A024450 -id:A024450 - cp's OEIS frontend

A157492 Apply partial sum operator twice to sequence of squares of the first n primes.

4, 17, 55, 142, 350, 727, 1393, 2420, 3976, 6373, 9731, 14458, 20866, 29123, 39589, 52864, 69620, 90097, 115063, 145070, 180406, 221983, 270449, 326836, 392632, 468629, 555235, 653290, 763226, 885931, 1024765, 1180760, 1355524, 1549609

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

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

Cf. A001223, A014148, A014150, A061789, A069482, A129701.

Partial sums of A024450.

ListTools:-PartialSums(ListTools:-PartialSums([seq(ithprime(i)^2,i=1..100)])); # Robert Israel, May 14 2019

s0=s1=0;lst={};Do[p=Prime[n];s0+=p^2;s1+=s0;AppendTo[lst,s1],{n,5!}];lst
Nest[Accumulate,Prime[Range[40]]^2,2] (* Harvey P. Dale, Jan 01 2020 *)

A223937 a(n) is the sum of the cubes of the first A122140(n) primes.

Original entry on oeis.org

8, 4696450, 7024453131396, 17761740387522, 155912686127038650, 87598780898450312031408, 2147216863131055036604400, 2908950240914054780101441371333254159676520, 384422969812280951687876430655304031054262132, 6187047308209705064673104196645071104957480508

Offset: 1

Robert Price, Mar 29 2013

nonn

Paul W. Dyson, Table of n, a(n) for n = 1..18 (terms 1..10 from Robert Price, 11 from Paul W. Dyson, 12..15 from Bruce Garner, 16 from Paul W. Dyson, 17 from Bruce Garner)
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n), A122140.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Title corrected by Hugo Pfoertner, Feb 09 2021

A240860 a(n) = Sum_{i=1..n} (-1)^{i+1} prime(i)^2, where prime(k) denotes the k-th prime: alternating sum of the squares of the first n primes.

Original entry on oeis.org

4, -5, 20, -29, 92, -77, 212, -149, 380, -461, 500, -869, 812, -1037, 1172, -1637, 1844, -1877, 2612, -2429, 2900, -3341, 3548, -4373, 5036, -5165, 5444, -6005, 5876, -6893, 9236, -7925, 10844, -8477, 13724, -9077, 15572, -10997, 16892, -13037, 19004, -13757

Offset: 1

Timothy Varghese, May 06 2014

sign

For n even this is the negative of the sum of (3^2 - 2^2) + (7^2 - 5^2) + ... + (prime(n)^2 - prime(n-1)^2). But this is half of the terms in the sum of (3^2 - 2^2) + (5^2 - 3^2) + (7^2 - 5^2) + ... + (prime(n)^2 - prime(n-1)^2) which has a sum that telescopes to prime(n)^2 - 4. Thus a good estimate of a(n) (half the terms) is prime(n)^2/2 (half the square of the n-th prime) which works well up to n = 10000. For odd n, add prime(n)^2 to the estimate for even n.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Timothy Varghese, Alternating sums, MathOverflow, May 2014.

Cf. A000040, A008347, A024450.

a(n) = sum(i=1, n, (-1)^(i+1)*prime(i)^2); \\ Michel Marcus, May 09 2014

A357251 a(n) = Sum_{1<=i<=j<=n} prime(i)*prime(j).

Original entry on oeis.org

4, 19, 69, 188, 496, 1029, 2015, 3478, 5778, 9519, 14479, 21768, 31526, 43609, 59025, 79218, 105178, 135739, 173795, 219164, 271140, 333629, 406171, 491878, 594698, 711959, 842151, 988848, 1150168, 1330177, 1548617, 1791098, 2063454, 2359107, 2698231, 3064708, 3470396, 3918157, 4404795, 4938846

Offset: 1

J. M. Bergot and Robert Israel, Sep 20 2022

nonn

a(n) is the sum of products of unordered pairs of (not necessarily distinct) elements from the first n primes.

