A109770 -id:A109770 - cp's OEIS frontend

A109791 a(n) = prime(n^4).

2, 53, 419, 1619, 4637, 10627, 21391, 38873, 65687, 104729, 159521, 233879, 331943, 459341, 620201, 821641, 1069603, 1370099, 1731659, 2160553, 2667983, 3260137, 3948809, 4742977, 5653807, 6691987, 7867547, 9195889, 10688173, 12358069

Offset: 1

Jonathan Vos Post, Aug 14 2005

nonn

Since the prime number theorem is the statement that prime[n] ~ n * log n as n -> infinity [Hardy and Wright, page 10] we have a(n) = prime(n^4) is asymptotically (n^4)*log(n^4) = 4*(n^4)*log(n).

			a(1) = prime(1^4) = 2,
a(2) = prime(2^4) = 53,
a(3) = prime(3^4) = 419, etc.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..140

Cf. A000040, A000583, A011757, A109724, A109770.

[NthPrime(n^4): n in [1..50] ]; // Vincenzo Librandi, Apr 22 2011

Prime[Range[30]^4] (* Harvey P. Dale, Jun 07 2017 *)

a(n)=prime(n^4) \\ Charles R Greathouse IV, Oct 03 2013

[nth_prime(n^4) for n in (1..30)] # G. C. Greubel, Dec 09 2018

a(n) = A000040(A000583(n)) for n > 0.

A109789 a(n) = prime(1^3) + prime(2^3) + prime(3^3) + ... + prime(n^3).

Original entry on oeis.org

2, 21, 124, 435, 1126, 2447, 4756, 8427, 13946, 21865, 32822, 47575, 66978, 91787, 123106, 161979, 209636, 267195, 336226, 418025, 514162, 626453, 756526, 906243, 1077772, 1272815, 1493676, 1742527, 2021958, 2334541, 2682248, 3068341

Offset: 1

Jonathan Vos Post, Aug 14 2005

nonn

Analogy with prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724). If we take the cumulative sum of A055875 including the 0th value of 1, the 3rd value becomes prime(0^3) + prime(1^3) + prime(2^3) + prime(3^3) = 1 + 2 + 19 + 103 = 125 = 5^3.

			a(1) = 2 because prime(1^3) = prime(1) = 2;
a(2) = 21 because prime(1^3) + prime(2^3) = prime(1) + prime(8) = 2 + 19;
a(3) = 124 because prime(1^3) + prime(2^3) + prime(3^3) = prime(1) + prime(8) + prime(27) = 2 + 19 + 103;
a(4) = 435 because prime(1^3) + prime(2^3) = prime(1) + prime(8) + prime(27) + prime(64) = 2 + 19 + 103 + 311.
a(6) = 2 + 19 + 103 + 311 + 691 + 1321 = 2447 (which is prime).
a(28) = 2 + 19 + 103 + 311 + 691 + 1321 + 2309 + 3671 + 5519 + 7919 + 10957 + 14753 + 19403 + 24809 + 31319 + 38873 + 47657 + 57559 + 69031 + 81799 + 96137 + 112291 + 130073 + 149717 + 171529 + 195043 + 220861 + 248851 = 1742527 (which is prime).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A055875, A011757, A109724, A109770.

a(n) = sum(k=1, n, prime(k^3)); \\ Michel Marcus, Apr 17 2021

Cumulative sums of A055875(n) for n>0.

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007

A109796 a(n) = prime(1^4) + prime(2^4) + ... + prime(n^4).

Original entry on oeis.org

2, 55, 474, 2093, 6730, 17357, 38748, 77621, 143308, 248037, 407558, 641437, 973380, 1432721, 2052922, 2874563, 3944166, 5314265, 7045924, 9206477, 11874460, 15134597, 19083406, 23826383, 29480190, 36172177, 44039724

Offset: 1

Jonathan Vos Post, Aug 15 2005

nonn

Analog of prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724). For a(n) to be prime for n > 1 it is necessary but not sufficient that n == 0 (mod 4).

