A122781 -id:A122781 - cp's OEIS frontend

A129521 Numbers of the form pq, p and q prime with q=2p-1.

6, 15, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

All terms are Fermat 4-pseudoprimes, i.e., satisfy 4^n == 4 (mod n). See A020136 and A122781.

David W. Wilson, Table of n, a(n) for n = 1..1000
D. E. Iannucci, When the small divisors of a natural number are in arithmetic progression, INTEGERS, Electronic Journal of Combinatorial Number Theory, #77, 2018. See p. 9.

Cf. A005382, A005383.
Subsequence of A006881, A129510, and A122781.
Intersection of A000384 and A001358, "hexagonal semiprimes". - Wesley Ivan Hurt, Jul 04 2013
Cf. A141768, A191311, A191592, A245365, A259676, A259677.

a129521 n = p * (2 * p - 1) where p = a005382 n
-- Reinhard Zumkeller, Nov 10 2013

[2*n^2-n: n in [0..1000]|IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 27 2010

p = Select[Prime[Range[155]], PrimeQ[2# - 1] &]; p (2p - 1) (* Robert G. Wilson v, Sep 11 2011 *)

forprime(p=2,10000,q=2*p-1;if(isprime(q),print1(p*q,", ")))

a(n) = A005382(n)*A005383(n).

A122780 Nonprimes k such that 3^k == 3 (mod k).

Original entry on oeis.org

1, 6, 66, 91, 121, 286, 561, 671, 703, 726, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7107, 7381, 8205, 8401, 8646, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345

Offset: 1

Farideh Firoozbakht, Sep 11 2006

Theorem: If q!=3 and both numbers q and (2q-1) are primes then k=q*(2q-1) is in the sequence. 6, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, ... is the related subsequence.

The terms > 1 and coprime to 3 of this sequence are the base-3 pseudoprimes, A005935. - M. F. Hasler, Jul 19 2012 [Corrected by Jianing Song, Feb 06 2019]

			66 is composite and 3^66 = 66*468229611858069884271524875811 + 3 so 66 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001567, A005935, A122014.

Cf. A122781, A122782, A122783, A122784, A122785, A122786.

isA122780 := proc(n)
    if isprime(n) then
        false;
    else
        modp( 3 &^ n,n) = modp(3,n) ;
    end if;
end proc:
for n from 1 do
    if isA122780(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 15 2012

Select[Range[30000], ! PrimeQ[ # ] && Mod[3^#, # ] == Mod[3, # ] &]
Join[{1},Select[Range[20000],!PrimeQ[#]&&PowerMod[3,#,#]==3&]] (* Harvey P. Dale, Apr 30 2023 *)

is_A122780(n)={n>0 & Mod(3, n)^n==3 & !ispseudoprime(n)} \\ M. F. Hasler, Jul 19 2012

A020136 Fermat pseudoprimes to base 4.

Original entry on oeis.org

15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071, 2465, 2701, 2821, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795, 4859, 5461, 5551, 6601, 6643, 7957, 8321, 8481, 8695, 8911, 9061, 9131

Offset: 1

David W. Wilson

nonn

If q and 2q-1 are odd primes, then n=q*(2q-1) is in the sequence. So for n>1, A005382(n)*(2*A005382(n)-1) form a subsequence (cf. A129521). - Farideh Firoozbakht, Sep 12 2006
Primes q and 2q-1 are a Cunningham chain of the second kind. - Walter Nissen, Sep 07 2009
Composite numbers n such that 4^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Chris Caldwell, Cunningham chain.
Chris Caldwell et al., Top Twenty Cunningham Chains (2nd kind).
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Index entries for sequences related to pseudoprimes

Subsequence of A122781.
Contains A001567 (Fermat pseudoprimes to base 2) as a subsequence.
Cf. A005382, A129521.

Select[Range[9200], ! PrimeQ[ # ] && PowerMod[4, # - 1, # ] == 1 &] (* Farideh Firoozbakht, Sep 12 2006 *)

isok(n) = (Mod(4, n)^(n-1)==1) && !isprime(n) && (n>1); \\ Michel Marcus, Apr 27 2018

A129494 Composite numbers k such that 4^k mod k is a power of 4 greater than 1.

