A002997 -id:A002997 - 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 *)

A153580 Terms of A083737 which are not Carmichael numbers (A002997).

Original entry on oeis.org

721801, 873181, 4504501, 8646121, 9006401, 9863461, 10403641, 12322133, 14609401, 15913261, 18595801, 18736381, 20234341, 21397381, 22066201, 22369621, 22885129, 25326001, 25696133, 28312921, 36307981, 42702661

Offset: 1

Ray Chandler & Artur Jasinski, Dec 28 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2871 (calculated from the b-file at A083737)

Cf. A002997, A083737, A122780, A153508, A153513, A153514, A153515, A153519, A153581.

Select[Range[5*10^7], ! PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, # - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 && Mod[ #, CarmichaelLambda[ # ]] != 1 &] (* Ray Chandler, Dec 28 2008 *)

A153581 Pseudoprimes to bases 2,3,5 and 7 which are not Carmichael numbers (A002997).

Original entry on oeis.org

721801, 8646121, 10403641, 22885129, 36307981, 42702661, 46094401, 48064021, 52204237, 79398901, 80918281, 81954133, 114329881, 116151661, 143168581, 170782921, 188985961, 217145881, 220531501, 282707461, 299671921, 303373801, 326695141, 353815801, 361307521

Offset: 1

Ray Chandler & Artur Jasinski, Dec 28 2008

nonn

Terms congruent to 5 (mod 6): 468950021, 493108481, 659846021, 5936122901, 8144063621, ... - Robert G. Wilson v, Sep 03 2014

Terms not congruent to 1 (mod 12): 468950021, 493108481, 643767931, 659846021, 773131927, 5779230451, 5936122901, 7294056727, 8144063621, 9671001451, ... - Robert G. Wilson v, Sep 03 2014

Amiram Eldar, Table of n, a(n) for n = 1..1077 (terms 1..125 from Robert G. Wilson v)

Cf. A002997, A083737, A122780, A153508, A153513, A153514, A153515, A153519, A153580.

fQ[n_] := ! PrimeQ[n] && PowerMod[2, n - 1, n] == 1 && PowerMod[3, n - 1, n] == 1 && PowerMod[5, n - 1, n] == 1 && PowerMod[7, n - 1, n] == 1 && Mod[n, CarmichaelLambda[n]] != 1; Select[ Range[ 365000000], fQ] (* Ray Chandler, Dec 28 2008; corrected by Robert G. Wilson v, Sep 01 2014 *)

Terms a(8) and onward from Robert G. Wilson v, Sep 01 2014

A262043 Carmichael numbers (A002997) that are not absolute Euler pseudoprimes (A033181).

Original entry on oeis.org

561, 1105, 2821, 6601, 8911, 10585, 29341, 52633, 62745, 63973, 101101, 115921, 126217, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 410041, 512461, 552721, 658801, 748657, 825265, 852841, 1024651, 1033669, 1082809, 1152271, 1193221, 1461241, 1569457, 1909001

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

These are composite numbers n such that b^(n-1) == 1 (mod n) and abs(b^((n-1)/2) mod n) <> 1 for every b coprime to n.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..131 from Daniel Lignon)

Cf. A002997 (Carmichael numbers), A033181 (absolute Euler pseudoprimes).

A265237 Carmichael numbers (A002997) that are the sum of two squares.

Original entry on oeis.org

1105, 2465, 10585, 29341, 46657, 115921, 162401, 252601, 278545, 294409, 314821, 410041, 488881, 530881, 552721, 1461241, 1909001, 2433601, 3224065, 3581761, 4335241, 5148001, 5310721, 5444489, 5632705, 6054985, 6189121, 7207201, 7519441, 8134561, 8355841

