A335543 -id:A335543 - cp's OEIS frontend

A335544 Numbers with more abundant divisors than deficient divisors.

216, 240, 288, 360, 432, 480, 504, 540, 576, 600, 648, 672, 720, 792, 840, 864, 936, 960, 972, 1008, 1056, 1080, 1120, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1728, 1800, 1872, 1920, 1944, 2016, 2112, 2160, 2240, 2268, 2304

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

This sequence is infinite. For example, 216*p is a term for all primes p.
The least odd term of this sequence is a(16317321) = 638512875.
Apparently, this sequence has an asymptotic density of about 0.025.

			216 is a term since it has 8 abundant divisors, {12, 18, 24, 36, 54, 72, 108, 216}, and only 7 deficient divisors, {1, 2, 3, 4, 8, 9, 27}.

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

Cf. A000203, A005100, A005101, A080224, A080226, A335543.

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

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;

A357462 Numbers whose sum of deficient divisors is equal to their sum of nondeficient divisors.

Original entry on oeis.org

6, 28, 30, 42, 66, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 308, 318, 330, 354, 364, 366, 390, 402, 426, 438, 462, 474, 476, 496, 498, 510, 532, 534, 546, 570, 582, 606, 618, 642, 644, 654, 678, 690, 714, 726, 750, 762, 786, 798, 812, 822, 834

Offset: 1

Amiram Eldar, Sep 29 2022

nonn

Numbers k such that A187793(k) = A187794(k) + A187795(k).
All the terms are nondeficient numbers (A023196).
All the perfect numbers (A000396) are terms.
This sequence is infinite: if k = 2^(p-1)*(2^p-1) is an even perfect number and q > 2^p-1 is a prime, then k*q is a term.
Since the total sum of divisors of any term is even, none of the terms are squares or twice squares.
Are there odd terms in this sequence? There are none below 10^10.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 6, 63, 605, 6164, 61291, 614045, 6139193, 61382607, 613861703, ... . Apparently, the asymptotic density of this sequence exists and equals 0.06138... .

			6 is a term since the sum of its deficient divisors, 1 + 2 + 3 is equal to 6, its only nondeficient divisor.
30 is a term since the sum of its deficient divisors, 1 + 2 + 3 + 5 + 10 + 15 = 36 is equal to the sum of its nondeficient divisors, 6 + 30 = 36.

Cf. A187793, A187794, A187795, A335543, A357460.
Subsequence of A023196 and A028983.
A000396 is a subsequence.

q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, #, -#] &] == 0; Select[Range[1000], q]

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

A357461 Odd numbers whose number of deficient divisors is equal to their number of nondeficient divisors.

Original entry on oeis.org

3010132125, 4502334375, 5065535475, 6456074625, 8813660625, 9881746875, 15395254875, 15452011575, 16874983125, 18699305625, 19814169375, 19909992375, 21380506875, 25366375125, 26643400875, 26746594875, 28943578125, 31562182575, 33074966925, 34315506225, 35300640375

Offset: 1

Amiram Eldar, Sep 29 2022

nonn

The odd terms of A357460.
If there are no odd perfect numbers, then this sequence is also the subsequence of the odd terms of A335543.
The first 100 terms are all divisible by 4725 = 3^3 * 5^2 * 7.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A005101, A005231 and A357460.

Cf. A335543.

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

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A335544 Numbers with more abundant divisors than deficient divisors.

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

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A357462 Numbers whose sum of deficient divisors is equal to their sum of nondeficient divisors.

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A357461 Odd numbers whose number of deficient divisors is equal to their number of nondeficient divisors.

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