A014117 -id:A014117 - cp's OEIS frontend

A108498 Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.

0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 0, 0, 6, 0, 4, 0, 2, 0, 0, 0, 5, 3, 0, 2, 6, 0, 8, 0, 0, 6, 2, 0, 0, 0, 6, 2, 0, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0, 6, 12, 6, 2, 0, 0, 0, 6, 12, 6, 2

Offset: 1

Henry Bottomley, Jun 06 2005

			Rows start: 0; 0,0; 0,2,0; 0,2,0,0; 0,2,3,4,0; 0,0,0,0,0,0; 0,2,3,4,5,6,0; etc.
T(7,3) = 3+9+27+81+243+729+2187 mod 7 = 3279 mod 7 = 3.

Cf. A014117, A059721, A108048, A108497, A108499.

T(n, k+n)=T(n, k). T(n, 0)=T(n, 1)=T(n, n)=T(1, k)=T(2, k)=T(6, k)=T(42, k)=T(1806, k)=0. T(p, k)=k for p prime and 1

A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 3, 6, 11, 4, 13, 14, 5, 16, 17, 6, 19, 12, 1, 22, 23, 8, 25, 26, 9, 28, 29, 58, 31, 32, 11, 34, 35, 12, 37, 38, 13, 24, 41, 2, 43, 44, 15, 46, 47, 16, 49, 30, 17, 52, 53, 18, 45, 56, 19, 58, 59, 116, 61, 62, 3, 64, 65, 22, 67, 68, 23

Offset: 1

Michel Lagneau, Apr 27 2012

nonn

Sum_{k=1..n} k^n (mod n) = 0 if n odd.
Properties of this sequence:
a(n) = 1 for n = 1, 3, 21, 903, ...
a(n) = n if n not divisible by 3;
a(3*n) = n except for n = 7, 10, 14, 20, 21, 26, 28, 30, 35, ...
a(21*n) = n, except for n = 10, 20, 26, 30, 40, 43, 50, 52, ...
a(903*n) = n, except for n = 10, ....
It appears that a(A007018(n)/2) = 1 and conjecturally a(m*A007018(n)/2) = m for a majority of value m.
No, a(A007018(n)/2) <> 1 for n > 4. (For example, a(A007018(5)/2) = a(1631721) = 1807.) - Jonathan Sondow, Oct 18 2013
0 < a(n) < 10 for n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 21, 24, 27, 42, 63, 84, 105, 126, 147, 168, 189, 903, 1806, 2709, 3612, 4515, 5418, 6321, 7224, 8127, .... Search limit was 25000. - Robert G. Wilson v, Jun 18 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..2499 from Michel Lagneau)
J. M. Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv 1309.7941, 2013.
Mathematics Stack Exchange, 1^n + 2^n + ... + (p-1)^n mod p = ?

Cf. A007018, A014117, A031971, A229307, A229311.

for n from 1 to 100 do: s:=sum('k^(2*n)', 'k'=1..2*n)
: x:=irem(s,2*n): printf(`%d, `,x):od:
# second Maple program:
a:= n-> add(k&^(2*n) mod (2*n), k=1..2*n) mod (2*n):
seq(a(n), n=1..100);

Table[Mod[Total[PowerMod[Range[2*n], 2*n, 2*n]], 2*n], {n, 100}] (* T. D. Noe, Apr 28 2012 *)

a(n) = A031971(2n) mod 2n. - Jonathan Sondow, Oct 18 2013

A302345 Primes p such that the set { 1+2p, 1+6p, 1+14p, 1+42p, 1+86p, 1+258p, 1+602p, 1+1806p } does not contain any primes.

Original entry on oeis.org

67, 97, 127, 163, 307, 317, 337, 349, 409, 521, 523, 547, 643, 709, 757, 811, 839, 857, 919, 967, 997, 1021, 1069, 1087, 1093, 1153, 1277, 1291, 1297, 1301, 1399, 1429, 1459, 1483, 1619, 1627, 1637, 1697, 1709, 1721, 1741, 1789, 1877, 1933, 1949, 1999, 2017, 2029, 2083, 2131, 2179, 2239, 2269, 2311, 2383, 2389, 2437, 2503, 2539, 2557, 2591, 2659, 2671, 2707, 2731

Offset: 1

Max Alekseyev, Apr 05 2018

nonn

For each term p, the solutions n to the congruence 1^n + 2^n + ... + n^n == p (mod n) form a subset of A014117 U p*A014117. In particular, there are at most 10 solutions for each such p.

The coefficients { 2, 6, 14, 42, 86, 258, 602, 1806 } are the even divisors of 1806 = 2 * 3 * 7 * 43.

