A392615 a(n) is the least k >= 0 such that gcd(A143293(n)+k, A002110(n+1)) = 1, where A002110 are the primorial numbers, and A143293 give their partial sums.

0, 2, 2, 2, 2, 8, 8, 8, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset

0,2

Comments

The least k >= 0 such that A020639(A143293(n+1)+k) > A000040(n+1).

The least k >= 0 such that A267263(A143293(n)+k)==n+1 and A324198(A143293(n)+k) = 1.

The first term > 14 is a(11980) = 32. See A392862 and A392864.

Links

Antti Karttunen, Table of n, a(n) for n = 0..12000

Index entries for sequences related to primorial numbers.

Formula

a(n) <= A392616(n).

Prog

(PARI)
A002110(n) = prod(i=1, n, prime(i));
A143293(n) = { if(n==0, return(1)); my(P=1, s=1); forprime(p=2, prime(n), s+=P*=p); s; };
A392615(n) = { my(b=A143293(n), pr=A002110(1+n)); for(k=0, oo, if(1==gcd(b+k, pr), return(k))); };
(PARI)
\\ Slower program:
A267263(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += !!d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
A392615(n) = if(!n, n, my(b=A143293(n)); for(k=b, oo, if(A267263(k)==n+1 && 1==A324198(k), return(k-b))));

Crossrefs

Cf. A000040, A002110, A020639, A143293, A324198, A392614, A392616, A392862, A392864.

Sequence in context: A013598 A100943 A152660 * A339163 A058787 A353392

Adjacent sequences: A392612 A392613 A392614 * A392616 A392617 A392618

Keyword

nonn

Author

Antti Karttunen, Jan 24 2026

Status

approved