A098561 -id:A098561 - cp's OEIS frontend

A024450 Sum of squares of the first n primes.

4, 13, 38, 87, 208, 377, 666, 1027, 1556, 2397, 3358, 4727, 6408, 8257, 10466, 13275, 16756, 20477, 24966, 30007, 35336, 41577, 48466, 56387, 65796, 75997, 86606, 98055, 109936, 122705, 138834, 155995, 174764, 194085, 216286, 239087, 263736, 290305, 318194

Offset: 1

Clark Kimberling

nonn

It appears that the only square in this sequence is 4. Checked 10^11 terms. a(10^11) = 247754953701579144582110673365391267. - T. D. Noe, Sep 06 2005
a(2n-1) is divisible by 2, a(3n+1) is divisible by 3, a(4n-3) is divisible by 4, a(6n+1) is divisible by 6, a(8n-3) is divisible by 8, a(12n+1) is divisible by 12, a(24n-11) is divisible by 24. - Alexander Adamchuk, Jun 15 2006
The sequence is best looked at in base 12, with X for 10 and E for 11: 4, 11, 32, 73, 154, 275, 476, 717, X98, 1479, 1E3X, 289E, 3860, 4941, 6082, 7823, 9844, EX25, 12546, 15447, 18548, 20089, 2406X, 2876E, 320E0, 37E91, 42152, 488E3, 53754, 5E015, 68416, 76337, 85178, 94399, X51EX, E643E, 108760, 120001. Since the squares of all primes greater than 3 are always 1 mod 12, the sequence obeys the rule a(n) mod 12 = (n-1) mod 12 for n>=2. The rule gives a(2n-1) = (2n-2) mod 12 and so a(2n-1) must be divisible by 2. a(3n+1) = (3n) mod 12 so a(3n+1) is divisible by 3. The other rules are proved similarly. Remember: base 12 is a research tool! - Walter Kehowski, Jun 24 2006
For all primes p > 3, we have p^2 == 1 (mod m) for m dividing 24 (and only these m). Using a covering argument, it is not hard to show that all terms except a(24k+13) are nonsquares. Hence in the search for square a(n), only 1 out of every 24 terms needs to be checked. - T. D. Noe, Jan 23 2008

N. J. A. Sloane, Table of n, a(n) for n = 1..5000
Carlos Rivera, Puzzle 128: Sum of consecutive squared primes a square, The Prime Puzzles & Problems Connection.
Vladimir Shevelev, Asymptotics of sum of the first n primes with a remainder term
Eric Weisstein's World of Mathematics, Prime Sums
Eric Weisstein's World of Mathematics, Prime Zeta Function
OEIS Wiki, Sums of powers of primes divisibility sequences

Partial sums of A001248.
Cf. A024447, A024525, A098561, A098562.
Cf. A007504 (sum of the first n primes).

a024450 n = a024450_list !! (n-1)
a024450_list = scanl1 (+) a001248_list
-- Reinhard Zumkeller, Jun 08 2015

[&+[NthPrime(k)^2: k in [1..n]]: n in [1..40]]; // Vincenzo Librandi, Oct 11 2018

[n le 1 select 4 else Self(n-1) + NthPrime(n)^2: n in [1..80]]; // G. C. Greubel, Jan 30 2025

A024450:=n->add(ithprime(i)^2, i=1..n); seq(A024450(n), n=1..100); # Wesley Ivan Hurt, Nov 09 2013

Table[ Sum[ Prime[k]^2, {k, 1, n} ], {n, 40} ]
Accumulate[Prime[Range[40]]^2] (* Harvey P. Dale, Apr 16 2013 *)

s=0;forprime(p=2,1e3,print1(s+=p^2", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = norml2(primes(n)); \\ Michel Marcus, Nov 26 2020

from sympy import prime, primerange
def a(n): return sum(p*p for p in primerange(1, prime(n)+1))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Apr 13 2021

a(n) = A007504(n)^2 - 2*A024447(n). - Alexander Adamchuk, Jun 15 2006
a(n) = Sum_{i=1..n} prime(i)^2. - Walter Kehowski, Jun 24 2006
a(n) = (1/3)*n^3*log(n)^2 + O(n^3*log(n)*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). - Vladimir Shevelev, Aug 02 2013
a(n) = a(n-1) + prime(n)^2, with a(1) = 4. - G. C. Greubel, Jan 30 2025

A098562 Primes that are the sum of the squares of the first k primes for some k.

