A109724 -id:A109724 - cp's OEIS frontend

A122207 Primes of the form A109724[n] = A007504[n^2] or the sum of the first n^2 primes.

2, 17, 8893, 24133, 768373, 1583293, 2180741, 3875933, 6426919, 173472547, 289093219, 741938801, 2738357903, 2895147163, 3058653607, 17056871941, 24109439269, 26602406177, 29286422621, 62073696299, 65420584637, 68902997299

Offset: 1

Alexander Adamchuk, Aug 25 2006

nonn

Numbers n such that A109724[n] = A007504[n^2] is a prime are listed in A122208[n] = {1,2,8,10,22,26,28,32,36,78,88,110,150,152,154,...}.

			Prime 17 = 2 + 3 + 5 + 7 = a(2) is equal to the sum of the first 4 = 2^2 primes.
Prime 24133 = a(4) is equal to sum of the first 100 = 10^2 primes.

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A122208, A109724, A007504.

s=0;Do[s=Sum[Prime[k],{k,1,n^2}];If[PrimeQ[s],Print[{n,n^2,s}]],{n,1,154}]
Select[Table[Total[Prime[Range[n^2]]],{n,500}],PrimeQ] (* Harvey P. Dale, Jul 20 2024 *)

a(n) = A109724[ A122208[n] ] = A007504[ A122208[n]^2 ].

More terms from Ray Chandler, Dec 02 2018

A122208 Numbers n such that the sum of the first n^2 primes A109724(n) = A007504(n^2) is a prime.

Original entry on oeis.org

1, 2, 8, 10, 22, 26, 28, 32, 36, 78, 88, 110, 150, 152, 154, 232, 252, 258, 264, 316, 320, 324, 368, 376, 426, 496, 516, 532, 608, 644, 666, 686, 764, 828, 832, 880, 932, 958, 1020, 1090, 1096, 1106, 1122, 1156, 1174, 1206, 1264, 1280, 1282, 1290, 1296, 1326

Offset: 1

Alexander Adamchuk, Aug 25 2006

nonn

Corresponding primes that are equal to the sum of the first a(n)^2 primes are listed in A122207(n) = {2, 17, 8893, 24133, 768373, 1583293, 2180741, 3875933, 6426919, 173472547, 289093219, 741938801, 2738357903, 2895147163, 3058653607, ...}. - Robert G. Wilson v, Sep 29 2006

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A122207, A109724, A007504.

s = 0; t = {}; Do[s = s + Sum[Prime@k, {k, (n - 1)^2 + 1, n^2}]; If[PrimeQ@s, AppendTo[t, n]], {n, 1341}]; t (* Robert G. Wilson v *)

A122207(n) = A109724( a(n) ) = A007504( a(n)^2 ). - Robert G. Wilson v, Sep 29 2006

More terms from Robert G. Wilson v, Sep 29 2006

A055875 a(0)=1, a(n) = prime(n^3).

Original entry on oeis.org

1, 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, 279431, 312583, 347707, 386093, 427169, 470933, 517553

Offset: 0

Steven Pigeon (pigeon(AT)iro.umontreal.ca), Jul 14 2000

nonn

A sequence of increments for Shell sort that produces good results. A bit better than Sedgewick's A036562 and A003462.

David A. Corneth, Table of n, a(n) for n = 0..2200 (first 1001 terms from Ivan Panchenko)
Index entries for sequences related to sorting
Andrew Booker, Nth Prime Page.

Cf. A033844, A036562, A003462, A036567, A036569, A055876.
Sequences used for Shell sort: A003462, A033622, A036562, A036564, A036569, A055875.
Cf. A011757, A109724.

[NthPrime(n^3): n in [0..50] ]; // Vincenzo Librandi, Apr 22 2011

{1}~Join~Array[Prime[#^3] &, 35] (* Michael De Vlieger, Apr 13 2021 *)

first(n) = { my(res = vector(n), t = 0); forprime(p = 2, oo, t++; if(ispower(t, 3, &i), print1([i, p]", "); res[i] = p; if(i >= n, return(concat(1, res))))) } \\ David A. Corneth, Apr 13 2021

a(n) = A000040(A000578(n)), n>0.

More terms from Jonathan Vos Post, Aug 13 2005

A109722 Sum of first 2n primes.

Original entry on oeis.org

0, 5, 17, 41, 77, 129, 197, 281, 381, 501, 639, 791, 963, 1161, 1371, 1593, 1851, 2127, 2427, 2747, 3087, 3447, 3831, 4227, 4661, 5117, 5589, 6081, 6601, 7141, 7699, 8275, 8893, 9523, 10191, 10887, 11599, 12339, 13101, 13887, 14697, 15537, 16401, 17283

Offset: 0

Giovanni Teofilatto, Aug 10 2005

Bisection of A007504.

Ray Chandler, Table of n, a(n) for n = 0..10000
Romeo Meštrović, Curious conjectures on the distribution of primes among the sums of the first 2n primes, arXiv:1804.04198 [math.NT], 2018.

