A357251 -id:A357251 - cp's OEIS frontend

A357252 Primes in A357251.

19, 14479, 43609, 406171, 711959, 1330177, 2698231, 3918157, 18987169, 26135339, 194727347, 269998639, 975929347, 5005853669, 8430389621, 24830247671, 36372313009, 69703708967, 93194681917, 126628534313, 139478926201, 304123612349, 359101509211, 384305009171, 387550106843, 400722388999

Offset: 1

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

nonn

Primes that are, for some N, the sum of p*q for all pairs of primes (p,q) with p <= q <= N.

			a(2) = 14479 is in the sequence because 14479 = A357251(11) is the sum of p*q for primes p <= q <= 31 and is prime.

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

Cf. A357251.

p:= 2: a:= 4: s:= 2: R:= NULL: count:= 0:
for i from 1 while count < 100 do
  p:= nextprime(p);
  s:= s + p;
  a:= a + p*s;
  if isprime(a) then count:= count+1; R:= R,a; fi
od:
R;

A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 5, 4, 0, 1, 10, 19, 8, 0, 1, 17, 69, 65, 16, 0, 1, 28, 188, 410, 211, 32, 0, 1, 41, 496, 1726, 2261, 665, 64, 0, 1, 58, 1029, 7182, 14343, 11970, 2059, 128, 0, 1, 77, 2015, 20559, 93345, 112371, 61909, 6305, 256, 0, 1, 100, 3478, 54814, 360612, 1139166, 848506, 315850, 19171, 512, 0

Offset: 0

Alois P. Heinz, Apr 27 2021

			A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0].
A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0].
Square array A(n,k) begins:
  1,  1,    1,     1,      1,        1,        1, ...
  0,  2,    5,    10,     17,       28,       41, ...
  0,  4,   19,    69,    188,      496,     1029, ...
  0,  8,   65,   410,   1726,     7182,    20559, ...
  0, 16,  211,  2261,  14343,    93345,   360612, ...
  0, 32,  665, 11970, 112371,  1139166,  5827122, ...
  0, 64, 2059, 61909, 848506, 13379332, 89131918, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-4 give: A000007, A000079, A001047(n+1), A016273, A025931.
Rows n=0-2 give: A000012, A007504, A357251.
Main diagonal gives A332967.
Cf. A000040, A074140, A124960, A324939, A325054.

A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1,
     If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)

A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).

A(n,k) = A124960(n+k,k) for k >= 1.

cp's OEIS Frontend

A357252 Primes in A357251.

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