A087788 -id:A087788 - cp's OEIS frontend

A064257 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,41.

306, 7686, 9900, 24168, 32778, 68448, 86160, 107070, 112236, 148398, 172998, 207930, 217770, 221706, 231546, 250488, 277548, 292800, 303378, 306576, 329208, 354300, 380130, 398580, 488616, 490338, 492798, 501408, 533388, 559218, 567828, 605220, 619980, 640890

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

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

Cf. A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 41}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

Offset corrected and more terms added by Amiram Eldar, Oct 17 2019

A064258 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,43.

Original entry on oeis.org

966, 12576, 36570, 37860, 71916, 78366, 103650, 126096, 132546, 242970, 279606, 316500, 387966, 392610, 404220, 407316, 418926, 459690, 498390, 500196, 523416, 524706, 554376, 604170, 707886, 729300, 749940, 755616, 769806, 840756, 897516, 903450, 974400, 1048446

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

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

Cf. A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 43}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

Offset corrected and more terms added by Amiram Eldar, Oct 17 2019

A064259 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,45.

Original entry on oeis.org

336, 606, 1236, 3036, 7536, 9066, 12576, 17256, 18786, 19416, 22026, 27966, 28596, 30576, 33636, 35616, 43986, 47136, 48486, 49476, 52806, 53526, 59106, 60726, 63246, 71706, 80526, 83136, 86286, 89976, 96096, 97986, 98886, 103836, 105096, 116256, 118686, 119046

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

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

Cf. A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 45}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

Offset corrected and more terms added by Amiram Eldar, Oct 17 2019

A064260 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,47.

Original entry on oeis.org

210, 2466, 4158, 5850, 14028, 116958, 156156, 166026, 176178, 188868, 208608, 225528, 232860, 241320, 252036, 284748, 290106, 290670, 345378, 350736, 399240, 439566, 444078, 448308, 450000, 498786, 582540, 669678, 708030, 722976, 746100, 809268, 813216, 860028

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

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

Cf. A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 47}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

Offset corrected and more terms added by Amiram Eldar, Oct 17 2019

A064261 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,49.

Original entry on oeis.org

660, 2718, 12420, 57990, 82980, 89448, 127080, 142368, 174120, 184998, 202638, 216750, 229980, 233508, 234978, 280548, 297600, 329940, 331998, 341700, 384918, 406380, 412848, 421080, 433428, 455478, 503988, 505458, 533388, 578370, 608358, 675978, 768588, 779760

Offset: 1

N. J. A. Sloane, Sep 23 2001

nonn

am+1, bm+1, cm+1 are primes and am | (N-1), bm | (N-1), cm |(N-1).

Harvey Dubner (harvey(AT)dubner.com), personal communication, Jun 27 2001.

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

Cf. A087788.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; Select[Range[10^5], AllTrue[(v = {1, 2, 49}*# + 1), PrimeQ] && carmQ[Times @@ v] &] (* Amiram Eldar, Oct 17 2019 *)

Offset corrected and more terms added by Amiram Eldar, Oct 17 2019

A328937 The number of imprimitive 3-Carmichael numbers (A087788 and A328935) below 10^n.

Original entry on oeis.org

4, 11, 25, 59, 127, 252, 471, 928, 1734, 3462, 6615, 12725, 24396, 46877, 89854, 173331, 334737, 647265, 1253176

Offset: 6

Amiram Eldar, Oct 31 2019

Granville and Pomerance conjectured that most Carmichael numbers are imprimitive, i.e. lim_{n->oo} a(n)/A132195(n) = 1.

			a(6) = 4 since there are 4 imprimitive 3-Carmichael numbers below 10^6: 294409, 399001, 488881, 512461.

J. M. Chick, Carmichael number variable relations: three-prime Carmichael numbers up to 10^24, arXiv preprint arXiv:0711.2915 [math.NT] (2007).
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Cf. A002997, A087788, A132195, A328935.

A006931 Least Carmichael number with n prime factors, or 0 if no such number exists.

