A024450 -id:A024450 - cp's OEIS frontend

A024447 Sum of the products of the primes taken 2 at a time from the first n primes.

0, 6, 31, 101, 288, 652, 1349, 2451, 4222, 7122, 11121, 17041, 25118, 35352, 48559, 65943, 88422, 115262, 148829, 189157, 235804, 292052, 357705, 435491, 528902, 635962, 755545, 890793, 1040232, 1207472, 1409783, 1635103, 1888690, 2165022, 2481945

Offset: 1

Clark Kimberling

nonn

a(n) is the 2nd elementary symmetric function of the first n+1 primes.

Using the identity that (x_1 + x_2 + ... + x_n)^2 - (x_1^2 + x_2^2 + ... + x_n^2) is the sum of the products taken two at a time, a(n) can be expressed with the sum of the primes and the sum of the prime squared. Since they both have asymptotic formulas, this yields an asymptotic formula for this sequence. - Timothy Varghese, May 06 2014

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

Cf. A007504, A024448, A024449, A024450.

Primes:= [seq](ithprime(i),i=1..100):
(map(`^`,ListTools:-PartialSums(Primes),2) - ListTools:-PartialSums(map(`^`,Primes,2)))/2; # Robert Israel, Sep 24 2015

a[1] = 0; a[n_] := a[n] = a[n-1] + Prime[n]*Total[Prime[Range[n-1]]];
Array[a, 35] (* Jean-François Alcover, Feb 28 2019 *)

/* Extra memory allocation could be required. */
Primes=List();
forprime(x=2,prime(500000),listput(Primes,x));
/* Keep previous lines global, before a(n) */
a(n)={my(p=vector(n,j,Primes[j]),s=0);forvec(y=vector(2,i,[1,#p]),s+=(p[y[1]]*p[y[2]]),2);s} \\ R. J. Cano, Oct 11 2015

a(1) = 0, a(n+1) = prime(n+1)*(sum of first n primes) + a(n), for n > 1.
a(n) = ((A007504(n))^2 - A024450(n))/2. - Timothy Varghese, May 06 2014
a(n) ~ (3*n^4*log^2(n) - 4*n^3*log^2(n))/24. - Timothy Varghese, May 06 2014

A048261 Numbers that are the sum of the squares of distinct primes.

Original entry on oeis.org

4, 9, 13, 25, 29, 34, 38, 49, 53, 58, 62, 74, 78, 83, 87, 121, 125, 130, 134, 146, 150, 155, 159, 169, 170, 173, 174, 178, 179, 182, 183, 194, 195, 198, 199, 203, 204, 207, 208, 218, 222, 227, 231, 243, 247, 252, 256, 289, 290, 293, 294, 298, 299, 302, 303

Offset: 1

Jud McCranie

nonn

17163 is the largest of 2438 positive integers that can't be expressed as the sum of squares of distinct primes. See A121518. - T. D. Noe, Aug 04 2006

			13 = 2^2 + 3^2.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry 17163.

T. D. Noe, Table of n, a(n) for n = 1..2000
Robert E. Dressler, Louis Pigno, and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40. MR 54 #7373.
Index entries for sequences related to sums of squares

Cf. A024450 (sum of squares of the first n primes).

nn=10; s={0}; Do[p=Prime[n]; s=Union[s,s+p^2], {n,nn}]; s=Select[s,0<#<=Prime[nn]^2&] (* T. D. Noe, Aug 04 2006 *)

It is easy to check that these 2438 numbers that are not the sum of distinct primes squared are all of the form sum_i e_i*q_i where e_i is 1 or -1 and the q_i's are distinct primes. - W. Edwin Clark, Oct 19 2003

A122103 Sum of the fifth powers of the first n primes.

Original entry on oeis.org

32, 275, 3400, 20207, 181258, 552551, 1972408, 4448507, 10884850, 31395999, 60025150, 129369107, 245225308, 392233751, 621578758, 1039774251, 1754698550, 2599294851, 3949419958, 5753649309, 7826720902, 10903777301, 14842817944

Offset: 1

Alexander Adamchuk, Aug 20 2006

a(n) is prime for n = {66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, ...} = A122125. Primes of this form are listed in A122126 = {32353461605953, 9874820441996857, 10821208357045699, ...}.

			a(2) = 275 because the first two primes are 2 and 3, the fifth powers of which are 32 and 243, and 32 + 243 = 275.
a(3) = 3400, because the third prime is 5, its fifth power if 3125 and 275 + 3125 = 3400.

OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A050997, A007504, A024450, A098999, A122102, A122125, A122126.

Table[Sum[Prime[k]^5, {k, n}], {n, 100}]

a(n)=sum(i=1,n,prime(i)) \\ Charles R Greathouse IV, Nov 30 2013

a(n) = sum(k = 1 .. n, prime(k)^5).

a(n) = 1/6*n^6*log(n)^5 + O(n^6*log(n)^4*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). For a generalization, see comment in A122102. - Vladimir Shevelev, Aug 14 2013

A133547 a(n) = sum of squares of first n odd primes.

Original entry on oeis.org

9, 34, 83, 204, 373, 662, 1023, 1552, 2393, 3354, 4723, 6404, 8253, 10462, 13271, 16752, 20473, 24962, 30003, 35332, 41573, 48462, 56383, 65792, 75993, 86602, 98051, 109932, 122701, 138830, 155991, 174760, 194081, 216282, 239083, 263732

Offset: 1

Artur Jasinski, Sep 16 2007

			a(3)=83 because 3^2+5^2+7^2=83.

Cf. A007148, A007504, A024450.

c = 2; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a

a(n) = A024450(n+1)-4. - Jason Yuen, Sep 23 2024

A024525 a(n) = 1^2 + prime(1)^2 + prime(2)^2 + ... + prime(n)^2.

Original entry on oeis.org

1, 5, 14, 39, 88, 209, 378, 667, 1028, 1557, 2398, 3359, 4728, 6409, 8258, 10467, 13276, 16757, 20478, 24967, 30008, 35337, 41578, 48467, 56388, 65797, 75998, 86607, 98056, 109937, 122706, 138835, 155996, 174765, 194086, 216287, 239088, 263737, 290306, 318195, 348124

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A024450.

Partial sums of A280076.

[1] cat [n le 1 select 5 else Self(n-1) + NthPrime(n)^2: n in [1..80]]; // G. C. Greubel, Jan 30 2025

Join[{1},Accumulate[Prime[Range[40]]^2]+1] (* Harvey P. Dale, Oct 24 2015 *)

def A024525(n):
    if n<2: return (1,5)[n]
    else: return A024525(n-1) + nth_prime(n)**2
print([A024525(n) for n in range(81)]) # G. C. Greubel, Jan 30 2025

a(n) = 1 + A024450(n), for n >= 1.

a(n) = a(n-1) + prime(n)^2, with a(0) = 1, a(1) = 5. - G. C. Greubel, Jan 30 2025

Corrected by Harvey P. Dale, Oct 24 2015

A127719 Floor of square root of sum of squares of n consecutive primes.

Original entry on oeis.org

2, 3, 6, 9, 14, 19, 25, 32, 39, 48, 57, 68, 80, 90, 102, 115, 129, 143, 158, 173, 187, 203, 220, 237, 256, 275, 294, 313, 331, 350, 372, 394, 418, 440, 465, 488, 513, 538, 564, 590, 616, 642, 670, 697, 724, 751, 780, 811, 843, 873

Offset: 1

Artur Jasinski, Jan 25 2007

nonn

Cf. A127720, A127721.

A024450 := proc(n) local i; add((ithprime(i))^2,i=1..n); end: A000196 := proc(n) floor(sqrt(n)); end: A127719 := proc(n) A000196(A024450(n)); end: for n from 1 to 30 do printf("%d, ",A127719(n)); od; # R. J. Mathar, Jan 28 2007

a = {}; k = 0; Do[k = k + (Prime[x])^2; AppendTo[a, Floor[Sqrt[k]]], {x, 1, 50}]; a

a(n) = A000196(A024450(n)). - R. J. Mathar, Jan 28 2007

A127720 Floor of square root of sum of squares of n odd consecutive primes.

Original entry on oeis.org

3, 5, 9, 14, 19, 25, 31, 39, 48, 57, 68, 80, 90, 102, 115, 129, 143, 157, 173, 187, 203, 220, 237, 256, 275, 294, 313, 331, 350, 372, 394, 418, 440, 465, 488, 513, 538, 564, 590, 616, 642, 670, 697, 724, 751, 780, 811, 843, 873

Offset: 1

Artur Jasinski, Jan 25 2007

nonn

Cf. A127719, A127721.

