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

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

Author

Amiram Eldar, Sep 29 2022

Keywords

• nonn

Comments

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... .

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences where any odd perfect numbers must occur.

Crossrefs

Cf. A187793, A187794, A187795, A335543, A357460.

Subsequence of A023196 and A028983.

A000396 is a subsequence.

Programs

Mathematica

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

PARI

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