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

1,1

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

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*# &]

Prog

(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; }

Crossrefs

Subsequence of A005101.

Cf. A007821, A320628, A360331.

Sequence in context: A101294 A280932 A039534 * A360356 A063347 A219807

Adjacent sequences: A360329 A360330 A360331 * A360333 A360334 A360335

Keyword

nonn

Author

Amiram Eldar, Feb 03 2023

Status

approved