A024450 := proc(n) local i ; add((ithprime(i))^2,i=1..n) ; end: Ax := proc(n) A024450(n+1)-4 ; end: A000196 := proc(n) floor(sqrt(n)) ; end: A127720 := proc(n) A000196(Ax(n)) ; end: for n from 1 to 30 do printf("%d, ",A127720(n)) ; od ; # R. J. Mathar, Jan 28 2007

a = {}; k = 0; Do[k = k + (Prime[x])^2; AppendTo[a, Floor[Sqrt[k]]], {x, 2, 50}]; a
Module[{nn=50},Floor[Sqrt[#]]&/@Accumulate[Prime[Range[2,nn+1]]^2]] (* Harvey P. Dale, Jul 27 2017 *)

a(n) = A000196(A024450(n+1) - 4). - R. J. Mathar, Jan 28 2007

A059804 Consider the line segment in R^n from the origin to the point v=(2,3,5,7,11,...) with prime coordinates; let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times v.v.

Original entry on oeis.org

1, 3, 9, 39, 87, 215, 391, 711, 1326, 1975, 2925, 4256, 5696, 7537, 9774, 12488, 16322, 20477, 24966, 30007, 35336, 41577, 48466, 56387, 65796, 75997, 86606, 98055, 109936, 122705, 138834, 155995, 174764, 194085, 216286, 239087, 263736, 290305

Offset: 2

N. J. A. Sloane and Vinay Vaishampayan, Feb 21 2001

v.v is given by A024450(n). For n >= 19, a(n) = A024450(n-1).

Officially these are just conjectures so far.

N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, Fat Struts: Constructions and a Bound, Proceedings Information Theory Workshop, Taormino, Italy, 2009. [Cached copy]
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, A Note on Projecting the Cubic Lattice, Discrete and Computational Geometry, Vol. 46 (No. 3, 2011), 472-478.
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, The Lifting Construction: A General Solution to the Fat Strut Problem, Proceedings International Symposium on Information Theory (ISIT), 2010, IEEE Press. [Cached copy]

Cf. A059774, A024450, A047896, A060453.

Cf. A137609 (where the minimum distance occurs along the line segment).

A077023 Integer values of sum of first k primes squared divided by k-th prime, for some k (A077022).

Original entry on oeis.org

2, 29, 284, 410066261, 941945317

Offset: 1

Randy L. Ekl, Zak Seidov, Oct 17 2002

Remainder, a(n), when sum of first n primes squared is divided by n-th prime in A072004. In A077023, 5 values of n such that a(n)=0 are given. In the sequence, the corresponding integer values of sum of first n primes squared divided by n-th prime are given. For primes <100,000,000, there are no other cases of a(n)=0.

			29 is a term because A024450(6)/prime(6) = 29;
284 is a term because A024450(17)/prime(17) = 284;
410066261 is a term because A024450(11156)/prime(11156) = 410066261;
941945317 is a term because A024450(16548)/prime(16548) = 941945317.

Cf. A024450, A072004, A077022.

a(n) = A024450(A077022(n))/prime(A077022(n)). - Michel Marcus, Jan 14 2023

A125907 Numbers k such that k divides 2^4 + 3^4 + 5^4 + ... + prime(k)^4.

Original entry on oeis.org

1, 2951, 38266951, 3053263643573, 3798632877308897

Offset: 1

Alexander Adamchuk, Feb 04 2007

No more terms to 10^13. - Charles R Greathouse IV, Mar 21 2011
a(4) is less than 10^13 contradicting the previous comment. It was found using the primesieve library by Kim Walisch and gmplib. - Bruce Garner, Feb 26 2021
a(6) > 4*10^15. - Paul W. Dyson, Nov 19 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.
Cf. A122102, A232869, A128168, A233893.

a(1) = 1; s = 2^4; Do[s = s + Prime[2n]^4+Prime[2n+1]^4; If[ Mod[s, 2n+1] == 0, Print[2n+1]], {n,1, 20000000}]

s=0; n=0; forprime(p=2, 4e9, s+=p^4; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 21 2011

a(4) from Bruce Garner, Feb 26 2021

a(5) from Paul W. Dyson, May 09 2024

cp's OEIS Frontend

A024447 Sum of the products of the primes taken 2 at a time from the first n primes.

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A048261 Numbers that are the sum of the squares of distinct primes.

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A122103 Sum of the fifth powers of the first n primes.

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A133547 a(n) = sum of squares of first n odd primes.

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A024525 a(n) = 1^2 + prime(1)^2 + prime(2)^2 + ... + prime(n)^2.

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A127719 Floor of square root of sum of squares of n consecutive primes.

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A127720 Floor of square root of sum of squares of n odd consecutive primes.

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A059804 Consider the line segment in R^n from the origin to the point v=(2,3,5,7,11,...) with prime coordinates; let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times v.v.

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