A167764 -id:A167764 - cp's OEIS frontend

A173291 Smallest prime p such that the concatenation of p and prime(n) is a prime, or 0 if no other number exists.

0, 2, 0, 3, 2, 3, 3, 7, 2, 2, 3, 3, 2, 7, 3, 3, 3, 7, 3, 2, 3, 3, 2, 3, 3, 5, 7, 5, 3, 2, 7, 2, 2, 19, 11, 7, 19, 3, 3, 9, 2, 3, 3, 7, 5, 37, 7, 31, 5, 3, 5, 2, 13, 2, 3, 41, 2, 3, 31, 2, 7, 2, 3, 2, 3, 11, 3, 13, 2, 7, 11, 3, 13, 3, 19, 2, 2, 13, 17, 37, 5, 13, 5, 3, 139, 5, 3, 3, 3, 3, 2, 5, 7, 3, 3

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 15 2010

If prime(n) has k digits then a(k) is the smallest prime(m) where 10^k * prime(m) + prime(n) is a prime.

In base 10, no prime can be prefixed to 2 or 5 to make another prime.

			a(2) = 2 because prime(2) = 3, and the concatenation of 2 and 3 gives the prime 23.
a(3) = 0 because prime(3) = 5 and there is no prime to concatenate with to give another prime.
a(4) = 3 because prime(5) = 7 but the concatenation with 2 gives 27 = 3^3, so it has to be 3 in order to give 37, which is prime.

John Derbyshire, Prime obsession. Joseph Henry Press, Washington, DC 2003
Marcus du Sautoy, Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004
Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005

Cf. A088606, A167764, A168327, A168417, A030469.

A167790 a(n) is the index k of k-th prime prime(k) in the smallest sum s(k)=2+3+...+prime(k)=t*prime(n) of first k primes where t is a true divisor and first occurrence of factor prime(n) (n=1,2,3,...)

Original entry on oeis.org

3, 10, 3, 5, 8, 49, 13, 23, 23, 7, 39, 29, 15, 10, 39, 25, 30, 151, 38, 19, 139, 27, 174, 21, 287, 422, 240, 24, 94, 22, 16, 173, 861, 231, 143, 140, 213, 902, 18, 134, 143, 310, 70, 58, 295, 550, 237, 210, 229, 57, 221, 271, 194, 540, 145, 718, 116, 184, 90, 71, 168

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 12 2009, Nov 13 2009

nonn

It is conjectured that the sequence is infinite

If t is not restricted to nontrivial divisors, the sequence becomes A111287. - R. J. Mathar, Nov 17 2009

			s(5)=2+3+5+7+11=28=2^2*7=4*prime(4) gives a(4)=5 as first occurrence of prime factor prime(4)=7;
s(8)=2+3+5+7+11+13+17+19=77=7*11=7*prime(5) gives a(5)=8 as first occurrence of prime factor prime(5)=11;
s(422)=2+3+5+...+2917=570145= 5 * 101 * 1129=5645*prime(26) gives a(26)=422 and demonstrates the numerical difficulties.

Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer 2005
Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996

Cf. A007504 (sum of first n primes).

Cf. A167764.

a(n) = min[2+3+...+prime(k)/t], where the minimum is taken with respect to k, the denominator t > 1 is an integer divisor of numerator s(k)=2+3+...+prime(k).

Extended by R. J. Mathar, Nov 17 2009

cp's OEIS Frontend

A173291 Smallest prime p such that the concatenation of p and prime(n) is a prime, or 0 if no other number exists.

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