0,1

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 multiple 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( prime(i)^e(i), i=1..oo ) 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)

M. F. Hasler, Table of n, a(n) for n = 0,...,100

Theorem: For any m>0 there is 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 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)

Writing c(n) for the n-th composite number A002808(n):

a(0) = prod(c(i),i=1..0)+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 isn't 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 isn't prime.

a(7) = p(7)+1 = 2903041 with p(n)=prod(c(i),i=1..n). (M. F. Hasler, Jun 16 2007)

(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)

Cf. A131100, A081546, A002808 (composite numbers), A073918.

Sequence in context: A086218 A138658 A067464 this_sequence A097496 A099657 A107633

Adjacent sequences: A081542 A081543 A081544 this_sequence A081546 A081547 A081548

hard,nonn,nice

Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 01 2003

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) Jun 13 2003

Terms beyond a(8) by M. F. Hasler (Maximilian.Hasler(AT)gmail.com) Jun 16 2007