A306414 -id:A306414 - cp's OEIS frontend

A174590 a(n) = (k-1)/lambda(k), the index of the n-th Carmichael number k.

7, 23, 48, 22, 47, 5, 45, 21, 44, 163, 342, 162, 43, 31, 1777, 314, 337, 161, 1753, 70, 2868, 1745, 421, 2487, 1363, 159, 39, 645, 950, 67, 198, 1358, 949, 158, 2303, 134, 305, 1692, 1733, 5731, 2794, 7107, 1732, 345, 1689, 2654, 1671, 1829, 947, 1353, 1557

Offset: 1

Michel Lagneau, Mar 23 2010, Mar 31 2010

nonn

The index of a Carmichael number k is i(k) = (k-1)/lambda(k).
Or, i(k) = (k-1)/lcm(p_1-1,p_2-1,...,p_j-1), where k = p_1*p_2*...*p_j. - Thomas Ordowski, Oct 15 2015
For composite k, lambda(k) divides k-1 iff k is a Carmichael number. - Thomas Ordowski, Oct 23 2015

			a(1)= 7 because A002997(1) = 561, and (561 - 1)/lambda(561) = 560/80 = 7.

Robert Israel, Table of n, a(n) for n = 1..10000
J. M. Chick, Carmichael number variable relations: three-prime Carmichael numbers up to 10^24, arXiv:0711.2915 [math.NT] 2007-2008, Table 1, p. 34.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), no. 238, 883-908.
R. G. E. Pinch, Carmichael numbers up to 10^18, April 2006.
Richard Pinch, Carmichael Numbers up to 10^20, Abstract, ANTS 7.
Richard Pinch, Carmichael Numbers up to 10^20, Poster, ANTS 7.

Cf. A002322 (the Carmichael lambda function), A002997, A011773, A306414.

with(numtheory) : for n from 2 to 2000000 do: if type(n,prime)=false and issqrfree(n)=true then  x:=factorset(n):n1:=nops(x):ii:=0:for j from 1 to n1 do:if irem(n-1, x[j]-1)=0  then ii:=ii+1:else fi:od:if ii=n1 then z:=(n-1)/lambda(n):printf(`%d, `,z):else fi:fi:od:

carNums = Select[Range[561, 3 10^6, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#]] == 1&];
a[n_] := (carNums[[n]] - 1)/CarmichaelLambda[carNums[[n]]];
Array[a, 60] (* Jean-François Alcover, Sep 05 2018 *)

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;
for(n=1, 1e7, if(n%2 && !isprime(n) && t(n) && n>1, print1((n-1)/(lcm(znstar(n)[2])), ", "))) \\ Altug Alkan, Oct 15 2015

a(n) = (A002997(n) - 1) / lambda(A002997(n)).

a(n) = (A002997(n) - 1) / A306414(n). - Jianing Song, Dec 12 2021

Edited by Michel Lagneau, Jul 31 2012

Further edits from N. J. A. Sloane, Oct 31 2015

A350084 a(n) = ord(2,A006935(n)/2), where ord(k,m) is the multiplicative order of k modulo m.

Original entry on oeis.org

1, 261, 165, 275, 13425, 1485, 1305, 32085, 825, 3465, 2093, 3135, 495, 495, 261, 847, 9405, 552189, 198561, 261, 579261, 2475, 6237, 166725, 111111, 3393, 3565, 25245, 18585, 4437, 891891, 309455, 37125, 4833, 2301585, 14355, 11781, 3315, 915, 84975, 35259

Offset: 1

Jianing Song, Dec 12 2021

nonn

List of ord(2,k) where k ranges over the odd numbers such that 2^(2*k-1) == 1 (mod k).

			A006935(2) = 161038, so a(2) = ord(2,161038/2) = 261.
A006935(3) = 215326, so a(3) = ord(2,215326/2) = 165.

Amiram Eldar, Table of n, a(n) for n = 1..1319

Cf. A006935, A347906, A350083, A306413, A306414.

list(lim) = my(v=[],d); forstep(k=1, lim, 2, if((2*k-1)%(d=znorder(Mod(2,k)))==0, v=concat(v,d))); v \\ gives a(n) for A347906(n) <= lim

a(n) = ord(2,A347906(n)) = (A006935(n) - 1) / A350083(n).

A367320 Carmichael numbers k such that (k-1)/lambda(k) > (m-1)/lambda(m) for all Carmichael numbers m < k, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

561, 1105, 1729, 29341, 41041, 63973, 172081, 825265, 852841, 1773289, 5310721, 9890881, 12945745, 18162001, 31146661, 93869665, 133205761, 266003101, 417241045, 496050841, 509033161, 1836304561, 1932608161, 2414829781, 4579461601, 9799928965, 11624584621, 12452890681

Offset: 1

Amiram Eldar, Nov 14 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..169 (terms below 10^22 calculated using data from Claude Goutier)
Amiram Eldar, Table of n, a(n), (a(n)-1)/A002322(a(n)) for n = 1..169.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Subsequence of A002997.

Cf. A002322, A174590, A226216, A306414, A367319.

seq[kmax_] := Module[{s = {}, r, rm = 0, lam}, Do[If[CompositeQ[k], lam = CarmichaelLambda[k]; If[Mod[k, lam] == 1, r = (k - 1)/lam; If[r > rm, rm = r; AppendTo[s, k]]]], {k, 9, kmax, 2}]; s]; seq[10^6]

lista(kmax) = {my(r, rm = 0, lam); forcomposite(k = 4, kmax, if(k % 2, lam = lcm(znstar(k)[2]); if(k % lam == 1, r = (k-1)/lam; if(r > rm, rm = r; print1(k, ", ")))));}

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A174590 a(n) = (k-1)/lambda(k), the index of the n-th Carmichael number k.

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A350084 a(n) = ord(2,A006935(n)/2), where ord(k,m) is the multiplicative order of k modulo m.

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A367320 Carmichael numbers k such that (k-1)/lambda(k) > (m-1)/lambda(m) for all Carmichael numbers m < k, where lambda is the Carmichael lambda function (A002322).

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