Original entry on oeis.org

13, 20477, 75997, 239087, 2210983, 3579761, 29194283, 40002073, 45448471, 55600481, 77290091, 108095623, 114986483, 155637463, 226226771, 302920139, 324657881, 519681709, 551321299, 618359839, 797005427, 944007487, 1039681147, 1124764853, 1923614047, 2135308631

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

These are the primes arising in A098561.

			From _K. D. Bajpai_, Dec 15 2014: (Start)
13 is in the sequence because the sum of the squares of the first 2 primes is 2^2 + 3^2 = 4 + 9 = 13, which is prime.
20477 is in the sequence because the sum of the squares of the first 18 primes is 2^2 + 3^2 + 5^2 + ... + 59^2 + 61^2 = 4 + 9 + 25 + ... + 3481 + 3721 = 20477, which is prime.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..15000

Cf. A098561 (corresponding n), A024450 (sum of squares of primes), A066525 (sums of cubes of primes), A013918 (sums of primes).

Cf. A000040, A006567. - Jonathan Vos Post, Aug 13 2009

Select[Table[Sum[Prime[k]^2, {k, 1, n}], {n, 1000}], PrimeQ]  (* K. D. Bajpai, Dec 15 2014 *)

s=0; forprime(p=2, 1e6, t=s+=p^2; if(isprime(t), print1(t,", "))) \\ K. D. Bajpai, Dec 15 2014

a(24)-a(26) from K. D. Bajpai, Dec 15 2014

a(42) in b-file corrected by Andrew Howroyd, Feb 28 2018

A098563 Numbers n such that the sum of the cubes of the first n primes is prime.

Original entry on oeis.org

4, 8, 38, 48, 98, 102, 118, 128, 130, 132, 156, 168, 172, 178, 180, 190, 202, 208, 308, 346, 358, 364, 424, 482, 540, 600, 602, 614, 646, 676, 722, 748, 768, 776, 782, 792, 838, 902, 1016, 1028, 1036, 1058, 1062, 1082, 1086, 1100, 1102, 1132, 1144, 1176

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

n must clearly be even.

			4 is a term as the sum of the cubes of the first four primes is 2^3 + 3^3 + 5^3 + 7^3 = 503, which is prime.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A066525 (corresponding primes), A098561 (sums of squares of primes), A013916 (sums of primes), A098999 (sums of cubes of primes).

with(numtheory): P:=proc(n) add(ithprime(k)^3, k=1..n): end:
A098563 := proc(n)local m: option remember: if(n=0)then return 0: fi: m:=procname(n-1)+2: while true do if(isprime(P(m)))then return m:fi: m:=m+2:od: end:
seq(A098563(n), n=1..50); # Nathaniel Johnston, Apr 21 2011

Select[Range[1000], PrimeQ[Sum[Prime[i]^3, {i, #}]] &] (* Carl Najafi, Aug 22 2011 *)

lista(nn) = {s = 0; ip = 0; forprime (p=1, nn, ip++; if (isprime(s+=p^3), print1(ip, ", ")););} \\ Michel Marcus, Aug 22 2015

A124225 Numbers n such that the sum of the first n primes is prime and the sum of the squares of the first n primes is also prime.

Original entry on oeis.org

2, 158, 192, 216, 356, 426, 548, 680, 1178, 1196, 1466, 1500, 1524, 2324, 2438, 2904, 2990, 3060, 3146, 3618, 3902, 4110, 4134, 4346, 4602, 5790, 5840, 6186, 6344, 6710, 6720, 6836, 6990, 7592, 7632, 7716, 7790, 7838, 8156, 8420, 8622, 8658, 8664, 9092

Offset: 1

Tanya Khovanova, Dec 13 2006

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 3618
Carlos Rivera, Puzzle 978. Improve this curio, Prime Puzzles and Problems Connection.

