A006972 -id:A006972 - cp's OEIS frontend

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

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.

A292353 Numbers n with a record number of Lucas-Carmichael numbers that can be generated from them using an adjusted version of Erdős's method.

Original entry on oeis.org

24, 72, 216, 240, 360, 720, 1440, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160

Offset: 1

Amiram Eldar, Sep 14 2017

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate Lucas-Carmichael numbers. Given a number n, let P be the set of primes p such that (p+1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == -1 (mod n) then c is a Lucas-Carmichael number.

The corresponding number of generated Lucas-Carmichael numbers are 1, 3, 5, 9, 21, 169, 681, 900, 1842, 7250, 29132, 77482, 932187, 4970111, 7456418.

			The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == -1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates Lucas-Carmichael numbers thus a(1)=24.

Cf. A006972, A287861, A292352.

a = {}; cmax = 0; Do[p = Select[Divisors[n] - 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c++], {j, 1, Length[ps]}]; If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

A292354 Numbers n with a record size of the largest Lucas-Carmichael number that can be generated from them using an adjusted version of Erdős's method.

Original entry on oeis.org

24, 48, 60, 144, 168, 240, 360, 720, 1440, 2520, 4320, 5040, 7560, 10080, 15120, 20160

Offset: 1

Amiram Eldar, Sep 14 2017

Erdős showed in 1956 how to construct Carmichael numbers from a given number n (see A287840). With appropriate sign changes the method can be used to generate Lucas-Carmichael numbers. Given a number n, let P be the set of primes p such that (p+1)|n but p is not a factor of n. Let c be a product of a subset of P with at least 3 elements. If c == -1 (mod n) then c is a Lucas-Carmichael number.

The corresponding largest Lucas-Carmichael numbers are 8855, 18095, 357599, 1010735, 406335215, 1087044101759, 4467427448759, ...

			The set of primes for n = 24 is P={2, 3, 5, 7, 11, 23}. One subset, {5, 7, 11, 23} have c == -1 (mod n): c = 5*7*11*23 = 8855. 24 is the least number that generates Lucas-Carmichael numbers thus a(1)=24.

Cf. A006972, A287862, A292352.

a = {}; cmax = 0; Do[p = Select[Divisors[n] - 1, PrimeQ]; pr = Times @@ p; pr = pr/GCD[n, pr]; ps = Divisors[pr]; c = 0; Do[p1 = FactorInteger[ps[[j]]][[;; , 1]]; If[Length[p1] < 3, Continue[]]; c1 = Times @@ p1; If[Mod[c1, n] == 1, c = Max[c, c1]], {j, 1, Length[ps]}]; If[c > cmax, cmax = c; AppendTo[a, n]], {n, 1, 1000}]; a

A292368 Numbers n with record number of primes p such that n*p is a Lucas-Carmichael number.

Original entry on oeis.org

1, 21, 55, 385, 49105, 136081, 701569, 2830465, 7996801, 29158921, 49268737, 52617601

Offset: 1

Amiram Eldar, Sep 15 2017

Given a number n and a prime number p such that n*p is a Lucas-Carmichael number, then (p+1)|(n-1), so the number of prime solution p given n is bounded by the number of divisors of (n-1).

The number of solutions is 0, 1, 2, 4, 5, 6, 7, 8, 10, 13, 15, 32.

			21 has one prime number, 19, such that 21*19 = 399 is a Lucas-Carmichael number. 55 has 2 prime numbers, 17 and 53, such that 55*17 = 935 and 55*53 = 2915 are Lucas-Carmichael numbers.

Cf. A006972.

lucasCarmichaelQ[n_]:=!PrimeQ[n] && Union[Transpose[FactorInteger[n]][[2]]] == {1} && Union[Mod[n + 1, Transpose[FactorInteger[n]][[1]]+1]]=={0};
numSol[n_]:=Module[{m = 0}, ds = Divisors[n-1]; Do[p = ds[[k]]-1; If[!PrimeQ[p], Continue[]]; If[! lucasCarmichaelQ[p*n], Continue[]]; m++, {k, 1, Length[ds]}]; m]; numSolmax = -1; seq = {}; nums = {};
Do[m = numSol[n]; If[m > numSolmax, AppendTo[seq, n]; AppendTo[nums, m]; Print[{n, m}]; numSolmax = m], {n, 1, 100000}]; seq

A299213 Lucas-Carmichael numbers whose prime factors do not divide any smaller Lucas-Carmichael number.

