A123373 a(n) = Sum_{i=1..n} (Sum_{j=1..n} prime(i)^prime(j)).

4, 48, 3598, 924780, 287358579128, 339575512147572, 836406636653653232322, 2225332017808171682043720, 21158384827910606570843063431876, 2570789828135881020104992992114519012237948

Offset: 1

Author

Alexander Adamchuk, Oct 12 2006

Keywords

• nonn

Comments

p divides a(p-1) for prime p = {2, 3, 5, 23, 29, ...}.

All terms are even.

2^2 divides a(n) for n = {1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 21, 25, 26, 29, 30, 33, 37, 41, 45, 48, 49, 50, 52, ...}.

2^3 divides a(n) for n = {2, 5, 8, 12, 13, 16, 18, 21, 29, 37, 45, 48, 50, 52, 53, 58, 61, 64, 69, 76, 77, 85, 88, 93, 94, 98, ...}.

2^4 divides a(n) for n = {2, 12, 13, 18, 29, 45, 53, 58, 64, 76, 77, 93, 102, 109, 112, 125, 128, 133, 141, 149, 150, 157, 170, ...}.

2^5 divides a(n) for n = {12, 13, 18, 53, 58, 64, 77, 93, 102, 112, 141, 149, 150, 173, 178, 188, 190, 196, 205, 232, 234, 237, ...}.

2^6 divides a(n) for n = {13, 58, 64, 102, 150, 173, 178, 190, 205, 232, 234, 237, 245, 277, 285, 290, 325, 333, 382, 429, 434, ...}.

2^7 divides a(n) for n = {13, 58, 64, 150, 173, 178, 205, 234, 325, 382, 472, 573, 592, 596, 621, 628, 653, 757, 796, 893, 950, ...}.

2^8 divides a(n) for n = {13, 58, 64, 178, 205, 325, 382, 573, 653, 757, ...}.

2^9 divides a(n) for n = {13, 58, 64, 178, 325, 382, 653, 757, ...}.

2^10 divides a(n) for n = {64, 178, ...}.

2^11 divides a(n) for n = {64, 178, ...}.

2^12 divides a(n) for n = {178, ...}.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..76

Crossrefs

Cf. A086787 (Sum_{i=1..n} (Sum_{j=1..n} i^j)), A122004.

Programs

Mathematica

Table[Sum[Sum[Prime[i]^Prime[j],{i,1,n}],{j,1,n}],{n,1,13}]

PARI

for(n=1,10, print1(sum(j=1,n, sum(k=1,n, (prime(k))^(prime(j)))), ", ")) \\ G. C. Greubel, Oct 25 2017

Python

from sympy import prime
def A123373(n): return sum(prime(i)**prime(j) for i in range(1,n+1) for j in range(1,n+1)) # Chai Wah Wu, Jan 08 2022