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,...)

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

Author

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

Keywords

• nonn

Comments

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

Examples

			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.
		

References

• 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

Crossrefs

Cf. A007504 (sum of first n primes).

Cf. A167764.

Formula

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).

Extensions

Extended by R. J. Mathar, Nov 17 2009