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A123855 a(n) = Sum_{j=1..n} Sum_{i=1..n} prime(i)^j. 7
2, 18, 208, 3730, 201092, 7335762, 526460272, 26465563878, 2363769149128, 487833920370774, 40049421223880084, 7972075784185713954, 1235006486302921316794, 124887894202756460238954 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p that divide a(p-1) are listed in A123856.

Nonprime numbers n that divide a(n-1) are listed in A123857.

It appears that 2^k divides a(2^k-1) for all k > 0 (confirmed for 0 < k < 10).

The summation over j can be carried out first and expressed analytically, leading to the given formula and Maple program. - M. F. Hasler, Nov 09 2006

LINKS

M. F. Hasler, Nov 09 2006, Table of n, a(n) for n = 1..25

FORMULA

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

a(p) = Sum_{i=1..p} (prime(i)^p - 1)/(prime(i) - 1)*prime(i). - M. F. Hasler, Nov 09 2006

EXAMPLE

a(1) = prime(1)^1 = 2.

a(2) = prime(1)^1 + prime(1)^2 + prime(2)^1 + prime(2)^2 = 2^1 + 2^2 + 3^1 + 3^2 = 18.

MAPLE

A123855 := p-> sum((ithprime(i)^p-1)/(ithprime(i)-1)*ithprime(i), i = 1 .. p); map(%, [$1..20]); # M. F. Hasler, Nov 09 2006

MATHEMATICA

Table[Sum[Sum[Prime[i]^j, {i, 1, n}], {j, 1, n}], {n, 1, 20}]

PROG

(PARI) vector(20, n, sum(i=1, n, sum(j=1, n, prime(i)^j )) ) \\ G. C. Greubel, Aug 08 2019

(MAGMA) [(&+[ (&+[ NthPrime(i)^j: j in [1..n]]): i in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 08 2019

(Sage) [sum(sum( nth_prime(i)^j for j in (1..n)) for i in (1..n)) for n in (1..20)] # G. C. Greubel, Aug 08 2019

CROSSREFS

Cf. A123856, A123857.

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

Sequence in context: A303381 A224881 A092882 * A121407 A153647 A052726

Adjacent sequences:  A123852 A123853 A123854 * A123856 A123857 A123858

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Oct 13 2006

STATUS

approved

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Last modified March 31 03:48 EDT 2020. Contains 333136 sequences. (Running on oeis4.)