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcén, Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n), Discrete Applied Mathematics 286 (2020), 3-9. Preprint: arXiv:1602.02407 [math.NT], 2016.

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), A226961 (m=3), A226962 (m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Select[Range[3000], PrimeQ[#] && AllTrue[{2, 6, 14, 42, 86, 258, 602, 1806}*# + 1, ! PrimeQ[#1] &] &] (* Amiram Eldar, Aug 09 2020 *)

{ is_A302345(p) = !vecmax( apply( x->ispseudoprime(1+x*p), 2*divisors(3*7*43) ) ); }

A330279 Numbers k such that x^k == k (mod k + 1) has multiple solutions for 0 <= x < k.

Original entry on oeis.org

27, 51, 65, 69, 75, 111, 123, 129, 147, 153, 171, 175, 185, 189, 195, 207, 231, 237, 243, 245, 267, 275, 279, 285, 291, 303, 309, 315, 321, 343, 363, 365, 369, 387, 395, 405, 411, 417, 425, 429, 435, 441, 447, 489, 495, 505, 507, 519, 531, 555, 567, 573, 591, 597

Offset: 1

Christopher Cormier, Dec 09 2019

nonn

All odd numbers have k^k == k (mod k + 1), but only some have other solutions in the least residue system (e.g. 3^27 and 19^27 == 27 (mod 28)).

Odd numbers k such that k and A000010(k+1) are not coprime. - Robert Israel, Jul 30 2023

			27 is in the list because x^27 == 27 (mod 28) has three solutions: 3, 19, and 27.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel et al, Even k such that x^k == -1 mod k has solutions

Cf. A000010, A014117, A096008, A195637.

select(t -> igcd(t,numtheory:-phi(t+1))>1, [seq(i,i=1..1000,2)]); # Robert Israel, Jul 30 2023

ok[k_] := Length[Select[Range[0, k-1], PowerMod[#, k, k + 1] == k &, 2]] > 1; Select[ Range@ 600, ok] (* Giovanni Resta, Dec 10 2019 *)

isok(k) = sum(i=0, k-1, Mod(i, k+1)^k == k) > 1; \\ Michel Marcus, Dec 10 2019

A242927 Numbers m such that k^m + (k+1)^m + ... + (k+m-1)^m is prime for some k.

Original entry on oeis.org

1, 2, 6, 42, 1806

Offset: 1

Derek Orr, May 26 2014

a(5) > 500. For m values < 500 not listed above, k has been checked for k <= 5000.
For the first four terms, the least k that makes k^m + (k+1)^m + ... + (k+m-1)^m prime is {2, 1, 4, 99} respectively.
For a(5) = 1806, k = 3081 yields a strong PRP with 6663 digits. - Don Reble, Mar 23 2018
The known terms a(1..5) coincide with the finite sequence A014117. - M. F. Hasler, May 20 2019

			k^1 = k is prime for k = 2 or any other prime (cf. A000040), so 1 is a term of this sequence.
k^2 + (k+1)^2 is prime for some k (e.g., k = 2 yields 13, see A027861 for the full list), so 2 is a term of this sequence.
k^3 + (k+1)^3 + (k+2)^3 = 3*(k+1)*(k^2+2*k+3) is never prime, therefore 3 is not a term of this sequence.
Similarly, the corresponding expression for m = 4 and m = 5 is a multiple of 2 and 5, respectively, and for all m = 7, ..., 41, the expression also shares a factor with m (and thus is a multiple of m whenever m is prime).
Index m = 110 is the smallest m > 42 for which the expression is not algebraically composite (the polynomial in k has content 1 and is irreducible over Q), but it does factor as (k(k+1)(k+2)(k+3)(k+4))^10 over Z_5, so is always a multiple of 5. Index m = 210 is the next one which is a similar case.
Index m = 231 is much like m = 110, but with a factor 7 instead of 5.
Index m = 330 again yields an irreducible polynomial with content 1, but as before one can show that it is always divisible by 5. And so on.

Cf. A000040, A027861, A014117.

k(n)=for(k=1,5000,if(ispseudoprime(sum(i=0,n-1,(k+i)^n)),return(k)))
for(n=1,500,if(k(n),print(n)))  \\ Edited by M. F. Hasler, Mar 23 2018

a(5) from Don Reble, Mar 23 2018

Example corrected and extended by M. F. Hasler, Apr 05 2018

A372771 Numbers m such that the congruence x^(m+1) == m (mod m+1) is solvable.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 70, 72, 73, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Juri-Stepan Gerasimov, May 12 2024

nonn

Includes all positive even numbers. - Robert Israel, Mar 14 2025

			9 is a term because x^(9+1) == 9 (mod 9+1) for x = 3 and x = 7, i.e., 3^10 = 59049 == 9 (mod 10) and 7^10 = 282475249 == 9 (mod 10).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014117, A330279, A371811.

