A002322 -id:A002322 - cp's OEIS frontend

A162578 Partial sums of A002322.

1, 2, 4, 6, 10, 12, 18, 20, 26, 30, 40, 42, 54, 60, 64, 68, 84, 90, 108, 112, 118, 128, 150, 152, 172, 184, 202, 208, 236, 240, 270, 278, 288, 304, 316, 322, 358, 376, 388, 392, 432, 438, 480, 490, 502, 524, 570, 574, 616, 636, 652, 664, 716, 734, 754, 760, 778

Offset: 1

Jonathan Vos Post, Jul 06 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000
Paul Erdős, Carl Pomerance, and Eric Schmutz, Carmichael's lambda function, Acta Arithmetica, Vol. 58, No. 4 (1991), pp. 363-385; alternative link.

Cf. A002322, A218342.

read("transforms3") ; a002322 := BFILETOLIST("b002322.txt") : A162578 :=proc(n) global a002322 ; local i; add(op(i,a002322),i=1..n) ; end: seq(A162578(n),n=1..120) ; # R. J. Mathar, Jul 16 2009

Accumulate[CarmichaelLambda[Range[60]]] (* Harvey P. Dale, Sep 21 2011 *)

a(n) = sum(i=1, n, lcm(znstar(i)[2])) \\ Felix Fröhlich, Jul 04 2018

a(n) = Sum_{k=1..n} A002322(k).

a(n) = (n^2/log(n)) * exp(B * (log(log(n))/log(log(log(n)))) * (1 + o(1))), where B = A218342 (Erdős et al., 1991). - Amiram Eldar, Dec 27 2022

a(13) corrected and more terms added by R. J. Mathar, Jul 16 2009

A231569 Composite numbers n such that lambda(n) divides 2n-2, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

4, 6, 8, 12, 15, 24, 28, 66, 91, 276, 435, 532, 561, 616, 703, 946, 1105, 1128, 1288, 1729, 1891, 2465, 2701, 2821, 2926, 3367, 5551, 6601, 8646, 8695, 8911, 10585, 11305, 11476, 12403, 13981, 15051, 15841, 16471, 18721, 19096, 23001, 26335, 29341, 30889

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

Contains the Carmichael numbers (A002997).

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, A002997, A231570, A231571, A231572, A231573.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[2 (# -1)/ CarmichaelLambda[#]] &]

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

A263028 Numbers n such that A002322(n) + 1 is a prime, where A002322 is Carmichael lambda.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83

Offset: 1

Vincenzo Librandi, Oct 12 2015

Complement of A263029.

Antti Karttunen, Table of n, a(n) for n = 1..39743

Cf. A002322, A263027, A263029, A296077 (characteristic function).

Cf. also A039698.

[1] cat [n: n in [2..100] | IsPrime(CarmichaelLambda(n)+1)];

Select[Range[1, 100], PrimeQ[CarmichaelLambda[#] + 1] &]

for(n=1, 1e3, if(isprime((1 + lcm(znstar(n)[2]))), print1(n", "))) \\ Altug Alkan, Oct 12 2015

More terms from Antti Karttunen, Dec 05 2017

A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

Original entry on oeis.org

1, 3, 7, 11, 21, 21, 43, 27, 61, 63, 111, 53, 157, 129, 87, 91, 273, 183, 343, 151, 175, 333, 507, 117, 521, 471, 547, 305, 813, 261, 931, 347, 447, 819, 483, 431, 1333, 1029, 631, 327, 1641, 525, 1807, 781, 681, 1521, 2163, 373, 2101, 1563, 1095, 1103, 2757

Offset: 1

Vladeta Jovovic, Apr 21 2001

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

Cf. A002322, A057660, A001001, A060640, A061259.

Cf. A000005, A027750, A141258.

a061258 n = sum $ zipWith (*) ds $ map a002322 ds
            where ds = a027750_row n
-- Reinhard Zumkeller, Sep 02 2014

a[n_] := DivisorSum[n, # * CarmichaelLambda[#] &]; Array[a, 100] (* Amiram Eldar, Apr 13 2024 *)

a(n) = sumdiv(n, d, d * lcm(znstar(d)[2])); \\ Amiram Eldar, Apr 13 2024

a(n) = Sum_{k = 1..A000005(n)} (A027750(n,k)*A002322(A027750(n,k))). - Reinhard Zumkeller, Sep 02 2014

A219175 a(n) = n mod lambda(n) where lambda is the Carmichael function (A002322).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 3, 2, 1, 0, 5, 2, 9, 4, 1, 2, 1, 0, 3, 2, 11, 0, 1, 2, 3, 0, 1, 0, 1, 4, 9, 2, 1, 0, 7, 10, 3, 4, 1, 0, 15, 2, 3, 2, 1, 0, 1, 2, 3, 0, 5, 6, 1, 4, 3, 10, 1, 0, 1, 2, 15, 4, 17, 6, 1, 0, 27, 2, 1, 0, 5

Offset: 1

Michel Lagneau, Nov 13 2012

nonn

a(n) = A068494(n) for n = 1..14.
a(k) = 1 for k = prime(n) > 2 or k = A002997(n).
a(n) is the smallest k >= 0 such that b^(n-k) == 1 (mod n) for every b coprime to n. - Thomas Ordowski, Jun 30 2017

			a(9) = 3 because lambda(9) = 6 and 9 == 3 mod 6.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A068494, A002322, A002997.

with(numtheory):for n from 1 to 100 do: x:=irem(n,lambda(n)): printf(`%d, `,x):od:

Table[Mod[n, CarmichaelLambda[n]], {n, 100}] (* T. D. Noe, