|
%I M2650 N1057 #34 Apr 14 2021 11:07:05
%S 2,3,7,15,27,41,62,85,115,150,186,229,274,323,380,443,509,577,653,733,
%T 818,912,1010,1114,1222,1331,1448,1572,1704,1845,1994,2138,2289,2445,
%U 2609,2774,2948,3127,3311,3502,3699,3900,4112,4324,4546,4775,5016,5255,5493
%N 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.
%C A perfect (or abundant) number with prime(n) as its lowest prime factor must be divisible by at least a(n) distinct primes.
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Amiram Eldar, <a href="/A001276/b001276.txt">Table of n, a(n) for n = 1..650</a>
%H Karl K. Norton, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa6/aa641.pdf">Remarks on the number of factors of an odd perfect number</a>, Acta Arith., 6 (1961), 365-374.
%F a(n) = li(prime(n)^2) + O(n^2/exp((log n)^(4/7 - e))) for any e > 0.
%F a(n) = pi(A001275(n)) - n + 1. - _Amiram Eldar_, Jul 12 2019
%e 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
%t 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 *)
%o (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
%Y Cf. A001275, A005101.
%Y Cf. A108227 (least number of prime factors for a (prime(n))-rough abundant number, counted with multiplicity).
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E Comment, formula, program, and new definition from _Charles R Greathouse IV_, May 10 2011
|
