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 1-10 of 17 results. Next

A123374 Primes p such that p^2 divides A086787(p+1).

Original entry on oeis.org

2, 5, 17, 151
Offset: 1

Views

Author

Alexander Adamchuk, Oct 13 2006

Keywords

Comments

p divides A086787(p+1) for all prime p, except p=3.
Primes p such that p^2 divides (3 + Sum_{j=1..p+1} B_j) * p + 1, where B_j is the j-th Bernoulli number. [Max Alekseyev, Feb 19 2011]
Next term, if it exists, exceeds 320000. [Max Alekseyev, Mar 13 2011]

Examples

			a(1) = 2 because 2^2 divides A086787(3) = 56.
a(2) = 5 because 5^2 divides A086787(6) = 82200, but 3^2 does not divide A086787(4) = 494.

Crossrefs

Cf. A086787.

A014741 Numbers k such that k divides 2^(k+1) - 2.

Original entry on oeis.org

1, 2, 6, 18, 42, 54, 126, 162, 294, 342, 378, 486, 882, 1026, 1134, 1314, 1458, 1806, 2058, 2394, 2646, 3078, 3402, 3942, 4374, 5334, 5418, 6174, 6498, 7182, 7938, 9198, 9234, 10206, 11826, 12642, 13122, 14154, 14406, 16002, 16254
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Comments

Also, numbers k such that k divides Eulerian number A000295(k+1) = 2^(k+1) - k - 2.
Also, numbers k such that k divides A086787(k) = Sum_{i=1..k} Sum_{j=1..k} i^j.
All terms greater than 1 are even; for a proof, see comment in A036236. - Max Alekseyev, Feb 03 2012
If k>1 is a term, then 3*k is also a term. - Alexander Adamchuk, Nov 03 2006
Prime numbers of the form a(m)+1 are given by A069051. - Max Alekseyev, Nov 14 2012
The number 2^m - 2 is a term of this sequence if and only if m - 1 is a term. - Thomas Ordowski, Jul 01 2024

Links

Crossrefs

Cf. A000295, A086787, A015919, A006517, A330382.

Programs

Formula

For n > 1, a(n) = 2*A014945(n-1). - Max Alekseyev, Nov 14 2012

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

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

Links

Crossrefs

Cf. A123857, A123855, A086787.

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 *)

A124405 a(n) = 1 + Sum_{i=1..n} Sum_{j=1..n} i^j.

Original entry on oeis.org

2, 9, 57, 495, 5700, 82201, 1419761, 28501117, 651233662, 16676686697, 472883843993, 14705395791307, 497538872883728, 18193397941038737, 714950006521386977, 30046260016074301945, 1344648068888240941018
Offset: 1

Views

Author

Alexander Adamchuk, Dec 14 2006

Keywords

Comments

p divides a(p) and a(p-1) for prime p.
p^2 divides a(p) for prime p in {5, 13, 563, ...} which seems to coincide with the Wilson primes (A007540).
p^2 divides a(p-1) for prime p in {3, 11, 107, ...} which seems to coincide with the odd primes in A079853.

Links

Crossrefs

Cf. A007540, A079853, A086787.

Programs

  • GAP
    List([1..30], n-> n+1 + Sum([2..n], j-> j*(j^n-1)/(j-1)) ); # G. C. Greubel, Dec 25 2019
  • Magma
    [0] cat [n+1 + (&+[j*(j^n-1)/(j-1): j in [2..n]]): n in [2..30]]; // G. C. Greubel, Dec 25 2019
  • Maple
    seq( n+1+add(j*(j^n-1)/(j-1), j=2..n), n=1..30); # G. C. Greubel, Dec 25 2019
  • Mathematica
    Table[Sum[i^j,{i,1,n},{j,1,n}]+1,{n,1,20}]
  • PARI
    vector(30, n, n+1 + sum(j=2,n, j*(j^n-1)/(j-1)) ) \\ G. C. Greubel, Dec 25 2019
  • Sage
    [n+1 + sum(j*(j^n-1)/(j-1) for j in (2..n)) for n in (1..30)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = 1 + Sum_{i=1..n} Sum_{j=1..n} i^j.
a(n) = n + 1 + Sum_{j=2..n} j*(j^n - 1)/(j-1).
a(n) = A086787(n) + 1.

Extensions

Edited by Max Alekseyev, Jan 29 2012

A215084 a(n) = sum of the sums of the k first n-th powers.

Original entry on oeis.org

0, 1, 6, 46, 470, 6035, 93436, 1695036, 35277012, 828707925, 21693441550, 626254969978, 19766667410282, 677231901484775, 25031756512858200, 992872579254244088, 42066929594261568840, 1896157095455962952169, 90601933352843530354170, 4574495282686422755339734, 243359175218492577008763726
Offset: 0

Views

Author

Olivier Gérard, Aug 02 2012

Keywords

Comments

First term a(0) may be computed as 1 by starting the inner sum at j=0 and taking the convention 0^0 = 1.

Examples

			a(3) = (1^3) + (1^3 + 2^3) + (1^3 + 2^3 + 3^3) = (1^3 + 1^3 + 1^3) + (2^3 + 2^3) + (3^3) = 3 * 1^3 + 2 * 2^3 + 1 * 3^3 = 46. - _David A. Corneth_, Jun 27 2018

Links

Crossrefs

Row sums of A215083.
Cf. A086787, A349116.

