A227319 - OEIS

%I #7 Jul 11 2013 11:01:09

%S 8,243,128,216,12167,1906624,6859,226981,3125,12167,1252726552,

%T 325660672,1331,19902511000,32768,537824,69934528000,704969,39304,

%U 42875,50653,751089429,79507,314432,21952,22665187,47437928,1605723211,10648,287496,5177717,7414875

%N Powers but not squares which are sum of consecutive primes less than 10^7 ordered according to the proximity of the first prime of the sum to the first prime: 2.

%C Having sequences with limits (10^7) is not OEIS policy. We make an exception here. - _T. D. Noe_, Jul 11 2013

%C There are 3 other important informative parameters (A 4-dimensional sequence from an informative point of view) for each term of the sequence : (b,l,k) where b is the base powered, l is the number of primes added and k is the k-th prime where the sum of the consecutive primes begin : (2,2,2), (3,5,13), (2,2,18), (6,2,28), (23,47,32), (124,704,34), (19,25,46), (61,233,47), (5,11,55), (23,27,74), (1078,15442,74), (688,8116,78), (11,3,85), (2710,57856,87), (2,48,99), (14,320,100), (4120,105616,111), (89,345,135), (34,42,139), (35,45,140), (37,51,143), (909,12023,149), (43,65,168), (68,186,170), (28,20,174), (283,2137,205), (362,3102,206), (1171,17211,247), (22,6,273), (66,126,277), (173,907,292), (195,1107,303)

%C The limit (of 10^7) in the name/definition of the sequence is necessary because no power with exponent greater than 2 has been found for sums of primes beginning with first prime. Or beginning with many other primes. Naturally this limit could be much widened and alter the sequence; but this is why I put it in the name.

%e 8 = 2^3 = 3 + 5; 243 = 3^5 = 41 + 43 + 47 + 53 + 59; 128 = 2^7 = 61 + 67; 216 = 6^3 = 107 + 109; 12167 = 23^3 = S(47,32) = Sum of 47 primes beginning with p(32); ... ; 11^3 = 1331 = 439 + 443 + 449; 5^5 = 3125 = S(11,55); 11^5 = 161051 = S(47,458); 2^13 = 8192 = 4093 + 4099;  3^13 = 1594323 = S(233,764); 2^17 = 131072 = S(40,443) = S(8,1896); 7^7 = 823543 = S(7^2,1917); 2^25 = S(1268,2269); 2001^3 = S(35209,2368).

%o (PARI) : n=10^7;v=vector(n);i=0;for(a=2,n,if(isprime(a),i++;v[i]=a));for(b=1,315,k=0;for(j=b,i,k=k+v[j];if(ispower(k,,&n)&!issquare(k),print1(k,", "))))

%Y Cf. A141092, A227249.

%K nonn,less

%O 1,1

%A _Robin Garcia_, Jul 06 2013