A227249 -id:A227249 - cp's OEIS frontend

A227319 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.

8, 243, 128, 216, 12167, 1906624, 6859, 226981, 3125, 12167, 1252726552, 325660672, 1331, 19902511000, 32768, 537824, 69934528000, 704969, 39304, 42875, 50653, 751089429, 79507, 314432, 21952, 22665187, 47437928, 1605723211, 10648, 287496, 5177717, 7414875

Offset: 1

Robin Garcia, Jul 06 2013

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

			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).

Cf. A141092, A227249.

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,", "))))

A227115 Powers but not squares which are sum of consecutive composites less than 10^7 ordered according to the proximity of the first composite of the sum to the first composite: 4.

Original entry on oeis.org

27, 10077696, 128, 32768, 8, 27, 1000, 1728, 5088448, 690807104, 27, 32, 512, 2048, 512, 6859, 4913, 243, 405224, 125, 3125, 2744, 98611128, 27000, 314432, 216, 1728, 1889568, 243, 2744, 512, 4913000

Offset: 1

Robin Garcia, Jul 04 2013

There are other informative data for each term of the sequence. They are (b,l,k) where b is the base to an odd power, l is the number of consecutive composites added, and k indicates the k-th composite c(k) from where the sums begin: (3,4,1), (6,4151,1), (2,10,2), (2,222,2), (2,1,3), (3,3,3), (10,30,7), (12,42,7), (172,2931,7), (884,35029,9), (3,1,17), (2,1,20), (2,13,20), (2,36,22), (2,12,23), (19,79,24), (17,59,31), (3,4,41), (74,772,42), (5,2,43), (5,37,43), (14,33,44), (462,13093,46), (30,162,47), (68,668,48), (6,3,50), (12,20,53), (18,1723,56), (3,3,57), (14,28,58), (2,6,59), (170,2827,60).

			We denote the n-th composite as c(n). Some of the odd powers are the sum of consecutive composites in several ways: 27 = 3^3 = c(1)+c(2)+c(3)+c(4) = c(3)+c(4)+c(5) = c(17) = 4 + 6 + 8 + 9 = 8 + 9 + 10. 243 = 3^5 = c(189) = c(90)+c(91) = c(57)+c(59)+c(59) = c(41)+c(42)+c(43)+c(44) = 121 + 122 = 80 + 81 + 82 = 58 + 60 + 62 + 63. 1000 = 10^3 is sum of 30 consecutive composites beginning with c(7) = 14. 1728 = 12^3 = Ramanujan taxicab minus 1 is sum of 42 consecutive composites beginning with c(7) = 14 and of 20 consecutive composites beginning with c(53) = 75.

Cf. A141092, A227249, A227319.

n1=10^7;v=vector(n1);i=0;for(a=2,n1,if(isprime(a),next,i++;v[i]=a));for(b=1,60,k=0;for(j=b,i,k=k+v[j];if(ispower(k,,&n)&ispower(k)%2==1,print1([k,n,ispower(k),j-b+1,b]," "))))

cp's OEIS Frontend

A227319 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.

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