OFFSET
0,1
COMMENTS
Let K(m) be the smallest possible K satisfying the Theorem. Conjecture: K(m) ~ m, i.e., a(k) ~ A002808(1)*...*A002808(k), only very few of the last factors will be insignificantly larger.
Let K(b,r) be the smallest possible K satisfying the Corollary, i.e., the index from which on all a(k)-1 are multiples of b^r. With the preceding conjecture, there are asymptotically at least (k+PrimePi(A002808(k)))/b multiples of b among the factors of a(k)-1, so this is an (asymptotic) lower bound on r.
Experimentally, a(k) = 1 + Product_{i>=1} prime(i)^e(i), with e(1)~k*3/2, e(2)~k*2/3, e(3)~k/4, e(4)~k/5, ... (Here ~ is not asymptotic equivalence.) Is there a simple formula? - M. F. Hasler, Jun 16 2007
LINKS
Jinyuan Wang, Table of n, a(n) for n = 0..300 (terms 0..99 from M. F. Hasler)
FORMULA
Theorem: For any m > 0 there is a K > 0 such that for all k > K, a(k)-1 is divisible by the first m composite numbers.
Corollary: For any b > 1, r > 0 there is a K > 0 such that for all k > K, a(k) == 1 (mod b^r). Taking b=10 shows that all a(k) > a(8) end in 0..01 with an increasing number of zeros. - M. F. Hasler, Jun 16 2007
EXAMPLE
Writing c(n) for the n-th composite number A002808(n):
a(0) = (Product_{i=1..0} c(i))+1 = 1+1 = 2 (empty product).
a(1) = c(1)+1 = 4+1 = 5.
a(2) = c(1)*c(4)+1 = 4*9+1 = 37, since c(1)*c(k)+1 is not prime for k < 4.
a(3) = c(1)*c(2)*c(3)+1 = 4*6*8 + 1 = 193.
a(4) = c(1)*c(2)*c(4)*c(5)+1 = 2161, nothing better since c(6)*c(3) > c(5)*c(4).
a(5) = c(1)*c(2)*c(3)*c(5)*c(6)+1 = 23041, none better since c(7)*c(4) > c(5)*c(6).
a(6) = c(1)*c(2)*c(3)*c(4)*c(5)*c(7)+1 = 241921, best since c(1)*...*c(6)+1 is not prime.
a(7) = p(7)+1 = 2903041 with p(n) = Product_{i=1..n} c(i). - M. F. Hasler, Jun 16 2007
PROG
(PARI) A081545(n, b=0 /*best*/, p=1 /*product*/, f=[]/*factors*/)={ if( #f<n, /* initialize f[1..n] to first n composites */ b=3; f=vector( n, i, while( isprime( b++ ), ); b ); p=prod( i=1, n-1, f[i] ); /* get upper limit by incrementing last factor until prime is found */ while( !isprime( 1+p*b ), while( isprime( b++ ), )); b=1+p*b; p*=f[n] ); if( isprime( 1+p ), /*print("best: " f); */ 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 = A081545( n-1, b, p*f[n], f), while( isprime( f[n]++ ), ) /* next composite */ ); b } /* then vector(30, n, A081545(n-1)) gives the first 30 terms */ \\ M. F. Hasler, Jun 16 2007
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Amarnath Murthy, Apr 01 2003
EXTENSIONS
More terms from Michel ten Voorde Jun 13 2003
Terms beyond a(8) by M. F. Hasler Jun 16 2007
STATUS
approved