Cf. A007504, A109723, A109724, A109725, A109726.

f[n_] := Sum[Prime[k], {k, n}]; Table[f[2n], {n, 0, 43}]
Join[{0},With[{nn=100},Take[Accumulate[Prime[Range[nn]]],{2,nn,2}]]] (* Harvey P. Dale, Dec 20 2021 *)

a(n) = A007504(2n).

Edited and extended by Ray Chandler, Aug 11 2005

A109723 Sum of the first 2n+1 primes.

Original entry on oeis.org

2, 10, 28, 58, 100, 160, 238, 328, 440, 568, 712, 874, 1060, 1264, 1480, 1720, 1988, 2276, 2584, 2914, 3266, 3638, 4028, 4438, 4888, 5350, 5830, 6338, 6870, 7418, 7982, 8582, 9206, 9854, 10538, 11240, 11966, 12718, 13490, 14288, 15116, 15968, 16840

Offset: 0

Giovanni Teofilatto, Aug 10 2005

Bisection of A007504.

Ray Chandler, Table of n, a(n) for n = 0..10000

Cf. A007504, A109722, A109724, A109725, A109726.

f[n_] := Sum[Prime[k], {k, n}]; Table[f[2n + 1], {n, 0, 42}]
Take[Accumulate[Prime[Range[160]]], {1,160,2}] (* Harvey P. Dale, Jan 11 2011 *)

a(n) = sum(k=1, 2*n+1, prime(k)); \\ Michel Marcus, Jan 31 2019

a(n) = A007504(2n+1).

Edited and extended by Ray Chandler, Aug 11 2005

A109725 Divide primes in groups with 2n+1 elements and add together.

Original entry on oeis.org

2, 15, 83, 281, 679, 1367, 2461, 4005, 6223, 9017, 12755, 17281, 22967, 29597, 37793, 47229, 57993, 70957, 85343, 101777, 119469, 141079, 163313, 188201, 216203, 247203, 280897, 316551, 355905, 398825, 445509, 494953, 549737, 605711, 665185, 730353, 801481

Offset: 0

Giovanni Teofilatto, Aug 10 2005

First difference of A109724.

Ray Chandler, Table of n, a(n) for n = 0..9999

Cf. A007504, A109722-A109726.

f[n_] := Sum[Prime[k], {k, n}]; Table[f[(n+1)^2] - f[n^2], {n, 0, 34}]
With[{nn=40},Total/@TakeList[Prime[Range[nn^2]],Range[1,2nn-1,2]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jan 05 2019 *)

from sympy import prime
def a(n): return sum(prime(i) for i in range(n*n+1, (n+1)**2+1))
print([a(n) for n in range(37)]) # Michael S. Branicky, Feb 15 2021

a(n) = A109724(n+1) - A109724(n) = A007504((n+1)^2) - A007504(n^2).

Edited and extended by Ray Chandler, Aug 11 2005

A109770 Prime(1^2) + prime(2^2) + ... + prime(n^2).

Original entry on oeis.org

2, 9, 32, 85, 182, 333, 560, 871, 1290, 1831, 2492, 3319, 4328, 5521, 6948, 8567, 10446, 12589, 15026, 17767, 20850, 24311, 28114, 32325, 36962, 42013, 47532, 53539, 60020, 67017, 74590, 82751, 91488, 100829, 110760, 121387, 132708, 144757

Offset: 1

Jonathan Vos Post, Aug 13 2005

nonn

			a(1) = 2 = prime(1^2).
a(2) = 9 = 2 + 7 = prime(1^2) + prime(2^2).
a(3) = 32 = 2 + 7 + 23 = prime(1^2) + prime(2^2) + prime(3^2).
a(4) = 32 = 2 + 7 + 23 + 53 = prime(1^2) + prime(2^2) + prime(3^2) prime(4^2).
a(x) = 2+7+23+53+97+151+227+311+419+541+661+827+1009+1193+1427+1619+1879+2143+2437+2741+3083+3461+3803+4211+4637+5051+5519+6007+6481+6997+7573+8161+8737+9341+9931+10627+11321+12049+12743+13499+14327 = 185326.

Cf. A011757, A109724.

Accumulate[Prime[Range[40]^2]] (* Harvey P. Dale, May 31 2014 *)

Cumulative sum of A011757.

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

Original entry on oeis.org

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)).

cp's OEIS Frontend

A122207 Primes of the form A109724[n] = A007504[n^2] or the sum of the first n^2 primes.

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A122208 Numbers n such that the sum of the first n^2 primes A109724(n) = A007504(n^2) is a prime.

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A055875 a(0)=1, a(n) = prime(n^3).

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A109722 Sum of first 2n primes.

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A109723 Sum of the first 2n+1 primes.

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A109725 Divide primes in groups with 2n+1 elements and add together.

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A109770 Prime(1^2) + prime(2^2) + ... + prime(n^2).

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