Offset: 1

Altug Alkan, Dec 06 2015

nonn

Carmichael numbers that are the sum of two distinct nonzero squares.
29341 is the first term for which neither of the squares can be the square of a prime.
Carmichael numbers that are not the sum of two squares start 561, 1729, 2821, 6601, 8911, 15841, ...
A Carmichael number m is a sum of two squares if and only if p == 1 (mod m) for every prime p|m. Observation, numerically checked by Amiram Eldar: the first 13 terms of this sequence are odd composites m such that m | EulerNumber(m-1) (A122045). - Thomas Ordowski, Mar 01 2020

			1105 is a term because 1105 = 23^2 + 24^2.
2465 is a term because 2465 = 41^2 + 28^2.
10585 is a term because 10585 = 37^2 + 96^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers

Cf. A002997, A004431, A122045.

t = Cases[Range[1, 10^7, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n]; Select[t, SquaresR[2, #] > 0 &] (* Michael De Vlieger, Dec 06 2015, after Artur Jasinski at A002997 *)

is(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1
is_c(n)={my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
for(n=1, 1e7, if(is(n)&&is_c(n), print1(n, ", ")))

A265628 Carmichael numbers (A002997) of the form k^3 + 1.

Original entry on oeis.org

1729, 46657, 2628073, 19683001, 110592000001, 432081216001, 2116874304001, 3176523000001, 312328165704192001, 12062716067698821000001, 211215936967181638848001, 411354705193473163968001, 14295706553536348081491001, 32490089562753934948660824001

Offset: 1

Altug Alkan, Dec 10 2015

For the first nine Carmichael numbers of the form k^3 + 1, the values of k + 1 are 13, 37, 139, 271, 4801, 7561, 12841, 14701, 678481 and only 14701 is not a prime number.

The sequence also includes: 32490089562753934948660824001, 782293837499544845175052968001, 611009032634107957276386802479001. - Daniel Suteu, Dec 25 2020

			2628073 is a term because it is a Carmichael number and 2628073 = 138^3 + 1.

G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A265285, A265328.

is_c(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=1, 1e10, if(is_c(k=n^3+1), print1(k, ", ")))

lista(kmax) = {my(m); for(k = 2, kmax, f = factor(k); for(i = 1, #f~, f[i, 2] *= 3); m = k^3 + 1; fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^3 + 1, ", "))); } \\ Amiram Eldar, May 02 2024

More terms from Alois P. Heinz, Dec 10 2015
a(10)-a(13) from Daniel Suteu, Dec 25 2020
a(14) from Daniel Suteu confirmed by Amiram Eldar, May 02 2024

A272754 Primes p such that p + 2 is a Carmichael number (A002997).

Original entry on oeis.org

1103, 2819, 6599, 29339, 41039, 52631, 62743, 172079, 188459, 278543, 340559, 488879, 656599, 670031, 1033667, 2100899, 3146219, 5048999, 6049679, 8719307, 10024559, 10402559, 10877579, 11119103, 12261059, 14913989, 15247619, 15829631, 15888311, 17315999, 17812079, 18900971, 25603199, 26921087

Offset: 1

Altug Alkan, May 05 2016

nonn

Because of Korselt's criterion, prime p is a member of this sequence if and only if p+2 is composite squarefree and q-1 divides p+1 for every prime q dividing p+2.

			1103 is a term because 1103 is prime and 1105 is a Carmichael number.

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

Cf. A002997, A135437.

Select[Cases[Range[1, 10^7, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]] - 2, PrimeQ] (* Michael De Vlieger, May 05 2016, after Artur Jasinski at A002997 *)

isA002997(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
lista(nn) = forprime(p=2, nn, if(isA002997(p+2), print1(p, ", ")));

A306414 a(n) = psi(A002997(n)), psi = A002322.

Original entry on oeis.org

80, 48, 36, 112, 60, 1320, 198, 504, 360, 180, 120, 288, 1224, 2024, 36, 240, 300, 720, 72, 2320, 60, 108, 600, 112, 216, 1980, 8568, 528, 420, 6120, 2268, 360, 540, 3360, 240, 4900, 2160, 396, 432, 144, 300, 120, 576, 2970, 612, 396, 648, 630, 1260, 1080, 1008

Offset: 1

Jianing Song, Feb 13 2019

nonn

By definition, A002997 lists the composite numbers k such that psi(k) divides k - 1. This sequence lists the values for psi(k) when k runs through A002997.

