A055553 -id:A055553 - cp's OEIS frontend

A306338 Carmichael numbers k such that phi(k) divides (k-1)*lambda(k).

561, 1105, 1729, 2465, 6601, 15841, 41041, 46657, 52633, 75361, 115921, 334153, 340561, 658801, 670033, 2455921, 2704801, 4903921, 5049001, 6049681, 6840001, 8355841, 9439201, 9582145, 9613297, 10402561, 11119105, 11205601, 11972017, 14469841, 15888313, 16778881

Offset: 1

Amiram Eldar and Thomas Ordowski, Feb 08 2019

nonn

Carmichael numbers k such that A034380(k) divides k-1.
A proper subset of Carmichael numbers in A173703.
The number of terms below 10^k for k=1,2,...,18 is 0, 0, 1, 5, 10, 15, 25, 56, 101, 184, 310, 508, 814, 1265, 1964, 2990, 4486, 6704. Cf. A055553.
Composite numbers k such that lcm(lambda(k),phi(k)/lambda(k)) divides k-1.
Problem: are there infinitely many such numbers?

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Carmichael numbers.

Cf. A000010, A002322, A002997, A034380, A173703.

Select[Range[3, 100000, 2], !PrimeQ[#] && Divisible[#-1, c = CarmichaelLambda[#]] && Divisible[c*(#-1), EulerPhi[#]] &]

A308086 Carmichael numbers c such that c-4, c-2 and c+2 are primes.

Original entry on oeis.org

656601, 11512252145095521, 35151891169379601, 89283676825965441, 209606994019068801, 584047819872236721, 627126355430628801, 1107574117930742001, 1152431453119654401, 2990125943388676401, 6919232969930803761

Offset: 1

Rick L. Shepherd, May 11 2019

nonn

Subsequence of A287591 (Carmichael numbers that are arithmetic means of cousin primes). Calculated from Amiram Eldar's table in that sequence. The Carmichael numbers here are contained within intervals defined by prime triples of the form (p, p+2, p+6); therefore, for each term, four consecutive odd numbers are prime, prime, Carmichael number (divisible by 3), then prime. None of the terms of A287591 available so far are contained within intervals defined by prime triplets of the form (p, p+4, p+6). Is that possible? If so, is it also possible for a Carmichael number to be immediately preceded and succeeded by twin primes, i.e., to be "contained" in a prime quadruplet? (Such Carmichael numbers would necessarily be multiples of 15.)

			656601 = 3*11*101*197 is a term because 656597 and 656599 are twin primes, 656601 is a Carmichael number, and 656603 is also a prime.

Amiram Eldar, Table of n, a(n) for n = 1..36 (terms below 10^22, calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Cf. A002997, A007530, A022004, A022005, A230715, A258801, A287591.

More terms from Amiram Eldar, Jul 02 2019

A309003 Carmichael numbers divisible by the sum of their prime factors, sopfr (A001414).

Original entry on oeis.org

3240392401, 13577445505, 14446721521, 84127131361, 203340265921, 241420757761, 334797586201, 381334973041, 461912170321, 1838314142785, 3636869821201, 10285271821441, 17624045440981, 18773053896961, 20137015596061, 24811804945201, 26863480687681, 35598629998801

Offset: 1

David James Sycamore, Jul 05 2019

nonn

Intersection of A002997 and A308643.

Intersection of A002997 and A036844.

			3240392401 = 29*37*41*73*1009, A001414(3240392401)=1189 = 29*41.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A001414, A036844, A046347, A308643.

sopfr(f) = f[, 1]~*f[, 2];
isCarmichael(n, f)= bittest(n, 0) && !for(i=1, #f~, (f[i, 2]==1 && n%(f[i, 1]-1)==1)||return) && (#f~>1);
isok(n) = my(f=factor(n)); isCarmichael(n, f) && !(n % sopfr(f)); \\ Michel Marcus, Jul 07 2019

A327787 a(n) is the smallest Carmichael number k such that gpf(p-1) = prime(n) for all prime factors p of k.

