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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. 3

%I #17 Oct 04 2021 02:07:06

%S 10,3,5,7,18,11,58,2,6,28,177,85,47,3,191,35,9,12,40,108,40,60,69,43,

%T 84,314,29,77,231,59,76,49,86,289,5,51,71,43,269,101,53,78,42,46,958,

%U 22,5,101,151,65,198,80,22,428,363,172,686,494,399,11,96,425,277,525

%N 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.

%C It is conjectured that this sequence is infinite.

%D Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer, 2005.

%D Marcus du Sautoy, Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen, 2004.

%H Charles R Greathouse IV, <a href="/A167764/b167764.txt">Table of n, a(n) for n = 1..10000</a>

%e a(2) = a(14) = 3 because 235 = 5 * 47 = prime(2+1) * prime(14+1) is the concatenation of the first 3 primes.

%e a(20) = 108 as prime(108) = 593 and the 283-digit concatenation "235...593" has prime factor 73 = prime(20+1).

%o (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

%o (Python)

%o from sympy import nextprime, prime

%o def a(n):

%o pn1 = prime(n+1)

%o k, pk, s = 1, 2, "2"

%o while int(s)%pn1:

%o k += 1; pk = nextprime(pk); s += str(pk)

%o return k

%o print([a(n) for n in range(1, 65)]) # _Michael S. Branicky_, Oct 03 2021

%K nonn,base

%O 1,1

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

%E Terms past a(10) and editing by _Charles R Greathouse IV_, Apr 27 2010

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