

A081545


Smallest prime which is 1 more than the product of n distinct composite numbers.


6



2, 5, 37, 193, 2161, 23041, 241921, 2903041, 55987201, 958003201, 17915904001, 250822656001, 5518098432001, 142216445952001, 2897001676800001, 90386452316160001, 1807729046323200001, 52563198423859200001
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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 nth 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, n1, 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 nth 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( n1, b, p*f[n], f), while( isprime( f[n]++ ), ) /* next composite */ ); b } /* then vector(30, n, A081545(n1)) gives the first 30 terms */ \\ M. F. Hasler, Jun 16 2007


CROSSREFS

Cf. A002808 (composite numbers), A073918, A081546, A131100.
Sequence in context: A202637 A138658 A067464 * A274074 A097496 A099657
Adjacent sequences: A081542 A081543 A081544 * A081546 A081547 A081548


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