Original entry on oeis.org

1729, 252601, 1152271, 1615681, 4335241, 172947529, 214852609, 79624621, 178837201, 775368901, 686059921, 985052881, 5781222721, 10277275681, 84350561, 5255104513, 492559141, 74340674101, 9293756581, 1200778753, 129971289169, 2230305949, 851703301, 8714965001, 6693621481

Offset: 2

Daniel Suteu, Sep 25 2019

nonn

The first term is the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2019

			a(2) = 1729 = (2*3 + 1)(2*2*3 + 1)(2*3*3 + 1).
a(3) = 252601 = (2*2*2*5 + 1)(2*2*3*5 + 1)(2*2*5*5 + 1).
a(4) = 1152271 = (2*3*7 + 1)(2*3*3*7 + 1)(2*3*5*7 + 1).
a(5) = 1615681 = (2*11 + 1)(2*3*3*11 + 1)(2*2*2*2*2*11 + 1).

Amiram Eldar, Table of n, a(n) for n = 2..2905 (calculated using data from Claude Goutier; terms 2..831 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Daniel Suteu, Terms and upper bounds for n = 2..10000 (values greater than 2^64 are upper bounds).
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Cf. A002997 (Carmichael numbers), A006530 (gpf), A001235.

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; gpf[n_] := FactorInteger[n][[-1, 1]]; g[n_] := If[Length[(u = Union[gpf /@ (FactorInteger[n][[;; , 1]] - 1)])] == 1, u[[1]], 1]; m = 5; c = 0; k = 0; v = Table[0, {m}]; While[c < m, k++ If[! carmQ[k], Continue[]]; If[(p = g[k]) > 1, i = PrimePi[p] - 1; If[i <= m && v[[i]] == 0, c++; v[[i]] = k]]]; v (* Amiram Eldar, Oct 08 2019 *)

use ntheory ":all"; sub a { my $p = nth_prime(shift); for(my $k = 1; ; ++$k) { return $k if (is_carmichael($k) and vecall { (factor($_-1))[-1] == $p } factor($k)) } }
for my $n (2..10) { print "a($n) = ", a($n), "\n" }

A328939 Carmichael numbers that are products of primes p for which each p-1 is squarefree.

Original entry on oeis.org

10267951, 72108421, 111291181, 139952671, 1588247851, 6004532941, 7200256261, 8815102297, 9001235881, 10884042841, 15989367241, 18500666251, 23729234761, 34268731321, 34558584607, 37870128451, 74689102411, 77538554731, 121254376891, 149842746691, 187054437571

Offset: 1

Amiram Eldar, Oct 31 2019

nonn

Shanks noted that among the first 300 Carmichael numbers only 3 are in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Paul Erdős, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), pp. 201-206.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.
Daniel Shanks, Solved and Unsolved Problems in Number Theory, 2nd ed., Chelsea Pub. Co., New York, 1978, p. 229.
Index entries for sequences related to Carmichael numbers.

Cf. A002997.

aQ[n_] := CompositeQ[n] && Divisible[n-1, CarmichaelLambda[n]] && AllTrue[FactorInteger[n][[;; , 1]] - 1, SquareFreeQ]; Select[Range[10^8], aQ]

A335336 Carmichael numbers k such that k+1 is divisible by gpf(k)+1, where gpf = A006530.

Original entry on oeis.org

687979968481, 1928376089641, 2638625591701, 3148470889201, 3152088903601, 14682521533681, 19656816822721, 37333372057201, 47003559452641, 80643055074121, 129235662445121, 140940741166849, 196945133626801, 336301807660741, 345186571310209, 439931062854361

Offset: 1

Daniel Suteu, Jun 04 2020

nonn

Are there any Carmichael numbers k with exactly four prime factors such that k+1 is divisible by gpf(k)+1?

Richard J. McIntosh and Mitra Dipra found the following base 2 Fermat pseudoprimes with exactly four prime factors satisfying s-1 | k-1 and s+1 | k+1, where s is the largest prime factor of k: 988679226253951, 3143193486942417481, 44307784380481317090001.

			For k = 687979968481 = 13 * 29 * 71 * 181 * 211 * 673, which is a Carmichael number, we have gpf(k) = 673. Thereafter we have gpf(k)+1 = 2 * 337 and k+1 = 2 * 337 * 347 * 911 * 3229, satisfying gpf(k)+1 | k+1.

Amiram Eldar, Table of n, a(n) for n = 1..3150 (terms below 10^22, calculated using data from Claude Goutier; terms 1..459 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard J. McIntosh and Mitra Dipra, Carmichael numbers with p+1|n+1, Journal of Number Theory, Volume 147, February 2015, Pages 81-91.
Wikipedia, Carmichael number.
Wikipedia, Lucas-Carmichael number.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A006530, A006972, A329948.

