A002322 -id:A002322 - cp's OEIS frontend

A327979 a(n) = gcd(n, A002322(n)), where A002322 is Carmichael lambda, also known as psi.

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 6, 1, 4, 3, 2, 1, 2, 5, 2, 9, 2, 1, 2, 1, 8, 1, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 1, 4, 7, 10, 1, 4, 1, 18, 5, 2, 3, 2, 1, 4, 1, 2, 3, 16, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 5, 2, 1, 6, 1, 4, 27, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 8, 1, 14, 3, 20, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Oct 04 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A002322.

Cf. also A009191, A009194, A009195, A306695.

Array[GCD[#, CarmichaelLambda[#]] &, 100] (* Amiram Eldar, Oct 04 2019 *)

PARI
```
A327979(n) = gcd(n, lcm(znstar(n)[2]));
```

a(n) = gcd(n, A002322(n)).

A341858 Numbers k such that psi(k^2) = k, psi = A002322; indices of 1 in A341857.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 42, 60, 84, 156, 220, 420, 660, 780, 1092, 1806, 1860, 2436, 3612, 3660, 4620, 5060, 5460, 8268, 8580, 12180, 12324, 13020, 15180, 18060, 20460, 24180, 24492, 25620, 29820, 31668, 40260, 41340, 44220, 46956, 47580, 57876, 60060, 61620, 86268, 88620

Offset: 1

Jianing Song, Feb 21 2021

nonn

For all k we have k divides psi(k^2). This sequence gives those k such that the quotient is 1.
Apart from 5 exceptional terms, every term is the product of 4 and distinct odd primes. The exceptional terms are precisely the 5 terms in A014117.
Except for k = 1, 2, 6, 42, 1806, k is a term if and only if k = 4*(p_1)*(p_2)*...*(p_m), where p_1 < p_2 < ... < p_m are odd primes, (p_i)-1 | 4*(p_1)*(p_2)*...*(p_(i-1)) for all 1 <= i <= m.
The LCM of two terms is again in this sequence.
Is this sequence infinite? If this sequence is finite, it means that there exists a term of the form k = 4*(p_1)*(p_2)*...*(p_s), where p_1 < p_2 < ... < p_s are odd primes such that: for every (e_0, e_1, ..., e_s) in {0, 1}^(s+1), 2^((e_0)+1)*(p_1)^(e_1)*(p_2)^(e_2)*...*(p_s)^(e_s)+1 is either composite or equal to some p_i. That term must be divisible by all other terms, since there are no more odd primes q other than p_1, p_2, ..., p_s such that q-1 | k.
Numbers k such that b^k == 1 (mod k^2) for every b coprime to k. Proof: these are numbers k such that psi(k^2) divides k, which holds if and only if psi(k^2) = k. Subsequence of A124240 (see my comment there). If k is a term of the sequence and k+1 is prime, then k*(k+1) is also a term. - Thomas Ordowski, Jul 26 2024

			1092 = 4 * 3 * 7 * 13 is a term since 3-1 | 4, 7-1 | 4*3 and 13-1 | 4*3*7. Indeed, we have psi(1092^2) = 1092.
5060 = 4 * 5 * 11 * 23 is a term since 5-1 | 4, 11-1 | 4*5 and 23-1 | 4*5*11.

Amiram Eldar, Table of n, a(n) for n = 1..6034 (terms below 10^11; terms 1..249 from Jianing Song)
Jianing Song, Factorization of terms <= 15*10^6 other than 1, 2, 6, 42, 1806

Cf. A002322, A341857, A014117.
A229289 gives the set of prime factors of the terms.
Subsequence of A124240.

Select[Range[10^5], CarmichaelLambda[#^2] == # &] (* Paolo Xausa, Mar 11 2024 *)

isA341858(n) = (A002322(n^2)==n) \\ See A002322 for its program

A206941 a(n) = phi(lambda(n)), where phi = A000010, lambda = A002322.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 4, 2, 2, 2, 8, 2, 6, 2, 2, 4, 10, 1, 8, 4, 6, 2, 12, 2, 8, 4, 4, 8, 4, 2, 12, 6, 4, 2, 16, 2, 12, 4, 4, 10, 22, 2, 12, 8, 8, 4, 24, 6, 8, 2, 6, 12, 28, 2, 16, 8, 2, 8, 4, 4, 20, 8, 10, 4, 24, 2, 24, 12, 8, 6, 8, 4, 24, 2, 18

Offset: 1

N. J. A. Sloane, Feb 13 2012

nonn

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 55, Theorem 4.10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.

