A066606 -id:A066606 - cp's OEIS frontend

A066445 a(n) = 12^n mod n^12.

0, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 0, 13800864889148, 36652392292352, 96953703492618, 236674172846080, 568249606736865, 15868743229440, 1270692936983464, 2296447475122176, 1898727404471631, 8621343763677184

Offset: 1

Robert G. Wilson v, Dec 27 2001

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A066606.

Table[ Mod[12^n, n^12], {n, 1, 20} ]
Table[PowerMod[12,n,n^12],{n,20}] (* Harvey P. Dale, Apr 05 2019 *)

a(n) = { lift(Mod(12, n^12)^n) } \\ Harry J. Smith, Feb 14 2010

A066608 a(n) = 4^n mod n^4.

Original entry on oeis.org

0, 0, 64, 0, 399, 208, 1978, 0, 6265, 8576, 6978, 1792, 19075, 22864, 36199, 0, 17089, 87616, 31867, 107776, 83791, 142928, 195688, 225280, 201999, 302656, 362152, 304896, 401393, 546976, 612688, 0, 226279, 629152, 399674, 718336, 1132463

Offset: 1

Amarnath Murthy, Dec 22 2001

			a(7) = 1978 as 4^7 = 16384 = (7^4)*6 + 1978.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066606.

Table[PowerMod[4,n,n^4],{n,50}] (* Harvey P. Dale, Aug 25 2023 *)

a(n) = { lift(Mod(4, n^4)^n) } \\ Harry J. Smith, Mar 11 2010

More terms from Floor van Lamoen and Robert G. Wilson v, Dec 23 2001

A066609 a(n) = 5^n mod n^5.

Original entry on oeis.org

0, 25, 125, 625, 0, 73, 10897, 30177, 4508, 65625, 29672, 36433, 263034, 288873, 575000, 159681, 592030, 1485433, 1639363, 1240625, 250928, 928073, 4040001, 93601, 0, 10915033, 14288075, 16048657, 3176520, 4515625

Offset: 1

Amarnath Murthy, Dec 24 2001

nonn

			a(6) = 73 as 5^6 = 15625 = (6^5)*2 + 73.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066606.

Table[PowerMod[5,n,n^5],{n,30}] (* Harvey P. Dale, Jul 05 2023 *)

a(n) = { lift(Mod(5, n^5)^n) } \\ Harry J. Smith, Mar 11 2010

More terms from Robert G. Wilson v, Dec 26 2001

A066610 a(n) = 6^n mod n^6.

Original entry on oeis.org

0, 36, 216, 1296, 7776, 0, 44638, 106752, 511758, 466176, 1398612, 0, 4175671, 4282944, 2765826, 4259840, 20220503, 0, 36376760, 40062976, 25606125, 57077760, 109780662, 0, 144545126, 283401024, 0, 454885376, 299969829

Offset: 1

Amarnath Murthy, Dec 24 2001

nonn

			a(7) = 44638 as 6^7 = 279936 = (7^6)*2 + 44638.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066606.

Table[PowerMod[6,n,n^6],{n,30}] (* Harvey P. Dale, Jan 21 2019 *)

a(n) = { lift(Mod(6, n^6)^n) } \\ Harry J. Smith, Mar 11 2010

More terms from Robert G. Wilson v, Dec 26 2001

A188339 Primes p such that 2^p mod p^2 is prime.

Original entry on oeis.org

5, 53, 61, 193, 227, 257, 307, 317, 383, 457, 577, 601, 607, 653, 727, 751, 947, 1019, 1031, 1039, 1049, 1093, 1123, 1193, 1259, 1283, 1409, 1471, 1483, 1607, 1613, 1667, 1987, 2011, 2029, 2203, 2357, 2371, 2377, 2909, 2939, 3011, 3049, 3089, 3163

Offset: 1

Juri-Stepan Gerasimov, Mar 28 2011

nonn

			5 is a term because 5 is prime and (2^5 mod 5^2) = 32 mod 25 = 7 also prime.

Felix Fröhlich, Table of n, a(n) for n = 1..3119 (all terms up to 10^6)

Cf. A066606.

Select[Prime[Range[500]], PrimeQ[PowerMod[2,#, #^2]] &] (* Alonso del Arte, Mar 28 2011 *)

forprime(p=2, 10^3, if(isprime(2^p%p^2), print1(p, ", "))) \\ Felix Fröhlich, Jun 28 2014

A219219 Numbers k such that 2^k (mod k^2) is prime.

Original entry on oeis.org

5, 21, 53, 55, 61, 95, 111, 155, 165, 189, 193, 213, 221, 227, 245, 249, 257, 289, 291, 303, 305, 307, 317, 339, 345, 355, 363, 383, 385, 423, 429, 437, 457, 465, 477, 505, 577, 597, 601, 607, 621, 653, 655, 679, 705, 715, 727, 749, 751, 765, 781, 849, 889, 939

Offset: 1

Alex Ratushnyak, Nov 15 2012

nonn

Indices of primes in A066606.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A066606.

import java.math.BigInteger;
public class A219219 {
  public static void main (String[] args) {
    BigInteger b2 = BigInteger.valueOf(2);
    for (int n=1; ; n++) {
      BigInteger bn = BigInteger.valueOf(n);
      BigInteger pp  = b2.modPow(bn, bn.multiply(bn));
      if (pp.isProbablePrime(2)) {
          if (pp.isProbablePrime(80))
              System.out.printf("%d, ",n);
      }
    }
  }
}

a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 0, a(n-1))
      while not isprime(2 &^k mod k^2) do od; k
    end:
seq (a(n), n=1..100);  # Alois P. Heinz, Nov 17 2012

Flatten[Position[Table[PowerMod[2, k, k^2], {k, 1000}], ?(PrimeQ[#] &)]] (* _T. D. Noe, Nov 15 2012 *)
Select[Range[1000],PrimeQ[PowerMod[2,#,#^2]]&] (* Harvey P. Dale, Mar 29 2020 *)

from sympy import isprime
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    if isprime(pow(2, k, k*k)): alst.append(k)
  return alst
print(aupto(939)) # Michael S. Branicky, May 21 2021

cp's OEIS Frontend

A066445 a(n) = 12^n mod n^12.

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A066608 a(n) = 4^n mod n^4.

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A066609 a(n) = 5^n mod n^5.

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A066610 a(n) = 6^n mod n^6.

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Mathematica

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Extensions

A188339 Primes p such that 2^p mod p^2 is prime.

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Programs

Mathematica

PARI

A219219 Numbers k such that 2^k (mod k^2) is prime.

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Java

Maple

Mathematica

Python