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 1-6 of 6 results.

A124275 Terms of A123856 that are not terms of A124273.

Original entry on oeis.org

2, 5, 181
Offset: 1

Views

Author

Alexander Adamchuk, Oct 23 2006

Keywords

Comments

Primes that divide A123855(p-1) = Sum_{j=1..p-1} Sum_{i=1..p-1} prime(i)^j but not A124271(p) = Sum_{i=1..p} (prime(i)^p - 1)/(prime(i) - 1).
The next term if it exists is greater than 1000.
The next term A124275(4), if it exists, is larger than 25000. (Checked by calculating sequences A123856 and A124273 to 1200 terms.) - M. F. Hasler, Nov 10 2006

Examples

			A123856(n) begins {2, 3, 5, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, ...}.
A124273(n) begins {3, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 197, ...}.
Thus a(1) = 2, a(2) = 5, a(3) = 181.

Crossrefs

Cf. A123855, A123856, A124271, A124273.

Programs

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.

Links

Crossrefs

Cf. A123856, A123857.
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

A124271 a(n) = Sum_{i=1..n} (prime(i)^n - 1)/(prime(i) - 1).

Original entry on oeis.org

1, 7, 51, 611, 19839, 603331, 32981935, 1469991559, 108336139407, 17389027481287, 1334783150250945, 222909199163881075, 31099653342061054699, 2994181661163361882651, 387134597481460117602345, 92092112661138292186297999, 26679920606217066273305101055
Offset: 1

Views

Author

Alexander Adamchuk, Oct 23 2006

Keywords

Links

Crossrefs

Cf. A124272 (prime terms of this sequence), A124273 (primes p that divide a(p)), A124274 (nonprimes n that divide a(n)).
Similar sequence: A123855. See also A123856.

Programs

  • Magma
    [&+[(NthPrime(k)^n - 1) div (NthPrime(k) - 1): k in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Oct 21 2018
  • Mathematica
    Table[Sum[(Prime[i]^n-1)/(Prime[i]-1),{i,1,n}],{n,1,20}]
  • PARI
    a(n) = sum(i=1, n, (prime(i)^n - 1)/(prime(i) - 1)) \\ Jianing Song, Oct 20 2018

A124273 Primes p that divide A124271(p) = Sum_{i=1..p} (prime(i)^p - 1) / (prime(i) - 1).

Original entry on oeis.org

3, 7, 13, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 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, 647, 677
Offset: 1

Views

Author

Alexander Adamchuk, Oct 23 2006

Keywords

Comments

a(n) almost coincides with A123856(n). Up to 1000 there are only 3 terms of A123856(n) that are different from the terms of a(n), see A124275.

Links

Crossrefs

Cf. A123856, A124271, A124274 (nonprimes n that divide A124271(n)), A124275 (terms of A123856 that are not in this sequence).

Programs

  • Maple
    A124271_mod:=proc(n) option remember; local s,i,p; s:=0: for i to n do p:=ithprime(i) mod n: if p<>1 then s:=s+(p&^n - 1)/(p - 1) mod p fi od:s end; A124273 := proc(n::posint) option remember; local p; if n>1 then p:=nextprime( procname(n-1)) else p:=2 fi: while A124271_mod(p)<>0 do p:=nextprime( p ) od: p end # M. F. Hasler, Nov 10 2006
  • Mathematica
    Select[Prime@ Range@ 125, Divisible[Sum[(Prime[i]^# - 1)/(Prime[i] - 1), {i, 1, #}], #] &] (* Michael De Vlieger, Jul 17 2016 *)
  • PARI
    isA124273(p) = isprime(p)&&!sum(i=1, p, sum(j=0, p-1, Mod(prime(i), p)^j)) \\ Jianing Song, Oct 20 2018

A123857 Composite numbers m that divide A123855(m-1) = Sum_{i=1..m-1} Sum_{j=1..m-1} prime(i)^j.

Original entry on oeis.org

4, 8, 16, 32, 38, 64, 128, 205, 256, 316, 512, 736, 1024, 2048, 3776, 4096, 4916, 5888, 7736, 8192, 11138, 16384, 22287, 23308, 23924, 32768, 39538, 62336, 65536, 71936
Offset: 1

Views

Author

Alexander Adamchuk, Oct 13 2006, Oct 15 2006, Oct 22 2006

Keywords

Comments

Most listed terms a(n) are the powers of 2, except for n = 5,8,10,12,... Corresponding terms that are not powers of 2 are listed in A124238.
It appears that 2^k divides A123855(2^k-1) for all k > 0 (confirmed for 0 < k < 10).
Prime p that divide A123855(p-1) are listed in A123856.

Crossrefs

Cf. A123856, A123855, A086787, A124238.

Programs

  • Mathematica
    Do[f=Mod[Sum[Sum[PowerMod[Prime[i],j,n],{i,1,n-1}],{j,1,n-1}],n];If[f==0&&!PrimeQ[n],Print[n]],{n,2,512}]

Extensions

More terms from Max Alekseyev, Sep 13 2009

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

Cf. A123857, A123856, A123855.

Extensions

More terms from Max Alekseyev, Sep 13 2009
Showing 1-6 of 6 results.

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