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 A073918 Smallest prime which is 1 more than a product of n distinct primes: a(n) is a prime and a(n) - 1 is a squarefree number with n prime factors. 14
 2, 3, 7, 31, 211, 2311, 43891, 870871, 13123111, 300690391, 6915878971, 200560490131, 11406069164491, 386480064480511, 18826412648012971, 693386350578511591, 37508276737897976011, 3087649419126112110271, 183452981525059000664911, 11465419967969569966774411 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Apparently the same as record values of A055734: least k such that phi(k) has n distinct prime factors, where phi is Euler's totient function. If the Mathematica program is used for large n, then "fact" should be reduced to, say, 1.1 in order to increase the search speed. - T. D. Noe, Dec 17 2003 LINKS M. F. Hasler Jun 16 2007, Table of n, a(n) for n = 0..24 FORMULA From M. F. Hasler, Jun 16 2007 (Start): Conjecture: For any m > 0 there is K > 0 such that for all k > K, a(k)-1 is divisible by the first m primes. Corollary: For any m > 1 there is K > 0 such that for all k > K, a(k) = 1 (mod m). Conjecture 2: Let K(m) be the smallest possible K satisfying the above Conjecture. Then K(m) ~ m, i.e., a(k) ~ A002110(k), only very few of the last factors will be a bit larger. (End) Remark: the last "~" above was not intended to mean asymptotic equivalence. It appears that lim inf a(n)/A002110(n) = 1, but the lim sup might well be larger. It would be interesting to know whether it has a finite value. - M. F. Hasler, May 31 2018 EXAMPLE a(0) = 1 + 1 = 2 (empty product of zero primes). a(1) = 1 + 2 = 3. a(2) = 1 + 2*3 = 7. a(3) = 1 + 2*3*5 = 31. a(4) = 1 + 2*3*5*7 = 211. a(5) = 1 + 2*3*5*7*11 = 1 + 11# = 2311. a(6) = 1 + 2*3*5*7*11*19 = 43891, since 13# + 1 and 11#*17 + 1 = 17#/13 + 1 is not prime, and 17#/p + 1 is larger than a(6) for all p in {2, ..., 11}. The index of the smallest prime which is not a factor of a(n)+1 is (1, 2, 3, 4, 5, 6, 6, 7, 7, 9, 10, 12, 11, 12, 13, 15, 16, 15, 16, 18, 19, 20, 21, 22, 22, 23, 25, 27, 26, 29, 29, ...) for n = 0, 1, 2, ... - M. F. Hasler, May 31 2018 MATHEMATICA Generate[pIndex_, i_] := Module[{p2, t}, p2=pIndex; While[p2[[i]]++; Do[p2[[j]]=p2[[i]]+j-i, {j, i+1, Length[p2]}]; t=Times@@Prime[p2]; t= f[n+1] ) || !b = A073918( n-1, b, p*f[n], f), f[n]= nextprime( f[n]+1 ) ); b } \\ then, e.g.: apply(A073918, [0..30]). - M. F. Hasler Jun 16 2007 CROSSREFS Cf. A055734 (number of distinct prime factors of phi(n)). Cf. A073917, A098026. Cf. A000040 (primes), A002110 (primorial), A081545 (same with composite instead of primes). Sequence in context: A006862 A038710 A241196 * A096350 A018239 A066279 Adjacent sequences:  A073915 A073916 A073917 * A073919 A073920 A073921 KEYWORD nonn AUTHOR Amarnath Murthy, Aug 18 2002 EXTENSIONS More terms from Vladeta Jovovic, Aug 20 2002 Edited by M. F. Hasler, May 31 2018 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)