cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A202297 Product of the sum of the first n^2 primes by the sum of the first (n+1)^2 primes.

Original entry on oeis.org

0, 34, 1700, 38100, 403860, 2572620, 11863176, 43468984, 134426588, 364794428, 890218104, 1998186072, 4178379984, 8232956688, 15425693558, 27713583130, 47890427740, 80095432340, 130221623840, 206201325600, 318555575550, 481995772662, 715882366878, 1043383039482
Offset: 0

Views

Author

Stephen Balaban, Dec 15 2011

Keywords

Links

Crossrefs

Cf. A109724.

Programs

  • Clojure
    (defn prod-prime-matrix [n] (* (sum-matrix (first-n2-primes n)) (sum-matrix (first-n2-primes (inc n)))))
  • Magma
    A109724:=func; [0] cat [A109724(n)*A109724(n+1): n in [1..23]];  // Bruno Berselli, Dec 16 2011
  • Mathematica
    Table[(Plus@@Prime[Range[n^2]]) (Plus@@Prime[Range[(n + 1)^2]]), {n, 0, 19}] (* Alonso del Arte, Dec 16 2011 *)
  • PARI
    a(n) = vecsum(primes(n^2))* vecsum(primes((n+1)^2)); \\ Michel Marcus, Mar 20 2023

Formula

a(n) = A109724(n)*A109724(n+1).

Extensions

More terms from Bruno Berselli, Dec 16 2011

A202298 If A is the n X n matrix containing the first n^2 primes, a(n) is the sum of the elements of the square of A.

Original entry on oeis.org

4, 155, 3702, 39933, 244676, 1046455, 3635046, 10406049, 26595892, 60712839, 128248632, 253217949, 472633812, 837636667, 1431878468, 2356057659, 3756191658, 5844567389, 8865989698, 13147819241, 19100995732, 27324708263, 38402817766, 53116446341, 72537301810, 97894517685
Offset: 1

Views

Author

Stephen Balaban, Dec 15 2011

Keywords

Examples

			M2 = [2,3; 5,7], M2*M2 = [19,27; 45,64], so a(2) = 19 + 27 + 45 + 64 = 155.

Links

Crossrefs

Cf. A000040, A109724.

Programs

  • Maple
    A202298 := proc(n)
    local A,A2,r,c ;
    A := Matrix(n, n) ;
    for r from 0 to n-1 do
    for c from 0 to n-1 do
        A[r+1,c+1] := ithprime(1+r*n+c) ;
    end do:
    end do:
    A2 := A^2 ;
    add(add(A2[r,c],r=1..n),c=1..n) ;
end proc: # R. J. Mathar, Feb 09 2017
# alternative
N:= 50: # for a(1)..a(N)
P:= [seq(ithprime(i),i=1..N^2)]:
f:= proc(n) local M,e,u; M:= Matrix(n,n,P[1..n^2]);
   e:= Vector(n,1);
   e^%T . (M . (M . e));
end proc:
map(f, [$1..N]); # Robert Israel, Jan 11 2024
  • PARI
    a(n) = my(m = matrix(n,n, i, j, prime((i-1)*n+j))); my(mm = m^2); sum(k=1, n, vecsum(mm[k,])); \\ Michel Marcus, Jan 28 2017

Formula

a(n) = Sum_{j=1..n} (Sum_{i=1..n} prime(i+n*(j-1)) * Sum_{i=1..n} prime(j+n*(i-1))). - Robert Israel, Jan 11 2024

Extensions

More terms from Michel Marcus, Jan 28 2017

A343512 Numbers k such that Sum_{i=1..k} prime(i^3) is prime.

Original entry on oeis.org

1, 6, 28, 72, 90, 92, 96, 112, 118, 148, 160, 162, 184, 222, 282, 312, 314, 316, 330, 336, 390, 396, 418, 440, 444, 448, 472, 488, 524, 534, 552, 598, 604, 614, 638, 748, 758, 798, 824, 848, 906, 916, 970, 992, 1008, 1010, 1012, 1016, 1056, 1078, 1084, 1094, 1098
Offset: 1

Views

Author

Chai Wah Wu, Apr 17 2021

Keywords

Comments

Numbers n such that A109789(n) is prime. For n > 1, a(n) is even.

Examples

			72 is a term since Sum_{i=1..72} prime(i^3) = 94154923 is prime.

Links

Crossrefs

Cf. A055875, A011757, A109724, A109770, A109789.

A109801 Cumulative sum of squares of primes indexed by squares.

Original entry on oeis.org

4, 53, 582, 3391, 12800, 35601, 87130, 183851, 359412, 652093, 1089014, 1772943, 2791024, 4214273, 6250602, 8871763, 12402404, 16994853, 22933822, 30446903, 39951792, 51930313, 66393122, 84125643, 105627412, 131140013, 161599374
Offset: 1

Views

Author

Jonathan Vos Post, Aug 15 2005

Keywords

Comments

Related to Prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724).

Examples

			a(1) = 4 because (prime[1^2])^2 = (prime[1])^2 = 2^2.
a(2) = 53 because (prime[1^2])^2 + (prime[2^2])^2 = 2^2 + 7^2 = 4 + 49 = 53 (which is prime).
a(3) = 582 because (prime[1^2])^2 + (prime[2^2])^2 + (prime[3^2])^2 = 2^2 + 7^2 + 23^2 = 582.
a(4) = 582 because (prime[1^2])^2 + (prime[2^2])^2 + (prime[3^2])^2 + (prime[4^2])^2 = 2^2 + 7^2 + 23^2 + 53^2 = 3391 (which is prime).
a(32) = a(31) + (prime[32^2])^2 = 345995122 + 8161^2 = 412597043 (which is prime).
a(34) = a(33) + (prime[34^2])^2 = 488932212 + 9341^2 = 576186493 (which is prime).

Crossrefs

Cf. A000040, A000290, A000583, A011757, A109724, A109770.

Programs

  • Mathematica
    Accumulate[Prime[Range[30]^2]^2] (* Harvey P. Dale, Mar 28 2012 *)

Formula

(Prime[1^2])^2 + (prime[2^2])^2 + ... + (prime[n^2])^2. a(n+1) = a(n) + (A011757(n+1))^2.

A340558 a(n) is the smallest prime that can be the apex of a triangle with n rows, all entries being distinct primes, and all row sums equal.

Original entry on oeis.org

2, 19, 53, 131, 269, 503, 853, 1361, 1999, 2879, 3989
Offset: 1

Views

Author

Michel Marcus, Jan 11 2021

Keywords

Comments

A109724(n) is a lower bound for the sum of terms of triangle.

Examples

			            19
For n=2: 3  5  11 , so a(2) = 19.

Links

Crossrefs

Cf. A047837, A109724.
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