Intersection of A098561 (Numbers n such that the sum of the squares of the first n primes is prime) and A013916 (Numbers n such that the sum of the first n primes is prime).

a013916:=func; a098561:=func; [n: n in [1..10000]|a013916(n) and a098561(n)];  // Bruno Berselli, Dec 28 2011

With[{nn=9100},Position[Thread[{Accumulate[Prime[Range[nn]]],Accumulate[ Prime[ Range[ nn]]^2]}],?(PrimeQ[ #[[1]]]&&PrimeQ[#[[2]]]&),1,Heads-> False]]//Flatten (* _Harvey P. Dale, Aug 18 2020 *)

s=0;t=0;n=0;forprime(p=2,1e6,s+=p;t+=p^2;n++;if(isprime(t)&&isprime(s),print1(n", "))) \\ Charles R Greathouse IV, Dec 28 2011

More terms from Bruno Berselli, Dec 28 2011

Definition rephrased by Harvey P. Dale, Aug 18 2020

A122209 Sum of squares of the first n^2 primes = A024450[n^2].

Original entry on oeis.org

4, 87, 1556, 13275, 65796, 239087, 710844, 1789395, 4083404, 8384727, 16156884, 29194283, 50363460, 82888311, 132264452, 204330315, 306450780, 450504551, 647579748, 913503459, 1262033828, 1725350295, 2318488092, 3072687971

Offset: 1

Alexander Adamchuk, Aug 25 2006

nonn

Prime a(n) are listed in A122210[n] = {239087,29194283,13459558559,2330212120559,591302115428891,...}. Corresponding numbers n such that a(n) is a prime are listed in A122211[n] = {6,12,30,66,156,180,228,336,366,...}.

Cf. A122210, A122211, A024450, A098561, A098562.

Table[Sum[Prime[k]^2,{k,1,n^2}],{n,1,50}]
Module[{nn=600,t},t=Accumulate[Prime[Range[nn]]^2];Table[t[[i]],{i,Range[ Floor[ Sqrt[nn]]]^2}]] (* Harvey P. Dale, Dec 21 2014 *)

a(n) = Sum[ Prime[k]^2, {k,1,n^2} ]. a(n) = A024450[n^2].

A122210 Primes in A122209[n].

Original entry on oeis.org

239087, 29194283, 13459558559, 2330212120559, 591302115428891, 1475383481009147, 6659290813076243, 78234869090622611, 134532153287171039, 1936272192837757871, 12491376574210826183, 25493310333833042507

Offset: 1

Alexander Adamchuk, Aug 26 2006

nonn

Sum of squares of the first n^2 primes is A122209[n] = A024450[n^2] = {4,87,1556,13275,65796,239087,710844,1789395,4083404,8384727,16156884,29194283,...}. Corresponding numbers n such that A122209[n] is prime are listed in A122211[n] = {6,12,30,66,156,180,228,336,366,558,750,840,894,978,...}.

Cf. A122209, A122211, A024450, A098561, A098562.

s=0;Do[p=Prime[n];k=Sqrt[n];s=s+p*p;If[PrimeQ[s]&&IntegerQ[k],Print[{k,n,s}]],{n,1,10^7}]

a(n) = A122209[ A122211(n) ] = A024450[ A122211(n)^2 ].

A122211 Numbers k such that the sum of squares of the first k^2 primes is a prime.

Original entry on oeis.org

6, 12, 30, 66, 156, 180, 228, 336, 366, 558, 750, 840, 894, 978, 1398, 1410, 1506, 1560, 1578, 1662, 1794, 1800, 1812, 1824, 1890, 1992, 2094, 2268, 2334, 2358, 2430, 2604, 2736, 2742, 2766, 2802, 2856, 2922, 3042, 3312, 3390, 3702, 3948, 3954, 3984, 4170, 4314

Offset: 1

Alexander Adamchuk, Aug 26 2006

nonn

Corresponding primes A122209(a(n)) = A024450(a(n)^2) are listed in A122210(n) = {239087, 29194283, 13459558559, 2330212120559, ...}. All a(n) are of the form 6*m, where m = {1, 2, 5, 11, 26, 30, 38, 56, 61, 93, 125, 140, 149, 163, 233, 235, 251, 260, 263, 277, 299, 300, ...}. Because A122209(2*m-1) is an even number and A122209(3*m-1) == A122209(3*m+1) == 0 (mod 3) for m >= 1. [Edited by Jinyuan Wang, Mar 23 2020]

Cf. A024450, A098561, A098562, A122209, A122210.

s=0;Do[p=Prime[n];k=Sqrt[n];s=s+p*p;If[PrimeQ[s]&&IntegerQ[k],Print[{k,n,s}]],{n,1,10^7}]

A122209(a(n)) = A024450(a(n)^2) = A122210(n).