It appears that 4 is the only square in the sequence.

			a(3) = 2*2 + 2*3 + 2*5 + 3*3 + 3*5 + 5*5 = 69.

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

Cf. A007504, A024447, A024450, A065762, A357252.
Partial sums of A143215.
Row n=2 of A343751.

P:= [seq(ithprime(i),i=1..100)]:
S:= ListTools:-PartialSums(P):
ListTools:-PartialSums(zip(`*`,P,S));

Accumulate[(p = Prime[Range[40]]) * Accumulate[p]] (* Amiram Eldar, Sep 20 2022 *)

from itertools import accumulate
from sympy import prime, primerange
def aupton(nn):
    p = list(primerange(2, prime(nn)+1))
    return list(accumulate(c*d for c, d in zip(p, accumulate(p))))
print(aupton(40)) # Michael S. Branicky, Sep 24 2022 after Amiram Eldar

a(n) = (A007504(n)^2 + A024450(n))/2.
a(n) = A024447(n) + A024450(n).
a(n) = A065762(n)/2. - Hugo Pfoertner, Sep 24 2022

A065762 a(n) = (sum of first n primes)^2 + sum of (squares of first n primes).

Original entry on oeis.org

8, 38, 138, 376, 992, 2058, 4030, 6956, 11556, 19038, 28958, 43536, 63052, 87218, 118050, 158436, 210356, 271478, 347590, 438328, 542280, 667258, 812342, 983756, 1189396, 1423918, 1684302, 1977696, 2300336, 2660354, 3097234, 3582196, 4126908, 4718214

Offset: 1

Terrel Trotter, Jr., Dec 04 2001

			a(4) = 376 because (2 + 3 + 5 + 7)^2 + (2^2 + 3^2 + 5^2 + 7^2) = 17^2 + (4 + 9 + 25 + 49) = 289 + 87 = 376.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A007504, A024450, A065595.

nn=50;With[{prs=Prime[Range[nn]]},Table[Total[Take[prs,n]]^2+ Total[Take[prs,n]^2],{n,nn}]] (* Harvey P. Dale, Aug 20 2011 *)

{ s=ss=0; for (n=1, 500, p=prime(n); s+=p; ss+=p^2; write("b065762.txt", n, " ", s^2 + ss) ) } \\ Harry J. Smith, Oct 30 2009

a(n) = A007504(n)^2 + A024450(n). - Michel Marcus, Oct 12 2015

More terms from Harvey P. Dale, Aug 20 2011

A072004 Remainder when sum of squares of first n primes is divided by n-th prime.

Original entry on oeis.org

0, 1, 3, 3, 10, 0, 3, 1, 15, 19, 10, 28, 12, 1, 32, 25, 0, 42, 42, 45, 4, 23, 77, 50, 30, 45, 86, 43, 64, 100, 23, 105, 89, 41, 87, 54, 133, 2, 59, 47, 147, 64, 174, 102, 65, 104, 7, 127, 107, 28, 210, 194, 106, 60, 159, 95, 119, 116, 104, 230, 224, 110, 183, 212, 287

Offset: 1

Randy L. Ekl, Jun 18 2002

			a[3] = 3 because s[3] = 2*2 + 3*3 + 5*5 = 38, p[3]=5 and q[3]= floor(38/5)=7, so a[3] = 38-5*7 = 3.

Cf. A024450 (s(n)).

Mod[#[[1]],#[[2]]]&/@With[{nn=70},Thread[{Accumulate[Prime[ Range[ nn]]^2], Prime[Range[nn]]}]] (* Harvey P. Dale, Aug 09 2015 *)

a(n) = sum(k=1, n, prime(k)^2) % prime(n); \\ Michel Marcus, Jan 14 2023

a(n) = s(n) - prime(n)*q(n), where s(n) = sum of squares of first n primes, prime(n) is n-th prime and q(n) is floor(s(n)/prime(n)).

