A002997 -id:A002997 - cp's OEIS frontend

A264024 a(n) = gcd(phi(k), k-1) / lambda(k), where k is n-th Carmichael number A002997(n) and lambda(k) = A002322(k).

1, 1, 12, 2, 1, 1, 9, 1, 4, 1, 6, 18, 1, 1, 1, 2, 1, 1, 1, 2, 12, 1, 1, 1, 1, 3, 3, 3, 50, 1, 18, 2, 1, 2, 1, 2, 5, 36, 1, 1, 2, 3, 4, 3, 3, 2, 3, 1, 1, 3, 3, 2, 4, 2, 5, 1, 4, 4, 4, 1, 1, 3, 40, 28, 1, 2, 4, 2, 4, 1, 2, 1, 2, 1, 33, 5, 50, 64, 1, 1, 3, 2, 1, 1, 12, 3, 1, 12, 1, 1, 1, 24, 1, 3, 128, 1, 6, 8, 5, 20, 3, 2, 2, 6, 4

Offset: 1

Thomas Ordowski, Nov 01 2015

nonn

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

Cf. A002322, A002997, A049559, A174590, A264012.

t = Cases[Range[1, 16 (10^6), 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n]; Table[GCD[EulerPhi@ t[[n]], t[[n]] - 1]/CarmichaelLambda@ t[[n]], {n, 105}] (* Michael De Vlieger, Nov 03 2015, after Artur Jasinski at A002997: alternatively use A002997 data for t *)

t(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1;
is(n)=n%2 && !isprime(n) && t(n) && n>1;
c(n)=gcd(eulerphi(n),n-1)/lcm(znstar(n)[2]);
for(n=1, 1e7, if(is(n), print1(c(n)", "))) \\ Altug Alkan, Nov 01 2015

a(n) = A049559(k)/A002322(k), where k = A002997(n).

More terms from Altug Alkan, Nov 01 2015

A265285 Carmichael numbers (A002997) k such that k-1 is a square.

Original entry on oeis.org

46657, 2433601, 67371265, 351596817937, 422240040001, 18677955240001, 458631349862401, 286245437364810001, 20717489165917230086401

Offset: 1

Altug Alkan, Dec 06 2015

This sequence contains all Carmichael numbers n such that for all primes p dividing n, p-1 divides n-1 and furthermore, n-1 is a square.

Numbers sqrt(a(n)-1) form a subsequence of A135590. - Max Alekseyev, Apr 25 2024

			46657 is a term because 46657 - 1 = 46656 = 216^2.
2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.

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.

Subsequence of A265237 and of A265328.

Cf. A002997, A135590, A265237, A303791.

isA002997:= proc(n) local F,p;
         if n::even or isprime(n)  then return false fi;
         F:= ifactors(n)[2];
         if max(seq(f[2],f=F)) > 1 then return false fi;
         andmap(f -> (n-1) mod (f[1]-1) = 0,  F)
end proc:
select(isA002997, [seq(4*i^2+1,i=1..10^6)]); # Robert Israel, Dec 08 2015

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(n) && issquare(n-1), print1(n, ", ")))

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

a(4)-a(5), using A002997 b-file, from Michel Marcus, Dec 07 2015
a(6) and a(7) from Robert Israel, Dec 08 2015
a(8) from Max Alekseyev, Apr 30 2018
a(9) from Daniel Suteu confirmed by Max Alekseyev, Apr 25 2024

A365022 The lesser of twin Carmichael numbers: a pair of consecutive Carmichael numbers (A002997) without a non-prime-power weak Carmichael number (A087442) between them.

Original entry on oeis.org

2465, 62745, 512461, 656601, 658801, 838201, 1033669, 2100901, 4903921, 5968873, 6049681, 8341201, 8719309, 9439201, 9582145, 9585541, 11119105, 11921001, 12261061, 15829633, 17236801, 26921089, 35571601, 36121345, 38624041, 41341321, 43286881, 43584481, 45877861

Offset: 1

Amiram Eldar, Aug 17 2023

nonn

The sequence of weak Carmichael numbers is A225498. The weak Carmichael numbers that are not powers of primes (A000961) are in A087442.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Mauro Fiorentini, Carmichael gemelli (numeri di) (in Italian).
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], 2013.