Original entry on oeis.org

399, 935, 565861139, 5778659039, 22824172799, 49569379679, 221511111527, 572531110799, 745012846679

Offset: 1

Tim Johannes Ohrtmann, Feb 05 2018

Also numbers whose number of occurrence in A253597 equals the number of their prime factors.

All known terms have only 3 prime factors. Does any term with more than 3 prime factors exist?

			565861139 = 193*1163*2521 and no smaller Lucas-Carmichael number is divisible by 193, 1163 or 2521.

Cf. A006972, A202562, A253597.

a=readvec("b006972.txt"); print1(399); for(b=2,10000, e=true; f=factor(a[b]); for(d=1,#f[, 1], for(c=1,b-1, if(a[c]%f[d,1]==0, e=false))); if(e==true, print1(", ",a[b])))

A329223 Poulet numbers (Fermat pseudoprimes to base 2) that are congruent to either 3 or 27 (mod 80) and each prime factor is congruent to 3 mod 80.

Original entry on oeis.org

51962615262396907, 330468624532072027, 2255490055253468347, 18436227497407654507

Offset: 1

Daniel Suteu, Nov 08 2019

If a term of this sequence is also a Carmichael number (A002997) and a Lucas-Carmichael number (A006972), then it would be a counterexample to Agrawal's conjecture, as Hendrick Lenstra and Carl Pomerance showed.
330468624532072027 is the only Carmichael number below 2^64 that is a term of this sequence. However, it is not a Lucas-Carmichael number.
The sequence also includes: 68435117188079800987, 164853581396047908970027, 522925572082528736632187, 1820034970687975620484907, 4263739170243679206753787, 4360728281510798266333387, 28541906071781213329174507, 33833150661360980271172507, 84444874320158644422192427, 175352076630428496579381067, 270136290676063386556053067, 615437738523352001584590187, 3408560627000081376639770587, 11260257876970792445537580187.
No term with 5 prime factors (which would be congruent to 3 mod 80) is known to the author.
Are all terms also strong pseudoprimes to base 2 (A001262)?

			51962615262396907 is a term because it is a Fermat pseudoprime to base 2 and it is congruent to 27 (mod 80) and all of its prime factors (643, 154723, 522306163) are congruent to 3 mod 80.

H. W. Lenstra, and Carl Pomerance, Remarks on Agrawal's conjecture, American Institute of Mathematics (2003), pp. 30-32.
Tomáš Váňa, Agrawal's Conjecture and Carmichael Numbers, student scientific conference, pp. 13-22.
Wikipedia, Agrawal's conjecture

Cf. A001567.

isok(n) = ((n%80==3) || (n%80==27)) && (Mod(2, n)^(n-1) == 1) || return(0); my(f=factor(n)[,1]); (#f > 1) && (#select(p->p%80==3, f) == #f);

A356821 Lucas-Carmichael numbers k that have an abundancy index sigma(k)/k that is larger than the abundancy indices of all smaller Lucas-Carmichael numbers.

Original entry on oeis.org

399, 6304359999, 408598269695, 517270926095, 20203946790335

Offset: 1

Amiram Eldar and Daniel Suteu, Aug 29 2022

The rounded values of sigma(k)/k are 1.604, 1.612, 1.666, 1.706, 1.752, ...

The sequence includes the smallest abundant Lucas-Carmichael number, which is <= 1012591408428327888883952080728349448745451794025524955777432246705535.

Cf. A000203, A004394, A006972.

Similar sequences: A328691, A329460.

lc = Import["https://oeis.org/A006972/b006972.txt", "Table"][[;; , 2]]; rm = 0; s = {}; Do[n = lc[[k]]; r = DivisorSigma[-1, n]; If[r > rm, AppendTo[s, n]; rm = r], {k, 1, Length[lc]}]; s

a(5) from Martin Ehrenstein, Jul 30 2023

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

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A292353 Numbers n with a record number of Lucas-Carmichael numbers that can be generated from them using an adjusted version of Erdős's method.

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A292354 Numbers n with a record size of the largest Lucas-Carmichael number that can be generated from them using an adjusted version of Erdős's method.

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A292368 Numbers n with record number of primes p such that n*p is a Lucas-Carmichael number.

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A299213 Lucas-Carmichael numbers whose prime factors do not divide any smaller Lucas-Carmichael number.

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PARI

A329223 Poulet numbers (Fermat pseudoprimes to base 2) that are congruent to either 3 or 27 (mod 80) and each prime factor is congruent to 3 mod 80.

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A356821 Lucas-Carmichael numbers k that have an abundancy index sigma(k)/k that is larger than the abundancy indices of all smaller Lucas-Carmichael numbers.

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