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

A125826 Numbers m that divide 2^7 + 3^7 + 5^7 + ... + prime(m)^7.

Original entry on oeis.org

1, 25, 1677, 21875, 538513, 1015989, 18522325, 1130976595, 1721158369, 561122374231, 1763726985077, 2735295422833, 7631117283951, 22809199833151, 46929434362563, 49217568518075, 151990420653423, 174172511353413, 1258223430425543

Offset: 1

Alexander Adamchuk, Feb 03 2007

See A232865 for prime(a(n)). - M. F. Hasler, Dec 01 2013
a(17) > 5.5*10^13. - Bruce Garner, Aug 30 2021
a(18) > 1.56*10^14. - Paul W. Dyson, Mar 02 2022
a(19) > 1.9*10^14. - Bruce Garner, Sep 18 2022

OEIS Wiki, Sums of powers of primes divisibility sequences.

Cf. A232865.
Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

s = 0; Do[s = s + Prime[n]^7; If[ Mod[s, n] == 0, Print[n]], {n, 25000}]

s=0; n=0; forprime(p=2, 4e9, s+=p^7; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 16 2011

More terms from Ryan Propper, Mar 26 2007
a(8)-a(9) from Charles R Greathouse IV, Mar 16 2011
a(10) from Paul W. Dyson, Jan 05 2021
a(11)-a(12) from Bruce Garner, Feb 26 2021
a(13) from Bruce Garner, Mar 23 2021
a(14) from Bruce Garner, May 19 2021
a(15)-a(16) from Bruce Garner, Aug 30 2021
a(17) from Paul W. Dyson, Mar 02 2022
a(18) from Bruce Garner, Sep 18 2022
a(19) from Paul W. Dyson, Jan 17 2024

A125827 Numbers m that divide 2^11 + 3^11 + 5^11 + ... + prime(m)^11.

Original entry on oeis.org

1, 25, 59, 2599, 6195, 421407, 11651191, 19293221, 255136097, 1820015683, 2183556659, 7993872143, 9850779563, 2006892138335, 2649677145789, 6645858099781, 318039538085101, 414996765110825

Offset: 1

Alexander Adamchuk, Feb 03 2007

a(17) > 8*10^12. - Bruce Garner, Mar 29 2021

a(19) > 5*10^14. - Paul W. Dyson, Dec 31 2024

OEIS Wiki, Sums of powers of primes divisibility sequences.

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

s = 0; Do[s = s + Prime[n]^11; If[ Mod[s, n] == 0, Print[n]], {n, 7000}]

s=0; n=0; forprime(p=2, 4e9, s+=p^11; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 20 2011

3 more terms from Stefan Steinerberger, Jun 06 2007
1 more term from Sean A. Irvine, Jan 26 2011
a(10)-a(13) from Charles R Greathouse IV, Mar 20 2011
a(14) from Paul W. Dyson, Jan 08 2021
a(15) from Bruce Garner, Mar 08 2021
a(16) from Bruce Garner, Mar 29 2021
a(17) from Paul W. Dyson, Jan 03 2023
a(18) from Paul W. Dyson, Dec 20 2024

cp's OEIS Frontend

A157492 Apply partial sum operator twice to sequence of squares of the first n primes.

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A223937 a(n) is the sum of the cubes of the first A122140(n) primes.

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A240860 a(n) = Sum_{i=1..n} (-1)^{i+1} prime(i)^2, where prime(k) denotes the k-th prime: alternating sum of the squares of the first n primes.

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A357251 a(n) = Sum_{1<=i<=j<=n} prime(i)*prime(j).

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A065762 a(n) = (sum of first n primes)^2 + sum of (squares of first n primes).

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A072004 Remainder when sum of squares of first n primes is divided by n-th prime.

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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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