},AllTrue[{Mean[Flatten[Table[#[[1]],#[[2]]]&/@fi]],Mean[fi[[;;,1]]]},IntegerQ]]; Select[Range[ 2,110],mdmaQ] (* Harvey P. Dale, Nov 16 2024 *)

Definition clarified by Harvey P. Dale, Nov 16 2024

A175418 Complement of A175397, where A175397 = numbers such that both arithmetic means of distinct and all prime factors are not integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 44, 45, 47, 49, 50, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99

Offset: 1

Jaroslav Krizek, May 09 2010

nonn

For these numbers hold that both arithmetic means of distinct and all prime factors are integers or only one of these means is an integer.

Includes all prime powers and odd semiprimes. - Robert Israel, Nov 10 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A174894, A176552, A175352, A175397, A176587.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  (add(t[1],t=F)/nops(F))::integer or (add(t[1]*t[2],t=F)/add(t[2],t=F))::integer
end proc:
select(filter, [$2..100]); # Robert Israel, Nov 10 2024

a(49) corrected by Robert Israel, Nov 10 2024

cp's OEIS Frontend

A175397 Numbers such that both arithmetic means of distinct and all prime factors are not integers.

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A174894 Numbers such that the arithmetic mean of their distinct prime factors and the arithmetic mean of all of their prime factors are both integers.

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A175418 Complement of A175397, where A175397 = numbers such that both arithmetic means of distinct and all prime factors are not integers.

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