Subsequence of A002997.

Cf. A000961, A087442, A225498, A365023 (greater counterparts), A365024.

npwcQ[n_] := Length[(p = FactorInteger[n][[;; , 1]])] > 1 && AllTrue[p, Divisible[n - 1, # - 1] &]; (* A087442 *)
seq[nmax_] := Module[{carmichaels = Select[Range[1, nmax, 2], CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &], s = {}, c1, c2}, Do[c1 = carmichaels[[k]] + 2; c2 = carmichaels[[k + 1]] - 2; While[c1 < c2, If[npwcQ[c1], Break[]]; c1 += 2]; If[c1 == c2, AppendTo[s, carmichaels[[k]]]], {k, 1, Length[carmichaels] - 1}]; s]; seq[10^6]

A365023 The greater of twin Carmichael numbers: a pair of consecutive Carmichael numbers (A002997) without a non-prime-power weak Carmichael number (A087442) between them.

Original entry on oeis.org

2821, 63973, 530881, 658801, 670033, 852841, 1050985, 2113921, 4909177, 6049681, 6054985, 8355841, 8719921, 9494101, 9585541, 9613297, 11205601, 11972017, 12262321, 15888313, 17316001, 26932081, 35703361, 36765901, 38637361, 41471521, 43331401, 43620409, 45890209

Offset: 1

Amiram Eldar, Aug 17 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000
Mauro Fiorentini, Carmichael gemelli (numeri di) (in Italian).
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], 2013.

Subsequence of A002997.

Cf. A000961, A087442, A225498, A365022 (lesser counterparts), A365024.

npwcQ[n_] := Length[(p = FactorInteger[n][[;; , 1]])] > 1 && AllTrue[p, Divisible[n - 1, # - 1] &]; (* A087442 *)
seq[nmax_] := Module[{carmichaels = Select[Range[1, nmax, 2], CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &], s = {}, c1, c2}, Do[c1 = carmichaels[[k]] + 2; c2 = carmichaels[[k + 1]] - 2; While[c1 < c2, If[npwcQ[c1], Break[]]; c1 += 2]; If[c1 == c2, AppendTo[s, carmichaels[[k+1]]]], {k, 1, Length[carmichaels] - 1}]; s]; seq[10^6]

A365024 Starts of runs of 3 consecutive Carmichael numbers (A002997) without a non-prime-power weak Carmichael number (A087442) between any two consecutive members.

Original entry on oeis.org

656601, 5968873, 9582145, 45877861, 67653433, 84311569, 171454321, 171679561, 193708801, 193910977, 230630401, 357277921, 367804801, 393122521, 393513121, 393716701, 395044651, 557160241, 703995733, 710382401, 775368901, 832060801, 833608321, 834244501, 939947009

Offset: 1

Amiram Eldar, Aug 17 2023

nonn

The second member in each triple is a term of both A365022 and A365023.

171454321 is the least start of 4 consecutive Carmichael numbers with this property, and 393122521 is the least start of 5, and also 6, consecutive Carmichael numbers with this property.

Amiram Eldar, Table of n, a(n) for n = 1..214
Amiram Eldar, List of triples.
Mauro Fiorentini, Carmichael gemelli (numeri di) (in Italian).
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], 2013.

Subsequence of A002997 and A365023.

Cf. A000961, A087442, A225498, A365022.

npwcQ[n_] := Length[(p = FactorInteger[n][[;; , 1]])] > 1 && AllTrue[p, Divisible[n - 1, # - 1] &]; (* A087442 *)
seq[indmax_] := Module[{carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]], s1 = s2 = {}, c1, c2, i}, Do[c1 = carmichaels[[k]] + 2; c2 = carmichaels[[k + 1]] - 2; While[c1 < c2, If[npwcQ[c1], Break[]]; c1 += 2]; If[c1 == c2, AppendTo[s1, carmichaels[[k]]]; AppendTo[s2, carmichaels[[k + 1]]]], {k, 1, Min[indmax, Length[carmichaels] - 1]}]; i = Position[Rest[s1] - Most[s2], 0] // Flatten; s1[[i]]]; seq[200]

