A328952 -id:A328952 - cp's OEIS frontend

A328953 Antiharmonic numbers (A020487) that are not arithmetic (A003601).

4, 9, 16, 25, 36, 50, 64, 81, 100, 117, 121, 144, 180, 196, 200, 225, 242, 256, 289, 324, 325, 400, 441, 450, 468, 484, 529, 576, 578, 625, 650, 676, 729, 784, 800, 841, 900, 968, 1024, 1058, 1089, 1156, 1225, 1280, 1296, 1300, 1444, 1476, 1521, 1600, 1620

Offset: 1

Jaroslav Krizek, Nov 17 2019

nonn

Numbers m such that the antiharmonic mean of the divisors of m is an integer but the arithmetic mean of the divisors of m is not an integer.
Numbers m such that B(m) = A001157(m) / A000203(m) is an integer but A(m) = A000203(m) / A000005(m) is not an integer.
Corresponding values of B(m): 3, 7, 11, 21, 21, 35, 43, 61, 63, 85, 111, 77, 91, 129, 119, 147, 185, 171, 273, 183, ...
Corresponding values of A(m): 7/3, 13/3, 31/5, 31/3, 91/9, 31/2, 127/7, 121/5, 217/9, 91/3, 133/3, ...

Complement of A277553 with respect to A020487.

Cf. A000005, A000203, A001157, A003601, A328952, A328954.

[m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]

Select[Range[1620], !Divisible[(sigma = DivisorSigma[1, #]), DivisorSigma[0, #]] && Divisible[DivisorSigma[2, #], sigma] &] (* Amiram Eldar, Nov 17 2019 *)

A328954 Numbers m that are neither arithmetic (A003601) nor antiharmonic (A020487).

Original entry on oeis.org

2, 8, 10, 12, 18, 24, 26, 28, 32, 34, 40, 48, 52, 58, 63, 72, 74, 75, 76, 80, 82, 84, 88, 90, 98, 104, 106, 108, 112, 120, 122, 124, 128, 130, 136, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178, 192, 194, 202, 208, 216, 218, 226, 228, 232, 234

Offset: 1

Jaroslav Krizek, Dec 03 2019

nonn

Numbers m such that neither the arithmetic mean of the divisors of m nor the antiharmonic mean of the divisors of m is an integer.
Numbers m such that neither A(m) = A000203(m)/A000005(m) nor B(m) = A001157(m)/A000203(m) is an integer.
Corresponding values of A(m): 3/2, 15/4, 9/2, 14/3, 13/2, 15/2, 21/2, 28/3, 21/2, 27/2, 45/4, 62/5, ...
Corresponding values of B(m): 5/3, 17/3, 65/9, 15/2, 35/3, 85/6, 425/21, 75/4, 65/3, 725/27, 221/9, ...

Cf. A000005, A000203, A001157, A003601, A277553, A328952, A328953.

[m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]

Select[Range[235], !Divisible[DivisorSigma[2, #], (s = DivisorSigma[1, #])] && !Divisible[s, DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 06 2019 *)

cp's OEIS Frontend

A328953 Antiharmonic numbers (A020487) that are not arithmetic (A003601).

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