select(m -> traperror(NumberTheory:-ModularRoot(m,m+1,m+1))::integer, [$1..200]); # Robert Israel, Mar 14 2025

Select[Range[1, 110], With[{m = #}, AnyTrue[Range[1, m + 1], PowerMod[#, m + 1, m + 1] == m &]] &] (* Robert P. P. McKone, May 14 2024 *)

Terms corrected by Robert P. P. McKone, May 14 2024

A108500 Number of values of k (1<=k<=n) where k^(n+1) != k mod n, or equivalently where sum_i{1<=i<=n} k^i != 0 mod n.

Original entry on oeis.org

0, 0, 1, 1, 3, 0, 5, 3, 5, 4, 9, 3, 11, 8, 11, 7, 15, 4, 17, 5, 13, 16, 21, 9, 19, 20, 17, 19, 27, 12, 29, 15, 29, 28, 31, 15, 35, 32, 31, 15, 39, 0, 41, 35, 37, 40, 45, 21, 41, 28, 47, 37, 51, 16, 43, 41, 49, 52, 57, 15, 59, 56, 47, 31, 61, 48, 65, 53, 65, 52, 69, 37, 71, 68, 63, 67

Offset: 1

Henry Bottomley, Jun 06 2005

nonn

			a(2)=0 since 1^3 = 1 mod 2 and 2^3 = 8 = 0 mod 2 = 2 mod 2.
a(3)=1 since 2^1+2^2+2^3 = 14 = 2 mod 3 != 0 mod 3 but 1^1+1^2+1^3 = 3 = 0 mod 3 and 3^1+3^2+3^3 = 39 = 0 mod 3.

Numbers of nonzeros in rows of A108497 or A108498.

a(n)=n-A108499(n). a(n)=0 iff n is in A014117. a(p)=p-2 for p prime.

A259111 a(n) = least number k > 1 such that 1^k + 2^k + ... + k^k == n (mod k).

Original entry on oeis.org

2, 4, 2, 8, 2, 3, 2, 16, 2, 4, 2, 3, 2, 4, 2, 32, 2, 3, 2, 5, 2, 4, 2, 3, 2, 4, 2, 7, 2, 3, 2, 64, 2, 4, 2, 3, 2, 4, 2, 5, 2, 3, 2, 8, 2, 4, 2, 3, 2, 4, 2, 8, 2, 3, 2, 7, 2, 4, 2, 3, 2, 4, 2, 128, 2, 3, 2, 8, 2, 4, 2, 3, 2, 4, 2, 8, 2, 3, 2, 5, 2, 4, 2, 3, 2, 4, 2, 11, 2, 3, 2, 8, 2, 4, 2, 3, 2, 4, 2, 5

Offset: 1

Derek Orr, Jun 18 2015

nonn

			Consider n=2:
Is k=2? 1^2 + 2^2 == 1 (mod 2). No.
Is k=3? 1^3 + 2^3 + 3^3 == 0 (mod 3). No.
Is k=4? 1^4 + 2^4 + 3^4 + 4^4 == 2 (mod 4). Yes. So a(2) = 4.
(Example corrected by _N. J. A. Sloane_, Jul 02 2019)

Alois P. Heinz, Table of n, a(n) for n = 1..8192

Cf. A014117, A226960-A226967.

a:= proc(n) local k; for k from 2 while
      add(i&^k mod k, i=1..k) mod k <> n mod k do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 18 2015

lnk[n_]:=Module[{k=2},While[Mod[Total[Range[k]^k],k]!=Mod[n,k],k++];k]; Array[ lnk,100] (* Harvey P. Dale, Jul 02 2019 *)

vector(100,n,k=2;while(sum(i=1,k,i^k)!=Mod(n,k),k++);k)

a(2^n) = 2^(n+1) for n >= 0.

Previous Showing 31-40 of 42 results. Next

cp's OEIS Frontend

A108495 a(n) = (n^7 - n)/6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A108496 a(n) = (n^43 - n)/42.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A108498 Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A182398 a(n) = (Sum_{k=1..2n} k^2n) mod 2n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A302345 Primes p such that the set { 1+2p, 1+6p, 1+14p, 1+42p, 1+86p, 1+258p, 1+602p, 1+1806p } does not contain any primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A330279 Numbers k such that x^k == k (mod k + 1) has multiple solutions for 0 <= x < k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A242927 Numbers m such that k^m + (k+1)^m + ... + (k+m-1)^m is prime for some k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A372771 Numbers m such that the congruence x^(m+1) == m (mod m+1) is solvable.

Views

Author

Keywords