OFFSET
1,1
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
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.
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
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Comment, formula, program, and new definition from Charles R Greathouse IV, May 10 2011
STATUS
approved