A352987 Carmichael numbers (A002997) that are overpseudoprimes to base 2 (A141232).

Original entry on oeis.org

65700513721, 168003672409, 459814831561, 13685652197857, 34477679139751, 74031531351121, 92327722290241, 206175669172201, 704077371354601, 1882982959757929, 2901482064497017, 3715607011189609, 5516564718607489, 5636724028491697, 6137426373439681, 14987802403246609

Offset: 1

Daniel Suteu, May 06 2022

nonn

If we define f(n) to be the smallest number in the sequence with n prime factors, then we have:
f(3) = 65700513721,
f(4) <= 84286331493236478328609,
f(5) <= 3848515708338676403444146123852434164444641.

Amiram Eldar, Table of n, a(n) for n = 1..450 (calculated using data from Claude Goutier; terms 1..102 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A141232.

Intersection of A291637 and A141232.

A232167 Number of composite integers k less than 10^n such that lambda(k) divides 2k-2, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

3, 9, 16, 31, 68, 149, 314, 724, 1670, 4063

Offset: 1

José María Grau Ribas, Mar 02 2014

Conjecture: A055553(n)/a(n) has a limit strictly smaller than 1 as n tends to infinity.

J. M. Grau and A. Oller-Marcén, On the congruence sum_{j=1}^{n-1} j^{k(n-1)} == -1 (mod n). k-strong Giuga and k-Carmichael numbers, arXiv:1311.3522 [math.NT], 2013.

Cf. A002322, A055553.

For[k = 4; cnt = 0, True, k++, If[CompositeQ[k] && Divisible[2k-2, CarmichaelLambda[k]], cnt++]; If[IntegerQ[n = Log[10, k+1]], Print[n, " ", cnt]]]; (* Jean-François Alcover, Feb 16 2019 *)

a(8)-a(10) from Giovanni Resta, Mar 03 2014

A263930 Number of quasi-Carmichael numbers less than 10^n.

Original entry on oeis.org

0, 2, 27, 165, 734, 3109, 11568, 40820, 137850, 457191

Offset: 1

Tim Johannes Ohrtmann, Oct 30 2015

For quasi-Carmichael numbers see A257750.

			a(1) = 0 because there are no quasi-Carmichael numbers below 10^1.
a(2) = 2 because there are two quasi-Carmichael numbers below 10^2, namely, 35 and 77.

Cf. A055553, A216929, A257750.

use ntheory ":all"; my($s,$e)=(0,1); forcomposites { say $e++," $s" if $ >= 10**$e; $s++ if is_quasi_carmichael($) } 1e7; # Dana Jacobsen, Apr 27 2017

a(8)-a(10) from Dana Jacobsen, Apr 27 2017

A328936 The number of imprimitive Carmichael numbers (A328935) below 10^n.

Original entry on oeis.org

4, 11, 25, 63, 134, 268, 508, 1013, 1901, 3773, 7208, 13834, 26353, 50343, 96122, 184354, 354218

Offset: 6

Amiram Eldar, Oct 31 2019

Granville and Pomerance conjectured that most Carmichael numbers are primitive, i.e. lim_{n->oo} a(n)/A055553(n) = 0.

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

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A055553, A328935.

a(20)-a(22) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 22 2024

cp's OEIS Frontend

A306338 Carmichael numbers k such that phi(k) divides (k-1)*lambda(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A308086 Carmichael numbers c such that c-4, c-2 and c+2 are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A309003 Carmichael numbers divisible by the sum of their prime factors, sopfr (A001414).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A327787 a(n) is the smallest Carmichael number k such that gpf(p-1) = prime(n) for all prime factors p of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Perl

A328939 Carmichael numbers that are products of primes p for which each p-1 is squarefree.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A335336 Carmichael numbers k such that k+1 is divisible by gpf(k)+1, where gpf = A006530.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A352987 Carmichael numbers (A002997) that are overpseudoprimes to base 2 (A141232).

Views

Author

Keywords

Comments

Links

Crossrefs

A232167 Number of composite integers k less than 10^n such that lambda(k) divides 2k-2, where lambda is the Carmichael lambda function (A002322).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A263930 Number of quasi-Carmichael numbers less than 10^n.

Views

Author

Keywords

Comments

Examples