Original entry on oeis.org

561, 41041, 825265, 321197185, 5394826801, 232250619601, 9746347772161, 1436697831295441, 60977817398996785, 7156857700403137441, 1791562810662585767521, 87674969936234821377601, 6553130926752006031481761, 1590231231043178376951698401

Offset: 3

N. J. A. Sloane and Richard Pinch

nonn

Alford, Grantham, Hayman, & Shallue construct large Carmichael numbers, finding upper bounds for a(3)-a(19565220) and a(10333229505). - Charles R Greathouse IV, May 30 2013

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 269, Pour la Science, Paris 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 3..35
W. R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue, Constructing Carmichael numbers through improved subset-product algorithms, arXiv:1203.6664 [math.NT], 2012-2013.
R. G. E. Pinch, The Carmichael numbers up to 10^15, Math. Comp. 61 (1993), no. 203, 381-391.
R. G. E. Pinch, The Carmichael numbers up to 10^17, arXiv:math/0504119 [math.NT], 2005.
R. G. E. Pinch, The Carmichael numbers up to 10^18, arXiv:math/0604376 [math.NT], 2006.
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A135717, A135719, A135720, A135721.

Cf. A087788, A141711, A074379, A112428, A112429, A112430, A112431, A112432.

(* Program not suitable to compute more than a few terms *)
A2997 = Select[Range[1, 10^6, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#] ] == 1&];
(First /@ Split[Sort[{PrimeOmega[#], #}& /@ A2997], #1[[1]] == #2[[1]]&])[[All, 2]] (* Jean-François Alcover, Sep 11 2018 *)

Korselt(n,f=factor(n))=for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1
a(n)=my(p=2,f);forprime(q=3,default(primelimit),forstep(k=p+2,q-2,2,f=factor(k);if(vecmax(f[,2])==1 && #f[,2]==n && Korselt(k,f), return(k)));p=q)
\\ Charles R Greathouse IV, Apr 25 2012

carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(1/Mod(m,l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n-1)%(p-1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p-1) == 1, list=concat(list, f(m*p, lcm(l, p-1), p+1, k-1))))); list); vecsort(Vec(f(1, 1, 3, k)));
a(n) = if(n < 3, return()); my(x=vecprod(primes(n+1))\2,y=2*x); while(1, my(v=carmichael(x,y,n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 24 2023

Corrected by Lekraj Beedassy, Dec 31 2002
More terms from Ralf Stephan, from the Pinch paper, Apr 16 2005
Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar.
Escape clause added by Jianing Song, Dec 12 2021

A033502 Carmichael numbers of the form (6k+1)(12k+1)(18k+1), where 6k+1, 12k+1 and 18k+1 are all primes.

Original entry on oeis.org

1729, 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, 11346205609, 13079177569, 21515221081, 27278026129, 65700513721, 71171308081, 100264053529, 168003672409, 172018713961, 173032371289, 464052305161

Offset: 1

Marc LeBrun

nonn

Also called Chernick's Carmichael numbers. The polynomial (6*k+1)*(12*k+1)*(18*k+1) is the simplest Chernick polynomial. [Named after the American physicist and mathematician Jack Chernick (1911-1971). - Amiram Eldar, Jun 15 2021]
The first term, 1729, is the Hardy-Ramanujan number and the smallest primary Carmichael number (A324316).
Dickson's conjecture implies that this sequence is infinite, as pointed out by Chernick.
All terms of this sequence are primary Carmichael numbers (A324316) having the following remarkable property. Let m be a term of A033502. For each prime divisor p of m, the sum of the base-p digits of m equals p. This property also holds for "almost all" 3-term Carmichael numbers (A087788), since they can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions, see Kellner 2019. - Bernd C. Kellner, Aug 03 2022

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A13, pp. 50-53.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269-274.
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Douglas E. Iannucci, When the small divisors of a natural number are in arithmetic progression, INTEGERS, Electronic Journal of Combinatorial Number Theory, Vol. 18 (2018), #77. See p. 9.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens, Vol. 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Values of k are given by A046025. Subsequence of A002997, A087788, and A324316.

Cf. A242980, A242981.

