A357462 - OEIS

%I #8 Sep 30 2022 04:25:13

%S 6,28,30,42,66,78,102,114,138,150,174,186,222,246,258,282,294,308,318,

%T 330,354,364,366,390,402,426,438,462,474,476,496,498,510,532,534,546,

%U 570,582,606,618,642,644,654,678,690,714,726,750,762,786,798,812,822,834

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

%C Numbers k such that A187793(k) = A187794(k) + A187795(k).

%C All the terms are nondeficient numbers (A023196).

%C All the perfect numbers (A000396) are terms.

%C 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.

%C Since the total sum of divisors of any term is even, none of the terms are squares or twice squares.

%C Are there odd terms in this sequence? There are none below 10^10.

%C 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... .

%H Amiram Eldar, <a href="/A357462/b357462.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>.

%e 6 is a term since the sum of its deficient divisors, 1 + 2 + 3 is equal to 6, its only nondeficient divisor.

%e 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.

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

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

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

%Y Subsequence of A023196 and A028983.

%Y A000396 is a subsequence.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 29 2022