Original entry on oeis.org

6, 12, 15, 20, 22, 24, 26, 28, 30, 34, 38, 40, 46, 48, 56, 58, 60, 62, 66, 69, 72, 74, 77, 80, 82, 84, 85, 86, 87, 88, 91, 93, 94, 96, 102, 104, 105, 106, 111, 117, 118, 120, 122, 123, 126, 129, 132, 134, 140, 141, 142, 144, 146, 158, 159, 166, 168, 170, 177, 178, 182

Offset: 1

Robert G. Wilson v, Apr 17 2007

Complement to composite numbers: 4, 8, 9, 10, 14, 16, 18, 21, 25, 27, 32, 33, 35, 36, 39, 42, 44, 45, 49, 50, 51, 52, 54, 55, 57, ... - R. J. Mathar, May 16 2008

			22 is a term since 4^22 mod 22 = 16.

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

Cf. A036236, A129492, A129493, A129495, A129496, A129497.

Contains A122781 except for 1 and 4.

[k:k in [2..200]| not IsPrime(k)  and  not IsZero(a)  and (PrimeDivisors(a) eq [2]) and  &+[j[1]*j[2]: j in Factorization(a) ] mod 4 eq 0 where a is 4^k mod k]; // Marius A. Burtea, Dec 04 2019

filter:= proc(n) local k,j;
  if isprime(n) then return false fi;
  k:= 4 &^ n mod n;
  j:= padic:-ordp(k,2);
  k>1 and j::even and k = 2^j
end proc:
select(filter, [$4..1000]); # Robert Israel, Dec 03 2019

Select[ Range@ 161, IntegerQ@ Log[4, PowerMod[4, #, # ]] &]

Corrected and extended by R. J. Mathar, May 16 2008

A122784 Nonprimes k such that 7^k == 7 (mod k).

Original entry on oeis.org

1, 6, 14, 21, 25, 42, 105, 133, 231, 301, 325, 525, 561, 703, 817, 1105, 1729, 1825, 2101, 2353, 2465, 2821, 3277, 3325, 3486, 3913, 4011, 4525, 4825, 5565, 5719, 5901, 6601, 6697, 7525, 8321, 8911, 9331, 10225, 10325, 10585, 10621, 11041, 11521

Offset: 1

Farideh Firoozbakht, Sep 12 2006

nonn

Theorem: If both numbers q and 2q-1 are primes then q*(2q-1) is in the sequence iff q=2 or mod(q,14) is in the set {1, 5, 13}. 6, 703, 18721, 38503, 88831, 104653, 146611, 188191,... are such terms.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005938, A121014, A122780, A122781, A122782, A122783, A122785, A122786.

Select[Range[20000], ! PrimeQ[#] && PowerMod[7, #, #] == Mod[7, #] &]
With[{nn=12000},Select[Complement[Range[nn],Prime[Range[PrimePi[ nn]]]], PowerMod[7,#,#]==Mod[7,#]&]] (* Harvey P. Dale, Jul 12 2012 *)

A175942 Odd numbers k such that 4^k == 4 (mod 3k) and 2^(k-1) == 4 (mod 3(k-1)).

Original entry on oeis.org

5, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543, 2579, 2819, 2879

Offset: 1

Alzhekeyev Ascar M, Oct 27 2010

nonn

Equivalently, integers k == 5 (mod 6) such that 4^k == 4 (mod k) and 2^(k-1) == 4 (mod k-1).
Equivalently, integers k == 5 (mod 6) such that both k and (k-1)/2 are primes or (odd or even) Fermat 4-pseudoprimes (A122781).
Contains terms k of A175625 such that k == 5 (mod 6).
Contains terms k of A303448 such that k == 5 (mod 6).
Many composite terms of this sequence are of the form A007583(m) = (2^(2m+1) + 1)/3 (for m in A303009). It is unknown if there exist composite terms not of this form.
Numbers k such that 2^(k-1) == 3k+1 (mod 3(k-1)k). This sequence contains all safe primes except 7. The term a(20) = 683 = 2*341+1 is the smallest prime that is not safe. - Thomas Ordowski, Jun 07 2021

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005385.