Programs

  • Mathematica
    Table[Sum[Sum[j^n, {j, 1, k}], {k, 0, n}], {n, 0, 20}]
a[n_] := (n+1)*HarmonicNumber[-1, -n] - HarmonicNumber[n, -n-1] + (n+1)*HarmonicNumber[n, -n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 05 2013 *)
Table[Total[Accumulate[Range[n]^n]],{n,0,20}] (* Harvey P. Dale, Mar 29 2020 *)
  • PARI
    a(n) = sum(k=1, n, sum(j=1, k, j^n)); \\ Michel Marcus, Jun 25 2018
  • PARI
    a(n) = sum(i=1, n, (n+1-i) * i^n); \\ David A. Corneth, Jun 27 2018

Formula

a(n) = Sum_{k=1..n} Sum_{j=1..k} j^n.
a(n) = Sum_{k=1..n} H_k^{-n} where H_k^{-n} is the k-th harmonic number of order -n.
a(n) = Sum_{k=1..n} (B(n+1, k+1) - B(n+1, 1))/(n+1), where B(n, x) are the Bernoulli polynomials. - Daniel Suteu, Jun 25 2018
G.f.: Sum_{k>=1} k^k*x^k/(1 - k*x)^2. - Ilya Gutkovskiy, Oct 11 2018
a(n) ~ c * n^n, where c = 1/(1 - 2*exp(-1) + exp(-2)) = 2.50265030107711874333... - Vaclav Kotesovec, Nov 06 2021

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

A124239 a(n) = Sum_{k=1..n} Sum_{m=1..n} (2*k - 1)^m.

Original entry on oeis.org

1, 14, 197, 3704, 90309, 2704470, 95856025, 3921108576, 181756280697, 9413656622446, 538727822713277, 33757715581666296, 2298714540642445405, 169016703698449309846, 13345320616706684277361, 1126219424250538393789824, 101160070702700567996590513, 9636001314414804672487492878
Offset: 1

Views

Author

Alexander Adamchuk, Oct 22 2006

Keywords

Comments

a(3) = 197 and a(11) = 538727822713277 are primes.
p divides a(p+1) for primes p > 3.
a(2*k-1) is odd. a(2*k) is even. a(2^k) is divisible by 2^(2*k - 1) for k > 0.
Numbers n such that a(n) is divisible by n are listed in A124240.

Links

Crossrefs

Cf. A124240, A124241.
Cf. also A068563, A086787, A123855.

Programs

  • Mathematica
    Table[Sum[(2k-1)^m,{k,1,n},{m,1,n}],{n,1,20}]
  • PARI
    a(n) = sum(k=1, n, sum(m=1, n, (2*k - 1)^m)); \\ Michel Marcus, Apr 24 2022

Formula

a(n) = Sum_{k=1..n} Sum_{m=1..n} (2*k - 1)^m.
a(n) = n + Sum_{k=2..n} (2*k - 1)*((2*k - 1)^n - 1)/(2*(k - 1)).

A349117 a(n) = Sum_{m=1..n} (Sum_{k=1..m} (Sum_{j=1..k} j^k)).

Original entry on oeis.org

1, 7, 49, 445, 5266, 77258, 1349554, 27306462, 627568355, 16142172173, 459332766227, 14324480721391, 485783513552956, 17798331858727376, 700589353757045796, 29484907446960975744, 1321168518044435497005, 62795290373559355285155, 3155553461189975793914005
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 08 2021

Keywords

Links

Crossrefs

Cf. A086787, A349116.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[j^k, {j, 1, k}], {k, 1, m}], {m, 1, n}], {n, 1, 20}]

Formula

a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393 = 1/A068996 = 1.581976706...

A123269 Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].

Original entry on oeis.org

1, 28, 7625731729896, 13407807929942597099574024998205985135931742965325158317510351105024878248924471298029103219186757034747676158536830429928105045387310278568778808509188348
Offset: 1

Views

Author

Alexander Adamchuk, Oct 09 2006

Keywords

Comments

The next term is too large to include.
Prime p divides a(p) for p = {2, 3, 7, 11, 23, 31, 43, 47, 59, 67, 71, 79, ...} = A039787[n] Primes p such that p-1 is squarefree. p^2 divides a(p) for prime p = {2,3}.

Crossrefs

Cf. A039787. Cf. A086787 - Sum[ i^j, {i, 1, n}, {j, 1, n} ].
Numbers n that divide a(n) are listed in A124391(n) = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 21, 22, 23, 24, 27, 28, 31, ...}.

Programs

  • Mathematica
    Table[Sum[i^j^k,{i,1,n},{j,1,n},{k,1,n}],{n,1,5}]
  • PARI
    a(n)=sum(i=1,n,sum(j=1,n,sum(k=1,n,i^j^k))) \\ Charles R Greathouse IV, May 15 2013

Formula

a(n) = Sum[ i^j^k, {i,1,n}, {j,1,n}, {k,1,n} ].
Showing 1-10 of 17 results. Next

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