A328954 -id:A328954 - cp's OEIS frontend

A328952 Arithmetic numbers (A003601) that are not antiharmonic (A020487).

3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102

Offset: 1

Jaroslav Krizek, Nov 17 2019

nonn

Numbers m such that the arithmetic mean of the divisors of m is an integer but the antiharmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m) / A000005(m) is an integer but B(m) = A001157(m) / A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 3, 4, 6, 7, 6, 6, 9, 10, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of B(m): 5/2, 13/3, 25/6, 25/4, 61/6, 85/7, 125/12, 65/6, 145/9, 181/10, 125/8, ...

Complement of A277553 with respect to A003601.

Cf. A000005, A000203, A001157, A328953, A328954.

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

Select[Range[100], Divisible[(sigma = DivisorSigma[1, #]), DivisorSigma[0, #]] && !Divisible[DivisorSigma[2, #], sigma] &]  (* Amiram Eldar, Nov 17 2019 *)

isok(m) = !(sigma(m) % numdiv(m)) && (sigma(m,2) % sigma(m)); \\ Michel Marcus, Nov 18 2019

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A328952 Arithmetic numbers (A003601) that are not antiharmonic (A020487).

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