A360332 Numbers k such that A360331(k) > 2*k.

56, 104, 112, 196, 208, 224, 304, 364, 368, 392, 416, 448, 464, 532, 608, 644, 728, 736, 784, 812, 832, 896, 928, 1036, 1064, 1184, 1204, 1216, 1288, 1316, 1352, 1372, 1376, 1456, 1472, 1484, 1504, 1568, 1624, 1664, 1696, 1708, 1792, 1856, 1952, 1976, 1988, 2044

Offset: 1

Author

Amiram Eldar, Feb 03 2023

Keywords

• nonn

Comments

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

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).

Includes all the abundant (A005101) terms of A320628.

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.

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... .

Links

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

Crossrefs

Subsequence of A005101.

Cf. A007821, A320628, A360331.

Programs

Maple

q:= n-> is(mul(`if`(isprime(numtheory[pi](i[1])), 1,
   (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2])>2*n):
select(q, [$1..2050])[];  # Alois P. Heinz, Feb 03 2023

Mathematica

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*# &]

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;}