OFFSET
0,1
COMMENTS
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)).
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