cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Robin Garcia, Jul 06 2013

Keywords

Comments

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.

Examples

			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).

Crossrefs

Cf. A141092, A227249.

Programs

  • 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,", "))))

A141278 Clusters of consecutive composites in A141089.

Original entry on oeis.org

25, 26, 48, 49, 114, 115, 123, 124, 212, 213, 287, 288, 332, 333, 342, 343, 398, 399, 415, 416, 440, 441, 446, 447, 470, 471, 488, 489, 510, 511, 512, 548, 549, 553, 554, 603, 604, 638, 639, 640, 648, 649, 675, 676, 771, 772, 785, 786, 818, 819, 836, 837
Offset: 1

Views

Author

Enoch Haga, Jun 21 2008

Keywords

Comments

A141089 contains composites A002808(k) such that the partial sum A053767(k) divides the partial product A036691(k). The sequence contains the subsequences of A141089 that contain two or more consecutive integers.

Examples

			The first pair of consecutive integers is (25,26) in A141089(6,7), the second (48,49) in A141089(9,10).
Triples of consecutive integers in A141089 are (510,511,512), (638,639,640), (889,890,891), (912,913,914), quadruples are (987,988,989,990), etc, all members included here.

Crossrefs

Cf. A141089 A141090 A141091 A141092 A141274 A141275.

Formula

Numbers A141089(i) such that either 1+A141089(i) = A141089(i+1) or A141089(i)-1 = A141089(i-1) or both.

Extensions

Edited by R. J. Mathar, Jul 08 2008

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

Views

Author

Robin Garcia, Jul 04 2013

Keywords

Comments

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).

Examples

			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.

Crossrefs

Cf. A141092, A227249, A227319.

Programs

  • PARI
    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]," "))))
Previous Showing 11-13 of 13 results.

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