A225005 Number of Carmichael numbers (A002997) less than 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 6, 9, 10, 15, 19, 23, 33, 45, 55, 69, 95, 130, 162, 214, 290, 375, 483, 656, 864, 1118, 1446, 1874, 2437, 3130, 4058, 5188, 6642, 8521, 11002, 14236, 18400, 23631, 30521, 39376, 50685, 65590, 84817, 109857, 141892, 183507, 237217, 307278, 398506, 517446, 672105, 873109, 1136472, 1479525, 1927138, 2513234, 3278553, 4279356

Offset: 1

Max Alekseyev, Apr 23 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..73 (terms 65..69 added using A182490; terms 70..73 calculated using data from Claude Goutier)
Jan Feitsma, The pseudoprimes below 2^64 - statistics. [Wayback Machine link]
Jan Feitsma and William Galway, Tables of pseudoprimes and related data.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Cf. A002997, A055553, A208276, A108797.

Partial sums of A182490.

A265328 Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).

Original entry on oeis.org

1729, 46657, 2433601, 2628073, 19683001, 67371265, 110592000001, 351596817937, 422240040001, 432081216001, 2116874304001, 3176523000001, 18677955240001, 458631349862401, 286245437364810001, 312328165704192001, 12062716067698821000001, 20717489165917230086401, 211215936967181638848001, 411354705193473163968001

Offset: 1

Altug Alkan, Dec 07 2015

From Antti Karttunen, Dec 08 2015: (Start)
The prime factorizations of the first six terms are:
7*13*19, 13*37*97, 17*37*53*73, 7*37*73*139, 13*37*151*271, 5*13*37*109*257
and the prime factorizations of the corresponding perfect powers (numbers one smaller) are:
(2^6 * 3^3), (2^6 * 3^6), (2^6 * 3^2 * 5^2 * 13^2), (2^3 * 3^3 * 23^3), (2^3 * 3^9 * 5^3), (2^8 * 3^6 * 19^2).
(End)
For each k in {22934100, 59553720, 74371320, 242699310, 3190927740, 9214178820, 84855997590}, which is a subset of A270840, k^3+1 is a Carmichael number. - Daniel Suteu, Aug 24 2019
Wagstaff (2024) found that there are no Carmichael numbers k below 10^21 such that k+1 is a perfect power. - Amiram Eldar, Dec 29 2024

			1729 = 7*13*19 is a term because 1729 - 1 = 1728 = 12^3, and 7-1 = 6, 13-1 = 12 and 19-1 = 18 can be all constructed from the primes available in 1728 = (2^6 * 3^3).
2433601 = 17*37*53*73 is a term because 2433601 - 1 = 2433600 = 1560^2, and 16, 36, 52 and 72 can be all constructed from the primes available in 2433600 = (2^6 * 3^2 * 5^2 * 13^2).
67371265 = 5*13*37*109*257 is a term because 67371264 = 8208^2, and 4 (= 2*2), 12 (= 2*2*3), 36 (= 2*2*3*3), 108 (= 2*2*3*3*3) and 256 (= 2^8) can be all constructed from the primes available in 67371264 = (2^8 * 3^6 * 19^2).

G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Contains A265285 as a subsequence.

Cf. A001597, A002997, A135590, A270840.

Select[Cases[Range[1, 10^7, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n], GCD @@ FactorInteger[# - 1][[All, 2]] > 1 &] (* Michael De Vlieger, Dec 14 2015, after Ant King at A001597 and Artur Jasinski at A002997 *)

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(n) && ispower(n-1), print1(n, ", ")))

use ntheory ":all"; foroddcomposites { say if is_power($-1) && is_carmichael($) } 1e8; # Dana Jacobsen, May 05 2017

More terms from Dana Jacobsen, May 05 2017

a(17) from Daniel Suteu confirmed, a(18)-a(20) added by Max Alekseyev, Apr 25 2024

A270267 Carmichael numbers (A002997) that are the sum of three consecutive primes.