			A002997(1) = 561, psi(561) = 80, 561 = 7*80 + 1.
A002997(2) = 1105, psi(1105) = 48, 1105 = 23*48 + 1.
A002997(3) = 1729, psi(1729) = 36, 1729 = 48*36 + 1.
A002997(4) = 2465, psi(2465) = 112, 2465 = 22*112 + 1.

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

Cf. A002997, A002322, A174590.

seq[kmax_] := Module[{s = {}, lam}, Do[If[CompositeQ[k], lam = CarmichaelLambda[k]; If[Mod[k, lam] == 1, AppendTo[s, lam]]], {k, 9, kmax, 2}]; s]; seq[10^6] (* Amiram Eldar, Nov 14 2023 *)

forstep(n=3, 1e6, 2, my(m=lcm(znstar(n)[2])); if((n-1)%m==0 && !isprime(n), print1(m, ", ")))

a(n) = A002322(A002997(n)).

a(n) = (A002997(n) - 1) / A174590(n). - Jianing Song, Dec 12 2021

A135719 a(n) is the index of the smallest Carmichael number (A002997) with n prime divisors, or 0 if no such number exists.

Original entry on oeis.org

1, 11, 40, 403, 1224, 4886, 19096, 120137, 485941, 2974628, 25293838

Offset: 3

Artur Jasinski, Nov 25 2007

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Tables relating to Carmichael numbers (All the Carmichael numbers up to 10^18).

Cf. A002997, A135717, A006931.

A002997(a(n)) = A006931(n). - M. F. Hasler, Apr 14 2015

a(8)-a(11) from Donovan Johnson, Feb 23 2012
a(12) from Amiram Eldar, Jul 08 2019
Escape clause added by Jianing Song, Dec 12 2021
a(13) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

A163956 Multiplicative order of 2 in Z/mZ with m = A002997(n).

Original entry on oeis.org

40, 24, 36, 56, 60, 660, 198, 252, 45, 180, 60, 144, 153, 1012, 36, 120, 300, 72, 36, 1160, 60, 36, 300, 56, 36, 660, 4284, 264, 420, 3060, 2268, 180, 540, 1680, 120, 4900, 1080, 396, 72, 72, 60, 60, 144, 2970, 612, 396, 324, 210, 180, 540, 504, 792, 198, 180

Offset: 1

A.K. Devaraj, Aug 28 2009

nonn

Related sequence: A162990. - A.K. Devaraj, Aug 31 2009

A. K. Devaraj, "Minimum universal exponent generalisation of Fermat's theorem" (ISSN 1550-3747)

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

Cf. A002326, A002997, A007733, A104016, A162990, A165139.

MultiplicativeOrder[2, #] & /@ Select[Range[1, 10^6, 2], CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jul 30 2020 *)

a(n) = A002326((A002997(n)-1)/2) = A007733(A002997(n)). - Amiram Eldar, Jul 30 2020

Corrected and extended by M. F. Hasler, Sep 23 2009
Edited by N. J. A. Sloane, Sep 23 2009, following suggestions from M. F. Hasler
More terms from Amiram Eldar, Jul 30 2020

cp's OEIS Frontend

A153515 Terms of A122782 which are not Carmichael numbers A002997.

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A153580 Terms of A083737 which are not Carmichael numbers (A002997).

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A153581 Pseudoprimes to bases 2,3,5 and 7 which are not Carmichael numbers (A002997).

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A262043 Carmichael numbers (A002997) that are not absolute Euler pseudoprimes (A033181).

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A265237 Carmichael numbers (A002997) that are the sum of two squares.

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A265628 Carmichael numbers (A002997) of the form k^3 + 1.

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A272754 Primes p such that p + 2 is a Carmichael number (A002997).

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A306414 a(n) = psi(A002997(n)), psi = A002322.

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