A069547 -id:A069547 - cp's OEIS frontend

A117131 Remainder when n^n is divided by the n-th prime number.

1, 1, 2, 4, 1, 12, 12, 7, 2, 6, 24, 26, 11, 6, 30, 15, 9, 41, 39, 32, 51, 64, 78, 4, 72, 43, 89, 25, 31, 109, 26, 62, 80, 36, 9, 72, 132, 140, 105, 52, 132, 135, 117, 147, 79, 193, 93, 49, 175, 76, 64, 187, 196, 9, 163, 221, 190, 62, 102, 280, 71, 87, 102, 268

Offset: 1

Axel Harvey, Jul 23 2006

			a(8)=7 because 8^8 is 16777216, the 8th prime is 19 and 16777216 modulo 19 is 7.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000312, A069547.

[Modexp(n, n, NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Mar 20 2019

a:= n-> n&^n mod ithprime(n):
seq(a(n), n=1..81);  # Alois P. Heinz, Mar 20 2019

Table[PowerMod[n, n, Prime[n]], {n, 100}] (* T. D. Noe, Aug 20 2013 *)

a(n) = lift(Mod(n, prime(n))^n); \\ Michel Marcus, Mar 20 2019

a(n) = mod(n^n, prime(n)).

More terms from Franklin T. Adams-Watters, Jul 26 2006

A167622 a(n) = n^3 mod n-th prime.

Original entry on oeis.org

1, 2, 2, 1, 4, 8, 3, 18, 16, 14, 29, 26, 24, 35, 38, 15, 16, 37, 25, 48, 63, 62, 49, 29, 8, 2, 10, 17, 82, 106, 73, 18, 43, 106, 112, 148, 99, 104, 34, 163, 6, 59, 51, 71, 111, 25, 11, 207, 63, 195, 74, 76, 180, 87, 96, 195, 121, 263, 122, 192, 15, 119, 149, 282, 124, 294

Offset: 1

Zak Seidov, Nov 07 2009

nonn

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000 (first 1000 terms from G. C. Greubel)

Cf. A069547, A167623.

(*1*) Table[PowerMod[n,3,Prime[n]],{n,100}]
(*2*) PowerMod[ #,3,Prime[ # ]]&/@Range[100]

a(n) = lift(Mod(n, prime(n))^3); \\ Michel Marcus, Jan 28 2025

from sympy import sieve
def A167622(n): return pow(n,3,sieve[n]) # Karl-Heinz Hofmann, Jan 28 2025

A167623 a(n) = n^3 mod (n-th prime squared).

Original entry on oeis.org

1, 8, 2, 15, 4, 47, 54, 151, 200, 159, 370, 359, 516, 895, 1166, 1287, 1432, 2111, 2370, 2959, 3932, 4407, 5278, 5903, 6216, 7375, 9074, 10503, 627, 1462, 13662, 15607, 17168, 662, 20674, 1054, 1355, 1734, 3541, 4142, 4839, 8566, 6545, 10686, 13507

Offset: 1

Zak Seidov, Nov 07 2009

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000 (first 1000 terms from G. C. Greubel)

Cf. A069547, A167622.

(*1*) Table[PowerMod[n,3,Prime[n]^2],{n,100}]
(*2*) PowerMod[ #,3,Prime[ # ]^2]&/@Range[100]

a(n) = lift(Mod(n, prime(n)^2)^3); \\ Michel Marcus, Jan 28 2025

from sympy import sieve
def A167623(n): return pow(n, 3, sieve[n]**2) # Karl-Heinz Hofmann, Jan 28 2025

A096459 Triangle read by rows: T(n,k) = n^2 mod prime(k), 1<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 4, 0, 1, 1, 2, 1, 1, 0, 4, 3, 0, 0, 1, 1, 3, 10, 1, 1, 4, 0, 5, 10, 15, 0, 1, 4, 1, 9, 12, 13, 7, 1, 0, 1, 4, 4, 3, 13, 5, 12, 0, 1, 0, 2, 1, 9, 15, 5, 8, 13, 1, 1, 1, 2, 0, 4, 2, 7, 6, 5, 28, 0, 0, 4, 4, 1, 1, 8, 11, 6, 28, 20, 33, 1, 1, 4, 1, 4, 0, 16, 17, 8, 24, 14, 21, 5, 0, 1, 1, 0

Offset: 1

Reinhard Zumkeller, Aug 12 2004

T(n,k)=0 iff k is a prime factor of n:
A001221(n) = number of zeros in n-th row;
T(n,1)=A000035(n);
T(n,2)=A011655(n) for n>1; T(n,3)=A070430(n) for n>2;
T(n,4)=A053879(n) for n>3; T(n,5)=A070434(n) for n>4;
T(n,6)=A070436(n) for n>5; T(n,7)=A054580(n) for n>6;
T(n,8)=A070441(n) for n>7; T(n,9)=A070445(n) for n>8;
T(n,10)=A070451(n) for n>9;
T(n,n)=A069547(n).

			Triangle begins:
1;
0, 1;
1, 0, 4;
0, 1, 1, 2;
1, 1, 0, 4, 3;
0, 0, 1, 1, 3, 10;
1, 1, 4, 0, 5, 10, 15;
......

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000290, A000040, A049759.

Table[Mod[n^2, Prime[k]], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, May 20 2017 *)

A097882 a(n) = floor( n^2/prime(n) ).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16

Offset: 1

Reinhard Zumkeller, Sep 02 2004

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A000290, A001221, A069547.

Table[Floor[n^2/Prime[n]], {n,1,50}] (* G. C. Greubel, May 21 2017 *)

for(n=1,50, print1(floor(n^2/prime(n)), ", ")) \\ G. C. Greubel, May 21 2017

a(n) = floor(A000290(n)/A000040(n)).

A000290(n) = a(n)*A000040(n) + A069547(n).

A152526 Numbers k with property that (k^2 mod prime(k)) < 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 13, 21, 27, 44, 104, 365, 5477, 16787, 20272, 446691, 1793239, 9806042, 198017575, 2830612574

Offset: 1

Zak Seidov, Oct 27 2009

Numbers k with property that A069547(k) < 10.

a(21) > 4.4*10^10, if it exists. - Amiram Eldar, Jul 04 2021

			(n, A069547(n)): (1,1),(2,1),(3,4),(4,2),(5,3),(8,7),(13,5),(21,3),(27,8),(44,6),(104,5),(365,7),(5477,7),(16787,8),(20272,3),(446691,5),(1793239,8),(9806042,2),(198017575,2),(2830612574, 6).
198017575^2 = 9384391*prime(198017575) + 2.

Cf. A069547 (n^2 mod n-th prime).

Do[m=PowerMod[n,2,Prime[n]];If[m<10,Print[{n,m}]],{n,2*10^8}]

a(20) from Amiram Eldar, Jul 04 2021

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A117131 Remainder when n^n is divided by the n-th prime number.

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A167622 a(n) = n^3 mod n-th prime.

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A167623 a(n) = n^3 mod (n-th prime squared).

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A096459 Triangle read by rows: T(n,k) = n^2 mod prime(k), 1<=k<=n.

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A097882 a(n) = floor( n^2/prime(n) ).

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A152526 Numbers k with property that (k^2 mod prime(k)) < 10.

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Mathematica

Extensions