A001275 Smallest prime p such that the product of q/(q-1) over the primes from prime(n) to p is greater than 2.

3, 7, 23, 61, 127, 199, 337, 479, 677, 937, 1193, 1511, 1871, 2267, 2707, 3251, 3769, 4349, 5009, 5711, 6451, 7321, 8231, 9173, 10151, 11197, 12343, 13487, 14779, 16097, 17599, 19087, 20563, 22109, 23761, 25469, 27259, 29123, 31081, 33029

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 a prime greater than or equal to a(n).

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. A001276.

Programs

Mathematica

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

PARI

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

Formula

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

a(n) = prime(A001276(n) + n - 1). - Amiram Eldar, Jul 12 2019

Extensions

Comment, formula, program, and new definition from Charles R Greathouse IV, May 09 2011