A001276 Smallest k such that the product of q/(q-1) over the primes from prime(n) to prime(n+k-1) is greater than 2.

2, 3, 7, 15, 27, 41, 62, 85, 115, 150, 186, 229, 274, 323, 380, 443, 509, 577, 653, 733, 818, 912, 1010, 1114, 1222, 1331, 1448, 1572, 1704, 1845, 1994, 2138, 2289, 2445, 2609, 2774, 2948, 3127, 3311, 3502, 3699, 3900, 4112, 4324, 4546, 4775, 5016, 5255, 5493

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

A perfect (or abundant) number with prime(n) as its lowest prime factor must be divisible by at least a(n) distinct primes.

In fact, a(n) is the least possible number of distinct prime factors for a (prime(n))-rough abundant number: (prime(n))^(e_n) * ... * (prime(n+a(n)-1))^(e_(n+a(n)-1)) is abundant for sufficiently large e_n, ..., e_(n+a(n)-1). - Jianing Song, Apr 13 2021

Examples

			Every odd abundant number has at least 3 distinct prime factors, and 945 = 3^3 * 5 * 7 has exactly 3, so a(2) = 3. - _Jianing Song_, Apr 13 2021
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..650
• Karl K. Norton, Remarks on the number of factors of an odd perfect number, Acta Arith., 6 (1961), 365-374.

Crossrefs

Cf. A001275, A005101.

Cf. A108227 (least number of prime factors for a (prime(n))-rough abundant number, counted with multiplicity).

Programs

Mathematica

a[n_] := Module[{p = Prime[n], r = 1, k = 0}, While[r <= 2, r *= p/(p - 1); p = NextPrime[p]; k++]; k]; Array[a, 50] (* Amiram Eldar, Jul 12 2019 *)

PARI

a(n)=my(pr=1.,k=0);forprime(p=prime(n),default(primelimit),pr*=p/(p-1);k++;if(pr>2,return(k))) \\ Charles R Greathouse IV, May 09 2011

Formula

a(n) = li(prime(n)^2) + O(n^2/exp((log n)^(4/7 - e))) for any e > 0.

a(n) = pi(A001275(n)) - n + 1. - Amiram Eldar, Jul 12 2019

Extensions

Comment, formula, program, and new definition from Charles R Greathouse IV, May 10 2011