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.

2, 3, 7, 31, 211, 2311, 43891, 870871, 13123111, 300690391, 6915878971, 200560490131, 11406069164491, 386480064480511, 18826412648012971, 693386350578511591, 37508276737897976011, 3087649419126112110271, 183452981525059000664911, 11465419967969569966774411

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

Max Alekseyev, Table of n, a(n) for n = 0..100 (terms for n = 0..24 from M. F. Hasler)

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<fact*base, AppendTo[s, t]; If[i<Length[p2], Generate[p2, i+1]]]]; fact=2; Table[pin=Range[n]; base=Times@@Prime[pin]; s={base}; Do[Generate[pin, j], {j, n}]; s=Sort[s]; noPrime=True; i=0; While[noPrime&&i<Length[s], i++; noPrime=!PrimeQ[1+s[[i]]]]; If[noPrime, -1, 1+s[[i]]], {n, 20}] - from T. D. Noe

Prog

(PARI) A073918(n, b=0 /*best*/, p=1 /*product*/, f=[]/*factors*/)={ if( #f<n, f=primes(n); p=factorback(f[^-1]); b=f[n]; /* get upper limit by incrementing last factor until prime is found */ while( !isprime( 1+p*b), b=nextprime(b+1)); b=1+p*b; p*=f[n] ); if( isprime( 1+p ), return( 1+p )); /* always p < b */ /* increase the n-th factor to recursively explore all solutions < b */ p /= f[n]; until( b <= 1+p*f[n] || ( n < #f && f[n] >= 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 * A018239 A096350 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