			a(1) = 2 because prime(1^4) = prime(1) = 2.
a(2) = 55 because prime(1^4) + prime(2^4) = prime(1) + prime(16) = 2 + 53.
a(3) = 474 because prime(1^4) + prime(2^4) + prime(3^4) = prime(1) + prime(16) + prime(81) = 2 + 53 + 419.
a(4) = 2093 because prime(1^4) + prime(2^4) + prime(3^4) + prime(4^4) = 2 + 53 + 419 + prime(256) = 2 + 53 + 419 + 1619.
a(8) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 = 77621 (which is prime).
a(12) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 + 65687 + 104729 + 159521 + 233879 = 641437 (which is prime).
a(28) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 + 65687 + 104729 + 159521 + 233879 + 331943 + 459341 + 620201 + 821641 + 1069603 + 1370099 + 1731659 + 2160553 + 2667983 + 3260137 + 3948809 + 4742977 + 5653807 + 6691987 + 7867547 + 9195889 = 53235613 (which is prime).
It is a coincidence that a(1), a(2) and a(3) are all palindromes.

Harvey P. Dale, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Biquadratic Number.

First differences are A109791.

Cf. A000040, A000290, A000583, A011757, A109724, A109770.

Accumulate[Table[Prime[n^4],{n,30}]] (* Harvey P. Dale, Feb 02 2019 *)

A109796(n)={
        sum(i=1,n,prime(i^4))
} /* R. J. Mathar, Mar 09 2012 */

a(n) = Sum_{i=1..n} A000040(A000583(i)).

A343512 Numbers k such that Sum_{i=1..k} prime(i^3) is prime.

Original entry on oeis.org

1, 6, 28, 72, 90, 92, 96, 112, 118, 148, 160, 162, 184, 222, 282, 312, 314, 316, 330, 336, 390, 396, 418, 440, 444, 448, 472, 488, 524, 534, 552, 598, 604, 614, 638, 748, 758, 798, 824, 848, 906, 916, 970, 992, 1008, 1010, 1012, 1016, 1056, 1078, 1084, 1094, 1098

Offset: 1

Chai Wah Wu, Apr 17 2021

nonn

Numbers n such that A109789(n) is prime. For n > 1, a(n) is even.

			72 is a term since Sum_{i=1..72} prime(i^3) = 94154923 is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A055875, A011757, A109724, A109770, A109789.

A109801 Cumulative sum of squares of primes indexed by squares.

Original entry on oeis.org

4, 53, 582, 3391, 12800, 35601, 87130, 183851, 359412, 652093, 1089014, 1772943, 2791024, 4214273, 6250602, 8871763, 12402404, 16994853, 22933822, 30446903, 39951792, 51930313, 66393122, 84125643, 105627412, 131140013, 161599374

Offset: 1

Jonathan Vos Post, Aug 15 2005

nonn

Related to Prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724).

			a(1) = 4 because (prime[1^2])^2 = (prime[1])^2 = 2^2.
a(2) = 53 because (prime[1^2])^2 + (prime[2^2])^2 = 2^2 + 7^2 = 4 + 49 = 53 (which is prime).
a(3) = 582 because (prime[1^2])^2 + (prime[2^2])^2 + (prime[3^2])^2 = 2^2 + 7^2 + 23^2 = 582.
a(4) = 582 because (prime[1^2])^2 + (prime[2^2])^2 + (prime[3^2])^2 + (prime[4^2])^2 = 2^2 + 7^2 + 23^2 + 53^2 = 3391 (which is prime).
a(32) = a(31) + (prime[32^2])^2 = 345995122 + 8161^2 = 412597043 (which is prime).
a(34) = a(33) + (prime[34^2])^2 = 488932212 + 9341^2 = 576186493 (which is prime).

Cf. A000040, A000290, A000583, A011757, A109724, A109770.

Accumulate[Prime[Range[30]^2]^2] (* Harvey P. Dale, Mar 28 2012 *)

(Prime[1^2])^2 + (prime[2^2])^2 + ... + (prime[n^2])^2. a(n+1) = a(n) + (A011757(n+1))^2.

cp's OEIS Frontend

A109791 a(n) = prime(n^4).

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A109789 a(n) = prime(1^3) + prime(2^3) + prime(3^3) + ... + prime(n^3).

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A109796 a(n) = prime(1^4) + prime(2^4) + ... + prime(n^4).

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A343512 Numbers k such that Sum_{i=1..k} prime(i^3) is prime.

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A109801 Cumulative sum of squares of primes indexed by squares.

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Mathematica

Formula