Select[Range[1,3001,2],PowerMod[4,#,3#]==4&&PowerMod[2,#-1,3(#-1)]==4&] (* Harvey P. Dale, Aug 04 2018 *)

Edited by Max Alekseyev, Apr 24 2018

A290543 Composite numbers n such that A290542(n) >= 2.

Original entry on oeis.org

28, 65, 66, 85, 91, 105, 117, 121, 124, 133, 145, 153, 154, 165, 185, 186, 190, 205, 217, 221, 231, 244, 246, 247, 259, 273, 276, 280, 286, 292, 301, 305, 310, 325, 341, 343, 344, 357, 364, 366, 369, 370, 377, 385, 396, 418, 425, 427, 429, 430, 435, 451

Offset: 1

Arkadiusz Wesolowski, Aug 05 2017

nonn

Is a(n) ~ n * log n as n -> infinity?

Cf. A001567, A006935, A122780, A122781, A122782, A122783, A122784, A122785, A122786, A290542.

lst:=[]; for n in [4..451] do if not IsPrime(n) then r:=Floor(Sqrt(n)); for k in [2..r] do if Modexp(k, n, n) eq k then Append(~lst, n); break; end if; end for; end if; end for; lst;

Select[Flatten@ Position[#, k_ /; k >= 2], CompositeQ] &@ Table[SelectFirst[Range[2, Sqrt@ n], PowerMod[#, n , n] == Mod[#, n] &], {n, 451}] (* Michael De Vlieger, Aug 09 2017 *)

A247307 Numbers of the form (4^k - 4)/k.

Original entry on oeis.org

0, 6, 20, 63, 204, 682, 2340, 381300, 1398101, 5162220, 71582788, 1010580540, 14467258260, 3059510616420, 2573485501354569, 9938978487990060, 148764065110560900, 510526106256177860940, 117943982401427236556700, 1799331452449680632120820

Offset: 1

Juri-Stepan Gerasimov, Sep 11 2014

nonn

Subsequence of A246445.
Generated by k = 1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 15, 17, 19, 23, 28, 29, 31,. ..
This set of k contains all terms of A122781 and all primes. [It contains the primes because j^p == j (mod p) for every integer j if p is prime; see e.g. the corollary 4.4 to the Lagrange theorem in Jones et al.]

			a(9) = 1398101 because (4^12 - 4)/12 = 1398101 for k = 12.

G. A. Jones and J. M. Jones, Congruences with a prime-power modulus, p 65-82 in "Elementary Number Theory", Springer Undergraduate Mathematics Series, (1988).

Cf. A020136, A064535, A122781, A246445, A247033.

lista(nn) = {for (k=1, nn, va = (4^k - 4)/k; if (type(va) == "t_INT", print1(va, ", ")););} \\ Michel Marcus, Sep 12 2014

cp's OEIS Frontend

A129521 Numbers of the form p*q, p and q prime with q=2*p-1.

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A122780 Nonprimes k such that 3^k == 3 (mod k).

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A020136 Fermat pseudoprimes to base 4.

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A129494 Composite numbers k such that 4^k mod k is a power of 4 greater than 1.

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A122784 Nonprimes k such that 7^k == 7 (mod k).

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A175942 Odd numbers k such that 4^k == 4 (mod 3*k) and 2^(k-1) == 4 (mod 3*(k-1)).

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A290543 Composite numbers n such that A290542(n) >= 2.

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Magma

Mathematica

A247307 Numbers of the form (4^k - 4)/k.

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A129521 Numbers of the form pq, p and q prime with q=2p-1.

A175942 Odd numbers k such that 4^k == 4 (mod 3k) and 2^(k-1) == 4 (mod 3(k-1)).