A167764 a(n) is the index k of k-th prime prime(k) in the smallest concatenation "2 3 ... prime(k)" where prime(n+1) is a factor.

10, 3, 5, 7, 18, 11, 58, 2, 6, 28, 177, 85, 47, 3, 191, 35, 9, 12, 40, 108, 40, 60, 69, 43, 84, 314, 29, 77, 231, 59, 76, 49, 86, 289, 5, 51, 71, 43, 269, 101, 53, 78, 42, 46, 958, 22, 5, 101, 151, 65, 198, 80, 22, 428, 363, 172, 686, 494, 399, 11, 96, 425, 277, 525

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 11 2009, Nov 13 2009

Keywords

• nonn
• base

Comments

It is conjectured that this sequence is infinite.

Examples

			a(2) = a(14) = 3 because 235 = 5 * 47 = prime(2+1) * prime(14+1) is the concatenation of the first 3 primes.
a(20) = 108 as prime(108) = 593 and the 283-digit concatenation "235...593" has prime factor 73 = prime(20+1).
		

References

• Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer, 2005.
• Marcus du Sautoy, Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen, 2004.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Programs

PARI

a(n)=my(p=prime(n+1),k=2,i=0); forprime(q=3,default(primelimit),i++; if(k%p==0,return(i)); k=k*10^#Str(q)+q) \\ Charles R Greathouse IV, Apr 27 2010

Python

from sympy import nextprime, prime
def a(n):
    pn1 = prime(n+1)
    k, pk, s = 1, 2, "2"
    while int(s)%pn1:
        k += 1; pk = nextprime(pk); s += str(pk)
    return k
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Oct 03 2021

Extensions

Terms past a(10) and editing by Charles R Greathouse IV, Apr 27 2010