A122782 -id:A122782 - cp's OEIS frontend

A153515 Terms of A122782 which are not Carmichael numbers A002997.

1, 4, 10, 15, 20, 65, 124, 190, 217, 310, 435, 781, 1541, 1891, 3565, 3820, 4123, 4495, 5461, 5611, 5662, 5731, 6735, 7449, 7813, 8029, 8290, 9881, 11041, 11476, 12801, 13021, 13333, 13981, 14981, 15751, 16297, 17767, 20345, 20710, 21361, 22791

Offset: 1

Artur Jasinski, Dec 28 2008

nonn

Are there entries in this sequence which are also in A153513 ?

Yes. This subsequence starts 721801, 873181, 4504501, 8646121, 9006401, 9863461, 10403641, 10680265,... (similar to A153580). - R. J. Mathar, Mar 30 2011

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

Cf. A002997, A122780, A153508, A153513.

Select[Range[10^4], !PrimeQ[#] && PowerMod[5, #, # ] == Mod[5, #] && Mod[#, CarmichaelLambda[#]] != 1 &] (* Amiram Eldar, Sep 19 2019 *)

A005936 Pseudoprimes to base 5.

Original entry on oeis.org

4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881, 11041, 11476, 12801, 13021, 13333, 13981, 14981, 15751, 15841, 16297, 17767, 21361, 22791, 23653, 24211, 25327, 25351, 29341, 29539

Offset: 1

N. J. A. Sloane

nonn

According to Karsten Meyer, 4 should be excluded, following the strict definition in Crandall and Pomerance. - May 16 2006
Theorem: If both numbers q and (2q - 1) are primes (q is in the sequence A005382) then n = q*(2q - 1) is a pseudoprime to base 5 (n is in the sequence) if and only if q is of the form 10k + 1. 1891, 88831, 146611, 218791, 721801, ... are such terms. This sequence is a subsequence of A122782. - Farideh Firoozbakht, Sep 14 2006
Composite numbers n such that 5^(n-1) == 1 (mod n).

R. Crandall and C. Pomerance, "Prime Numbers - A Computational Perspective", Second Edition, Springer Verlag 2005, ISBN 0-387-25282-7 Page 132 (Theorem 3.4.2. and Algorithm 3.4.3)
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 124, p. 43, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, A12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, T. D. Noe and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..92893 (terms a(1)-a(776) from R. J. Mathar, a(777)-a(1000) from T. D. Noe)
J. Bernheiden, Pseudoprimes (Text in German)
C. Pomerance & N. J. A. Sloane, Correspondence, 1991
F. Richman, Primality testing with Fermat's little theorem
Eric Weisstein's World of Mathematics, Fermat Pseudoprime
Index entries for sequences related to pseudoprimes

Pseudoprimes to other bases: A001567 (2), A005935 (3), A005937 (6), A005938 (7), A005939 (10).

Cf. A005382, A122782.

base = 5; t = {}; n = 1; While[Length[t] < 100, n++; If[! PrimeQ[n] && PowerMod[base, n-1, n] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Feb 21 2012 *)
Select[Range[30000],CompositeQ[#]&&PowerMod[5,#-1,#]==1&] (* Harvey P. Dale, Jul 21 2023 *)

More terms from David W. Wilson, Aug 15 1996

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

A123091 Numbers k such that k divides 5^k - 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 13, 15, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 124, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 217, 223, 227, 229, 233, 239

Offset: 1

Alexander Adamchuk, Nov 04 2006

nonn

All primes are the terms of a(n). Nonprimes in a(n) are listed in A122782(n) = {1,4,10,15,20,65,124,190,217,310,435,561,781,...}. All pseudoprimes to base 5 are the terms of a(n). They are listed in A005936(n) = {4,124,217,561,781,...}. Numbers n up to 10^6 such that n divides 5^n + 5 are {1,2,5,6,10,30,70,1565,2806,3126,51670,58290,214405,285286,378258}.

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

Cf. A122782 (nonprimes n such that 5^n==5 (mod n)).
Cf. A005936 (pseudoprimes to base 5).
Cf. A067946 (numbers n such that n divides 5^n-1).
Cf. A015951 (numbers n such that n | 5^n + 1).

Select[Range[1000], IntegerQ[(PowerMod[5,#,# ]-5)/# ]&]

is(n)=Mod(5,n)^n==5 \\ Charles R Greathouse IV, Nov 04 2016

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 *)

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 *)

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A153515 Terms of A122782 which are not Carmichael numbers A002997.

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A005936 Pseudoprimes to base 5.

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

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Maple

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PARI

A123091 Numbers k such that k divides 5^k - 5.

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PARI

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

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

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Magma

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