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 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. (Formerly M2650 N1057) 6
 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A001275, A005101. Cf. A108227 (least number of prime factors for a (prime(n))-rough abundant number, counted with multiplicity). Sequence in context: A276047 A209658 A098763 * A006884 A074742 A020873 Adjacent sequences: A001273 A001274 A001275 * A001277 A001278 A001279 KEYWORD nonn AUTHOR N. J. A. Sloane EXTENSIONS Comment, formula, program, and new definition from Charles R Greathouse IV, May 10 2011 STATUS approved

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Last modified September 25 18:55 EDT 2023. Contains 365648 sequences. (Running on oeis4.)