[n : k in [1..710] | IsPrime(a) and IsPrime(b) and IsPrime(c) and IsOne(n mod CarmichaelLambda(n)) where n is a*b*c where a is 6*k+1 where b is 12*k+1 where c is 18*k+1]; // Arkadiusz Wesolowski, Oct 29 2013

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; (6# + 1)(12# + 1)(18# + 1) & /@
Select[ Range@ 1000, PrimeQ[6# + 1] && PrimeQ[12# + 1] && PrimeQ[18# + 1] && CarmichaelNbrQ[(6# + 1)(12# + 1)(18# + 1)] &]

Definition corrected (thanks to Umberto Cerruti) by Bruno Berselli, Jan 18 2013

A112428 Carmichael numbers equal to the product of 5 primes.

Original entry on oeis.org

825265, 1050985, 9890881, 10877581, 12945745, 13992265, 16778881, 18162001, 27336673, 28787185, 31146661, 36121345, 37167361, 40280065, 41298985, 41341321, 41471521, 47006785, 67371265, 67994641, 69331969, 74165065

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

A subsequence is given by (6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) with n in A206349. - M. F. Hasler, Apr 14 2015

			a(1)=825265=5*7*17*19*73

R. J. Mathar, Table of n, a(n) for n = 1..14938

Cf. A002997, A014614, A206349.

Cf. A087788, A141711, A074379, A112429 - A112432, A006931.

is_A112428(n)=bigomega(n)==5&&is_A002997(n) \\ M. F. Hasler, Apr 14 2015

A112428 = A002997 intersect A014614. - M. F. Hasler, Apr 14 2015

Crossrefs added by M. F. Hasler, Apr 14 2015

A083876 Least pseudoprime to base 2 through base prime(n).

Original entry on oeis.org

341, 1105, 1729, 29341, 29341, 162401, 252601, 252601, 252601, 252601, 252601, 252601, 1152271, 2508013, 2508013, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 6733693, 6733693, 6733693

Offset: 1

Robert G. Wilson v, May 06 2003

nonn

Records: 341, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, .... - Robert G. Wilson v, May 11 2012
Conjecture: for n > 1, a(n) is the smallest Carmichael number k with lpf(k) > prime(n). It seems that such Carmichael numbers have exactly three prime factors. - Thomas Ordowski, Apr 18 2017
The conjecture is true if a(n) < A285549(n) for all n > 1. It holds for all a(n) < 2^64. - Max Alekseyev and Thomas Ordowski, Mar 13 2018
If prime(n) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= prime(n). - Thomas Ordowski, Mar 05 2018
By this conjecture in the second comment, a(n) <= A135720(n+1), with equality for n > 1 iff a(n) < a(n+1), namely for n = 2, 3, 5, 6, 12, 13, 15, 25, 28, 29, ... For such n, a(n) gives all terms of A300629 > 561. - Thomas Ordowski, Mar 10 2018

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert G. Wilson v)

Cf. A002997, A001567, A052155, A083737, A087788, A083739, A135720, A141768, A250199, A271221, A285549, A300629.

k = 4; Do[l = Table[ Prime[i], {i, 1, n}]; While[ PrimeQ[k] || Union[PowerMod[l, k - 1, k]] != {1}, k++ ]; Print[k], {n, 1, 29}]

isps(k, n) = {if (isprime(k), return (0)); my(nbok = 0); for (b=2, prime(n), if (Mod(b, k)^(k-1) == 1, nbok++, break)); if (nbok==prime(n)-1, return (1));}
a(n) = {my(k=2); while (!isps(k, n), k++); return (k);} \\ Michel Marcus, Apr 27 2018

cp's OEIS Frontend

A064257 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,41.

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A064258 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,43.

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A064259 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,45.

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A064260 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,47.

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A064261 Values of m such that N=(am+1)(bm+1)(cm+1) is a 3-Carmichael number (A087788), where a,b,c = 1,2,49.

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A328937 The number of imprimitive 3-Carmichael numbers (A087788 and A328935) below 10^n.

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A006931 Least Carmichael number with n prime factors, or 0 if no such number exists.

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A033502 Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1), where 6*k+1, 12*k+1 and 18*k+1 are all primes.

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A033502 Carmichael numbers of the form (6k+1)(12k+1)(18k+1), where 6k+1, 12k+1 and 18k+1 are all primes.