Cf. A000010, A002322, A077197.

a206941 = a000010 . a002322 -- Reinhard Zumkeller, Feb 18 2012

Table[EulerPhi@ CarmichaelLambda@ n, {n, 96}] (* Michael De Vlieger, Mar 18 2016 *)

a(n)=eulerphi(lcm(znstar(n)[2])) \\ Charles R Greathouse IV, Feb 21 2013

A207193 Auxiliary function for computing the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 2, 6, 10, 12, 4, 16, 18, 22, 20, 18, 28, 30, 8, 36, 40, 42, 46, 42, 52, 58, 60, 16, 66, 70, 72, 78, 54, 82, 88, 96, 100, 102, 106, 108, 112, 110, 100, 126, 32, 130, 136, 138, 148, 150, 156, 162, 166, 156, 172, 178, 180, 190, 192, 196, 198

Offset: 1

Reinhard Zumkeller, Feb 16 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000961, A002322, A025473, A025474, A085730.

a207193 1 = 1
a207193 n | p == 2 && e > 2 = 2 ^ (e - 2)
          | otherwise       = (p - 1) * p ^ (e - 1)
          where p = a025473 n; e = a025474 n

f[p_, e_] := If[p == 2 && e > 2, 2^(e-2), (p-1)*p^(e-1)]; s[n_] := If[n == 1, 1, If[PrimePowerQ[n], f @@ (FactorInteger[n][[1]]), Nothing]]; Array[s, 200] (* Amiram Eldar, Apr 05 2025 *)

a(n) = f(A000961(n)), where f(1) = 1, and f(p^e) = 2^(e-2) if p = 2 and e > 2, and f(p^e) = (p-1)*p^(e-1) otherwise.

A231571 Composite numbers n such that lambda(n) divides 4n-4, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

4, 6, 8, 10, 12, 15, 16, 20, 24, 28, 30, 40, 48, 52, 60, 66, 70, 80, 85, 91, 112, 120, 130, 176, 190, 208, 232, 240, 276, 280, 286, 364, 370, 435, 451, 496, 520, 532, 561, 616, 703, 742, 910, 946, 976, 1036, 1105, 1128, 1288, 1387, 1456, 1729, 1770, 1891

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

Contains the Carmichael numbers (A002997) and A231569.

Conjecture: the relative asymptotic density of the Carmichael numbers in this sequence exists, is positive and smaller than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. M. Grau and Antonio Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.

Cf. A002322, A231569, A231570, A231572, A231573.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[4 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (4*n-4)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

A246700 Table read by rows: trajectories under iteration of Carmichael's lambda function (cf. A002322).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 2, 1, 7, 6, 2, 1, 8, 2, 1, 9, 6, 2, 1, 10, 4, 2, 1, 11, 10, 4, 2, 1, 12, 2, 1, 13, 12, 2, 1, 14, 6, 2, 1, 15, 4, 2, 1, 16, 4, 2, 1, 17, 16, 4, 2, 1, 18, 6, 2, 1, 19, 18, 6, 2, 1, 20, 4, 2, 1, 21, 6, 2, 1, 22, 10, 4, 2, 1

Offset: 1

Reinhard Zumkeller, Sep 02 2014

Length of row n = A185816(n) + 1.

			.   |  1 | 1             | 13 | 13-12-2-1        | 25 | 25-20-4-2-1
.   |  2 | 2-1           | 14 | 14-6-2-1         | 26 | 26-12-2-1
.   |  3 | 3-2-1         | 15 | 15-4-2-1         | 27 | 27-18-6-2-1
.   |  4 | 4-2-1         | 16 | 16-4-2-1         | 28 | 28-6-2-1
.   |  5 | 5-4-2-1       | 17 | 17-16-4-2-1      | 29 | 29-28-6-2-1
.   |  6 | 6-2-1         | 18 | 18-6-2-1         | 30 | 30-4-2-1
.   |  7 | 7-6-2-1       | 19 | 19-18-6-2-1      | 31 | 31-30-4-2-1
.   |  8 | 8-2-1         | 20 | 20-4-2-1         | 32 | 32-8-2-1
.   |  9 | 9-6-2-1       | 21 | 21-6-2-1         | 33 | 33-10-4-2-1
.   | 10 | 10-4-2-1      | 22 | 22-10-4-2-1      | 34 | 34-16-4-2-1
.   | 11 | 11-10-4-2-1   | 23 | 23-22-10-4-2-1   | 35 | 35-12-2-1
.   | 12 | 12-2-1        | 24 | 24-2-1           | 36 | 36-6-2-1  .