More terms from Jinyuan Wang, Mar 23 2020

A160358 Indices of primes in A133547, i.e., numbers n such that the sum of the squares of the first n odd primes is prime.

Original entry on oeis.org

3, 5, 9, 11, 23, 29, 63, 65, 71, 95, 141, 159, 161, 173, 179, 183, 209, 219, 255, 299, 323, 341, 365, 371, 389, 393, 453, 485, 521, 567, 579, 605, 623, 633, 635, 639, 677, 701, 711, 723, 725, 747, 785, 827, 867, 945, 981, 993, 999, 1001, 1013, 1035, 1037, 1041

Offset: 1

M. F. Hasler, May 18 2009

nonn

All terms are necessarily odd. Thus one could also consider the sequence floor(a(n)/2) = (1,2,4,5,11,14,31,32,35,...). Other possible variations would be to list the index a(n)+1 of the largest prime in that sum, or, since this is always even, (a(n)+1)/2 = (2,3,5,6,12,15,32,33,36,...).

Cf. A098561, A133547.

s=0; for( i=2,1999, isprime(s+=prime(i)^2) & print1(i-1,","))

A160359(n) = A133547(a(n)) = A024450(a(n)+1) - 4.

A140596 Squares nearest to and < terms in A098562.

Original entry on oeis.org

9, 20449, 75625, 238144, 2208196, 3579664, 29192409, 39992976, 45441081, 55591936, 77281681, 108076816, 114982729, 155625625, 226201600, 302899216, 324648324, 519657616, 551310400, 618317956, 796989361, 943964176, 1039675536

Offset: 1

Enoch Haga, May 17 2008

			The first term of A098562 is 13, the prime sum of 2^2=4 and 3^2=9. The square just preceding 13 is 9, the first term of this sequence.

Carlos Rivera, Puzzle 128. Sum of consecutive squared primes a square, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 443. Sum of cubes of consecutive primes, The Prime Puzzles and Problems Connection.

Cf. A048760, A140597, A066525, A098561, A098562.

a(n) = A048760(A098562(n)). - Jason Yuen, Sep 30 2024

A140597 Squares nearest to and > terms in A098562.

Original entry on oeis.org

16, 20736, 76176, 239121, 2211169, 3583449, 29203216, 40005625, 45454564, 55606849, 77299264, 108097609, 115004176, 155650576, 226231681, 302934025, 324684361, 519703209, 551357361, 618367689, 797045824, 944025625, 1039740025

Offset: 1

Enoch Haga, May 17 2008

			The first term of A098562 is 13, the prime sum of 2^2 + 3^2, where 4+9=13. The square just exceeding 13 is 16, the first term of this sequence.

Carlos Rivera, Puzzle 128. Sum of consecutive squared primes a square, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 443. Sum of cubes of consecutive primes, The Prime Puzzles and Problems Connection.

Cf. A048761, A140596, A066525, A098561, A098562, A140250, A140251.

a(n) = A048761(A098562(n)). - Jason Yuen, Sep 30 2024

cp's OEIS Frontend

A024450 Sum of squares of the first n primes.

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A098562 Primes that are the sum of the squares of the first k primes for some k.

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A098563 Numbers n such that the sum of the cubes of the first n primes is prime.

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A124225 Numbers n such that the sum of the first n primes is prime and the sum of the squares of the first n primes is also prime.

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A122209 Sum of squares of the first n^2 primes = A024450[n^2].

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A122210 Primes in A122209[n].

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A122211 Numbers k such that the sum of squares of the first k^2 primes is a prime.

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