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.

Showing 1-3 of 3 results.

A124238 Terms of A123857 that are not powers of 2.

Original entry on oeis.org

38, 205, 316, 736, 3776, 4916, 5888, 7736, 11138, 22287, 23308, 23924, 39538, 62336, 71936
Offset: 1

Views

Author

Alexander Adamchuk, Oct 22 2006

Keywords

Comments

A123857 list composite indices n that divide A123855(n-1) = Sum[ Sum[ Prime[i]^j, {i,1,n-1}], {j,1,n-1}]. Corresponding indices n such that A123857(n) belongs to this sequence are 5,8,10,12,...

Crossrefs

Extensions

More terms from Max Alekseyev, Sep 13 2009

A123855 a(n) = Sum_{j=1..n} Sum_{i=1..n} prime(i)^j.

Original entry on oeis.org

2, 18, 208, 3730, 201092, 7335762, 526460272, 26465563878, 2363769149128, 487833920370774, 40049421223880084, 7972075784185713954, 1235006486302921316794, 124887894202756460238954
Offset: 1

Views

Author

Alexander Adamchuk, Oct 13 2006

Keywords

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

Examples

			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.
		

Crossrefs

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

Programs

  • Magma
    [(&+[ (&+[ NthPrime(i)^j: j in [1..n]]): i in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 08 2019
    
  • 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}]
  • PARI
    vector(20, n, sum(i=1,n, sum(j=1,n, prime(i)^j )) ) \\ 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

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

A123856 Primes p that divide A123855(p-1).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, 211, 223, 227, 229, 269, 317, 337, 349, 353, 379, 383, 389, 401, 421, 439, 449, 457, 463, 479, 521, 523, 541, 547, 563, 569, 571, 587, 599, 613, 617, 631, 643
Offset: 1

Views

Author

Alexander Adamchuk, Oct 13 2006

Keywords

Comments

A123855(n) = Sum_{j=1..n} Sum_{i=1..n} prime(i)^j.
Prime p = a(n) divides A123855(p-1).
Nonprime numbers n that divide A123855(n-1) are listed in A123857.
It appears that 2^k divides A123855(2^k-1) for all k>0 (confirmed for 0

Crossrefs

Programs

  • Maple
    A123855_mod := proc(n,p) option remember; local s,i,pi; s:=0: for i to n do pi:= ithprime(i) mod p: if pi=1 then s:=s+n mod p: else s := s+pi*(pi &^ n - 1)/(pi-1) mod p fi od end; A123856 := proc(n::posint) option remember; local p; if n>1 then p:=nextprime( procname(n-1)) else p:=2 fi: while A123855_mod(p-1,p)<>0 do p:=nextprime( p ) od: p end; # M. F. Hasler, Nov 10 2006
  • Mathematica
    fQ[p_] := Mod[ Sum[ PowerMod[ Prime@ i, j, p], {j, p - 1}, {i, p - 1}], p] == 0; Select[ Prime@ Range@ 117, fQ] (* Robert G. Wilson v, Jun 10 2011 *)
Showing 1-3 of 3 results.