%I #11 Feb 04 2023 14:14:22

%S 56,104,112,196,208,224,304,364,368,392,416,448,464,532,608,644,728,

%T 736,784,812,832,896,928,1036,1064,1184,1204,1216,1288,1316,1352,1372,

%U 1376,1456,1472,1484,1504,1568,1624,1664,1696,1708,1792,1856,1952,1976,1988,2044

%N Numbers k such that A360331(k) > 2*k.

%C Analogous to abundant numbers (A005101) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.

%C The least odd term is 7^4 * (13*19)^3 * (29*...*71)^2 * (73*...*281) = 2.411... * 10^105 (where the dots are for consecutive terms in A007821).

%C Includes all the abundant (A005101) terms of A320628.

%C There are terms that are not in A320628, and the least of them is 3 * m, where m is a term of A320628 with sigma(m) > 6. Such a number exists, and it should be a positive multiple of Product_{i=1..k} A007821(k) = 2 * 7 * ... * 11443 = 9.164... * 10^4148, where k = 1160 is the least number such that Product_{i=1..k} A007821(k)/(A007821(k)-1) > 6.

%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 1, 23, 215, 1997, 19231, 189457, 1873511, 18593697, ... . Apparently, the asymptotic density of this sequence equals 0.018... .

%H Amiram Eldar, <a href="/A360332/b360332.txt">Table of n, a(n) for n = 1..10000</a>

%p q:= n-> is(mul(`if`(isprime(numtheory[pi](i[1])), 1,

%p (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2])>2*n):

%p select(q, [$1..2050])[]; # _Alois P. Heinz_, Feb 03 2023

%t f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2000], s[#] > 2*# &]

%o (PARI) is(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}

%Y Subsequence of A005101.

%Y Cf. A007821, A320628, A360331.

%K nonn

%O 1,1

%A _Amiram Eldar_, Feb 03 2023