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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,...)

%I A167790 #17 May 10 2025 08:58:55
%S A167790 3,10,3,5,8,49,13,23,23,7,39,29,15,10,39,25,30,151,38,19,139,27,174,
%T A167790 21,287,422,240,24,94,22,16,173,861,231,143,140,213,902,18,134,143,
%U A167790 310,70,58,295,550,237,210,229,57,221,271,194,540,145,718,116,184,90,71,168
%N 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,...)
%C A167790 It is conjectured that the sequence is infinite
%C A167790 If t is not restricted to nontrivial divisors, the sequence becomes A111287. - _R. J. Mathar_, Nov 17 2009
%D A167790 Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer 2005
%D A167790 Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
%D A167790 Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996
%F A167790 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).
%e A167790 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;
%e A167790 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;
%e A167790 s(422)=2+3+5+...+2917=570145= 5 * 101 * 1129=5645*prime(26) gives a(26)=422 and demonstrates the numerical difficulties.
%Y A167790 Cf. A007504 (sum of first n primes).
%Y A167790 Cf. A167764.
%K A167790 nonn
%O A167790 1,1
%A A167790 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 12 2009, Nov 13 2009
%E A167790 Extended by _R. J. Mathar_, Nov 17 2009