A110585 Smallest number k of consecutive primes > p_n such that p_n^2 * p_(n+1) * p_(n+2) * ... * p_(n+k) is an abundant number.

1, 3, 7, 16, 29, 44, 65, 89, 120, 155, 192, 236, 282, 332, 390, 453, 520, 589, 666, 746, 832, 927, 1026, 1131, 1239, 1350, 1467, 1592, 1725, 1867, 2017, 2161, 2313, 2469, 2634, 2800, 2975, 3155, 3339, 3532, 3729, 3931, 4143, 4356, 4579, 4809, 5051, 5291

Offset

1,2

Comments

The sequence arose while solving puzzle 329 from Carlos Rivera's Prime Puzzles & Problems Connection site.

Links

Table of n, a(n) for n=1..48.

Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3.

Example

a(2)=3 because the second prime being 3, then 3^2 * 5 * 7 * 11 = 3465 and sigma(3465) - 2*3465 = 558, a positive number (i.e., 3465 is abundant), but 3^2 * 5 * 7 = 315 and sigma(315) - 2*315 = -6, a nonpositive number (i.e., 315 is not abundant).

Mathematica

abQ[n_] := DivisorSigma[1, n] > 2n; f[0] = 0; f[n_] := f[n] = Block[{k = f[n - 1]}, p = Fold[Times, Prime[n], Prime[ Range[n, n + k]]]; While[ !abQ[p], k++; p = p*Prime[n + k]]; k]; Table[ f[n], {n, 48}] (* Robert G. Wilson v *)

Prog

(PARI) forprime(p=2, 100, k=0; while(k++, if(sigma(n=p^2*prod(j=1, k, prime(j+primepi(p))))-n>n, print(k); break)))

Crossrefs

Cf. A005101.

Cf. A005231, A047802.

Sequence in context: A211379 A373104 A213180 * A184677 A224340 A240739

Adjacent sequences: A110582 A110583 A110584 * A110586 A110587 A110588

Keyword

nonn

Author

Igor Schein, Sep 13 2005

Extensions

Edited and extended by Robert G. Wilson v, Sep 15 2005

Status

approved