A098562 -id:A098562 - cp's OEIS frontend

A140596 Squares nearest to and < terms in A098562.

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

A024450 Sum of squares of the first n primes.

Original entry on oeis.org

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

A098561 Numbers n such that the sum of the squares of the first n primes is prime.

Original entry on oeis.org

2, 18, 26, 36, 68, 78, 144, 158, 164, 174, 192, 212, 216, 236, 264, 288, 294, 338, 344, 356, 384, 404, 416, 426, 500, 516, 518, 522, 534, 540, 548, 614, 678, 680, 782, 858, 866, 876, 878, 896, 900, 912, 950, 974, 996, 1064, 1080, 1082, 1100, 1122, 1158, 1160

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

a(n) must clearly be even.

			2 is a term as the sum of the squares of the first two primes is 2^2 + 3^2 = 13, which is prime.

Zak Seidov, Table of n, a(n) for n = 1..1050

Cf. A098562 (corresponding primes), A024450 (sums of squares of primes), A098563 (sums of cubes of primes), A013916 (sums of primes).

Select[Range[1000], PrimeQ[Sum[Prime[i]^2, {i, #}]] &] (* Carl Najafi, Aug 22 2011 *)

A117503 Primes among partial sums of floor(Pi*prime(k)), k=1,2,3,....

Original entry on oeis.org

613, 6229, 7607, 9679, 46133, 61469, 69191, 120067, 211663, 285049, 316697, 354323, 402371, 444979, 481109, 490313, 532709, 993907, 1055543, 1083721, 1237487, 1329701, 1409977, 1442899, 1484671, 1656199, 1700471, 1874767

Offset: 1

Enoch Haga, Mar 25 2006

Modeled on the same concept as cumulative sums of squared primes in A098562.

Cf. A117504, A098562.

Digits := 30 ; A117503 := proc(nmax) local a,pisum,p ; a := [] ; pisum := 0 ; p :=1 ; while nops(a) <=nmax do while true do pisum := pisum+floor(Pi*ithprime(p)) ; p := p+1 ; if isprime(pisum) then a := [op(a),pisum] ; break ; fi ; od : od : RETURN(a) ; end: a := A117503(30) ; # R. J. Mathar, Oct 26 2006

Select[Accumulate[Floor[Pi Prime[Range[800]]]],PrimeQ] (* Harvey P. Dale, Jun 06 2022 *)

10 Ct=1
20 B=nxtprm(B)
30 C=int(pi(B))
40 D=D+C
41 print Ct,B,C,D
50 if D=prmdiv(D) then print D:stop
55 Ct=Ct+1
60 goto 20

Define the sequence s as s(j) = Sum_{k=1..j} floor(Pi*prime(k)) for j >= 1; then a(n) is the n-th prime in the sequence s.

Edited by Jon E. Schoenfield, Sep 23 2018

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

A160359 Primes that are partial sums of squares of the odd primes.

Original entry on oeis.org

83, 373, 2393, 4723, 56383, 122701, 1789391, 1987849, 2706679, 7287991, 27833021, 41765519, 43570537, 55600477, 62283451, 67053359, 104720809, 122353547, 204330311, 347756707, 450504547, 540649093, 677774533, 715849843, 839886301

Offset: 1

M. F. Hasler, May 18 2009

nonn

There is necessarily an odd number of terms in the sum a(n) = 3^2+5^2+...+p(k)^2, i.e. the index k=A160358(n)+1 of the last prime in this sum is even.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A098562, A133547.

P:= select(isprime,[seq(i,i=3..10000,2)]):
S:= ListTools:-PartialSums(map(`^`,P,2)):
select(isprime,S); # Robert Israel, May 13 2024

Select[Accumulate[Prime[Range[2,400]]^2],PrimeQ] (* Harvey P. Dale, Jul 17 2021 *)

s=0; forprime( p=3,9999, isprime(s+=p^2) & print1(s","))

Equals A000040 intersect A133547.

A368850 Primes that are equal to the sum of the first k proper prime powers for some k.

Original entry on oeis.org

37, 89, 3391, 6547, 10271, 20233, 88397, 2256827, 6160597, 11073263, 14337313, 15797057, 18938809, 24514339, 28298057, 29442893, 33119963, 40078453, 118950121, 128935063, 135825923, 154641481, 209301217, 244837321, 342462997, 471596659, 498175681

Offset: 1

Ilya Gutkovskiy, Jan 07 2024

nonn

			37 is a term because 37 is a prime and 37 = 4 + 8 + 9 + 16 = 2^2 + 2^3 + 3^2 + 2^4.

Cf. A000040, A013918, A025475, A066525, A098562, A246547.

Select[Accumulate[Select[Range[5000000], PrimeOmega[#] > 1 && PrimePowerQ[#] &]], PrimeQ[#] &]

cp's OEIS Frontend

A140596 Squares nearest to and < terms in A098562.

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A140597 Squares nearest to and > terms in A098562.

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A024450 Sum of squares of the first n primes.

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A098561 Numbers n such that the sum of the squares of the first n primes is prime.

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A117503 Primes among partial sums of floor(Pi*prime(k)), k=1,2,3,....

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UBASIC

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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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