cp's OEIS Frontend

A175679 Numbers m such that arithmetic mean Ad(m) of divisors of m and arithmetic mean Ak(m) of numbers 1 <= k <= m are both integer.

%I A175679 #10 Oct 09 2023 09:24:18
%S A175679 1,3,5,7,11,13,15,17,19,21,23,27,29,31,33,35,37,39,41,43,45,47,49,51,
%T A175679 53,55,57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,99,101,
%U A175679 103,105,107,109,111,113,115,119,123,125,127,129,131,133,135,137,139,141
%N A175679 Numbers m such that arithmetic mean Ad(m) of divisors of m and arithmetic mean Ak(m) of numbers 1 <= k <= m are both integer.
%C A175679 Numbers m such that Ad(m) = A000203(m) / A000005(m) = A057020(m) / A057021(m) and Ak(m) = A000217(m) / A000027(m) = A145051(m) / A040001(m) are both integer.
%C A175679 Subsequence of A003601: {a(n)} = odd arithmetic numbers from A003601.
%C A175679 {a(n)} union A175678 = A003601 (arithmetic numbers).
%C A175679 From _Robert G. Wilson v_, Aug 09 2010: (Start)
%C A175679 All terms are odd because the second criterion is equivalent to n|T(n), where T(n) is the n-th triangular number, A000217(n).
%C A175679 Terms that are not prime are 1, 15, 21, 27, 33, 35, 39, 45, 49, 51, 55, 57, 65, 69, 77, 85, ..., .
%C A175679 Odd integers that are not terms: 9, 25, 63, 75, 81, 117, 121, 171, 175, 225, 243, 279, 289, ..., . (End)
%e A175679 a(4) = 7, Ad(7) = (1+7)/2 = 4, Ak(7) = (1+2+3+4+5+6+7)/7 = 4, Ad(7) and Ak(7) are both integer.
%t A175679 fQ[n_] := OddQ@n && Mod[DivisorSigma[1, n], DivisorSigma[0, n]] == 0; Select[ Range@ 142, fQ] (* _Robert G. Wilson v_, Aug 09 2010 *)
%K A175679 nonn
%O A175679 1,2
%A A175679 _Jaroslav Krizek_, Aug 07 2010
%E A175679 More terms from _Robert G. Wilson v_, Aug 09 2010