Reinhard Zumkeller, Rows n = 1..1001 of triangle, flattened

Cf. A002322, A185816, A375478.

import Data.List (genericIndex)
a246700 n k = genericIndex a246700_tabf (n - 1) !! (k-1)
a246700_row n = genericIndex a246700_tabf (n - 1)
a246700_tabf = [1] : f 2  where
   f x = (x : a246700_row (a002322 x)) : f (x + 1)

Array[Most[FixedPointList[CarmichaelLambda, #]] &, 25] (* Paolo Xausa, Aug 17 2024 *)

T(n,1) = n and T(n,k+1) = A002322(T(n,k)), k = 1..A185816(n).

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

Original entry on oeis.org

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

A303758 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 1, 3, 3, 4, 1, 4, 1, 5, 5, 2, 1, 6, 1, 2, 2, 6, 1, 6, 1, 1, 3, 2, 3, 7, 1, 3, 4, 7, 1, 8, 1, 4, 5, 2, 1, 8, 2, 2, 3, 6, 1, 4, 3, 9, 5, 2, 1, 9, 1, 2, 10, 4, 7, 5, 1, 5, 3, 8, 1, 11, 1, 2, 4, 6, 3, 9, 1, 10, 1, 2, 1, 12, 6, 3, 3, 6, 1, 10, 11, 4, 4, 2, 3, 2, 1, 4, 5, 5, 1, 7, 1, 12, 13

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of f, where f(1) = 0 and f(n) = A002322(n) for n > 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A002322.

Cf. also A303756, A303757.

a[1] = 1; a[n_] := With[{c = CarmichaelLambda[n]}, Select[Range[2, n], c == CarmichaelLambda[#]&] // Length];
Array[a, 1000] (* Jean-François Alcover, Sep 19 2020 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
Aux303758(n) = if(1==n,0,A002322(n));
v303758 = ordinal_transform(vector(up_to,n,Aux303758(n)));
A303758(n) = v303758[n];

Except for a(2) = 1, a(n) = A303756(n).

A304480 a(n) is the least m such that lambda(k) >= n for all k >= m where lambda is A002322, the Carmichael lambda function.

Original entry on oeis.org

1, 3, 25, 25, 241, 241, 505, 505, 505, 505, 505, 505, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 65521, 131041, 131041, 131041, 131041, 131041, 131041, 171865, 171865, 171865, 171865, 171865, 171865, 138181681, 138181681, 138181681, 138181681, 138181681, 138181681

Offset: 1

Michel Marcus, May 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Project Euler, Problem 533 - Minimum values of the Carmichael function
Wikipedia, Carmichael function

Cf. A002174, A002322, A002396, A141162, A143407, A335116, A335117.

minvl(n, v) = {vgt = select(x->(x>=n), v, 1); first = vgt[1]; vgtd = vector(#vgt-1, k, vgt[k+1] - vgt[k]); vgtdr = Vecrev(vgtd); vokdiff = select(x->x!=1, vgtdr, 1); if (#vokdiff, #v - vokdiff[1]+1, first);}
lista(nn) = {v = read("v002322.txt"); for (n=1, nn, print1(minvl(n, v), ", "););}

a(32) and beyond from Seiichi Manyama, May 24 2020

A329885 a(n) = A051903(n) mod A002322(n), where A051903 gives the maximal prime exponent of n, and A002322 is Carmichael's lambda (also known as psi).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 0, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 0, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Dec 11 2019

nonn

This differs from A051903 at n = 2, 4, 8, 12, 16, 24, 48, 80, 240. Are there any other such n? (None other found <= 201326592.)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A002322, A051903, A327295.

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A329885(n) = (A051903(n)%A002322(n));

cp's OEIS Frontend

A327979 a(n) = gcd(n, A002322(n)), where A002322 is Carmichael lambda, also known as psi.

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A341858 Numbers k such that psi(k^2) = k, psi = A002322; indices of 1 in A341857.

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A206941 a(n) = phi(lambda(n)), where phi = A000010, lambda = A002322.

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A207193 Auxiliary function for computing the Carmichael lambda function (A002322).

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A231571 Composite numbers n such that lambda(n) divides 4n-4, where lambda is the Carmichael lambda function (A002322).

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A246700 Table read by rows: trajectories under iteration of Carmichael's lambda function (cf. A002322).

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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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A303758 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

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