Original entry on oeis.org

252601, 410041, 1615681, 2113921, 10606681, 10877581, 11921001, 26932081, 44238481, 54767881, 82929001, 120981601, 128697361, 208969201, 246446929, 255160621, 278152381, 280067761, 311388337, 325546585, 334783585, 416964241, 533860309, 593234929, 672389641

Offset: 1

Altug Alkan, Mar 14 2016

nonn

In other words, Carmichael numbers of the form p + q + r where p, q and r are consecutive primes.
If a Carmichael number is the sum of n consecutive primes, it is so obvious that the minimum value of n is 3.
Intersection of A002997 and A034961.

			84191, 84199 and 84211 are consecutive primes and sum of them is 252601 that is a Carmichael number.
136657, 136691 and 136693 are consecutive primes and sum of them is 410041 that is a Carmichael number.
538553, 538561 and 538567 are consecutive primes and sum of them is 1615681 that is a Carmichael number.

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

Cf. A002997, A034961.

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}
a034961(n) = my(p=prime(n), q=nextprime(p+1)); p+q+nextprime(q+1);
for(n=1, 1e6, if(isA002997(a034961(n)), print1(a034961(n), ", ")));

More terms from Amiram Eldar, Jun 25 2019

A291637 Carmichael numbers (A002997) that are super-Poulet numbers (A050217).

Original entry on oeis.org

294409, 1299963601, 4215885697, 4562359201, 7629221377, 13079177569, 19742849041, 45983665729, 65700513721, 147523256371, 168003672409, 227959335001, 459814831561, 582561482161, 1042789205881, 1297472175451, 1544001719761, 2718557844481, 3253891093249, 4116931056001, 4226818060921, 4406163138721, 4764162536641, 4790779641001, 5419967134849, 7298963852041, 8470346587201

Offset: 1

Max Alekseyev and Thomas Ordowski, Aug 28 2017

nonn

Problem: are there infinitely many such numbers?
From Daniel Suteu, Sep 17 2020: (Start)
If we consider f(n) to be the smallest number in the sequence with n prime factors, then we have:
f(3) = 294409,
f(4) = 3018694485093841,
f(5) <= 521635331852681575100906881,
f(6) <= 2835402730651853232634509813787097410561,
f(7) <= 165784025660216242122027716057592895796242004385542265601. (End)

Amiram Eldar, Table of n, a(n) for n = 1..5328 (calculated using data from Claude Goutier; terms 1..983 from Max Alekseyev)
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 A178997 and A002997.

A299799 Carmichael numbers (A002997) that are Chebyshev pseudoprimes (A175530).

Original entry on oeis.org

443372888629441, 582920080863121, 39671149333495681, 842526563598720001, 2380296518909971201, 3188618003602886401, 33711266676317630401, 54764632857801026161, 122762671289519184001, 361266866679292635601, 4208895375600667752001, 7673096805497432749441

Offset: 1

Max Alekseyev, Feb 19 2018

Odd composite integer n is in this sequence if n == 1 or p (mod (p^2-1)/2) for every prime p|n.

No other terms below 10^22.

Intersection of A002997 and A175530.

Contains A175531 as a subsequence.

a(9) from Daniel Suteu confirmed and a(10) added by Max Alekseyev, Dec 16 2020

a(11)-a(12) from Max Alekseyev, Apr 21 2024

cp's OEIS Frontend

A264024 a(n) = gcd(phi(k), k-1) / lambda(k), where k is n-th Carmichael number A002997(n) and lambda(k) = A002322(k).

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A265285 Carmichael numbers (A002997) k such that k-1 is a square.

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A365022 The lesser of twin Carmichael numbers: a pair of consecutive Carmichael numbers (A002997) without a non-prime-power weak Carmichael number (A087442) between them.

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A365023 The greater of twin Carmichael numbers: a pair of consecutive Carmichael numbers (A002997) without a non-prime-power weak Carmichael number (A087442) between them.

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A365024 Starts of runs of 3 consecutive Carmichael numbers (A002997) without a non-prime-power weak Carmichael number (A087442) between any two consecutive members.

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A225005 Number of Carmichael numbers (A002997) less than 2^n.

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A265328 Carmichael numbers (A002997) k such that k-1 is a perfect power (A001597).

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A270267 Carmichael numbers (A002997) that are the sum of three consecutive primes.

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