A335544 -id:A335544 - cp's OEIS frontend

A335543 Numbers with an equal number of deficient and abundant divisors.

144, 324, 336, 756, 900, 1176, 1848, 2100, 2184, 2940, 3200, 3520, 4000, 4160, 4400, 5200, 5952, 10880, 11440, 12160, 12348, 12544, 13600, 14720, 15200, 16368, 17360, 18304, 18400, 18560, 19344, 19360, 19404, 22932, 23200, 27040, 28600, 29988, 33516, 40572, 47124

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

This sequence is infinite. For example, 3200*p is a term for all primes p >= 257.

The least odd term of this sequence is a(1273824) = 3010132125.

			144 is a term since it has 7 deficient divisors: {1, 2, 3, 4, 8, 9, 16} and 7 abundant divisors: {12, 18, 24, 36, 48, 72, 144}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A005100, A005101, A080224, A080226, A335544.

ab[n_] := DivisorSigma[1, n] - 2n; eqdivQ[n_] := Count[(abs = ab/@Divisors[n]), ?(# > 0 &)] == Count[abs, ?(# < 0 &)]; Select[Range[50000], eqdivQ]

Numbers k such that A080224(k) = A080226(k).

A357460 Numbers whose number of deficient divisors is equal to their number of nondeficient divisors.

Original entry on oeis.org

72, 108, 120, 168, 180, 252, 420, 528, 560, 624, 1188, 1224, 1368, 1400, 1404, 1632, 1656, 1824, 1836, 1960, 1980, 2040, 2052, 2088, 2208, 2232, 2280, 2340, 2484, 2664, 2760, 2772, 2784, 2856, 2952, 2976, 3060, 3096, 3132, 3192, 3200, 3276, 3348, 3384, 3420, 3432

Offset: 1

Amiram Eldar, Sep 29 2022

nonn

Numbers k such that A080226(k) = A341620(k).
This sequence is infinite: if p >= 17 is a prime then 72*p is a term.
The least odd term of this sequence is a(36126824) = A357461(1) = 3010132125.
Since the number of divisors of any term is even, none of the terms are squares.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 10, 131, 1172, 12003, 120647, 1199147, 11992293, 120089446, ... . Apparently, the asymptotic density of this sequence exists and is equal to about 0.012.

			72 is a term since it has 12 divisors, 6 of them (1, 2, 3, 4, 8 and 9) are deficient and 6 (6, 12, 18, 24, 36 and 72) are not.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A000037 and A005101.

Cf. A080226, A335543, A335544, A341620, A357461, A357462.

q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, 1, -1] &] == 0; Select[Range[3500], q]

is(n) = sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;

cp's OEIS Frontend

A335543 Numbers with an equal number of deficient and abundant divisors.

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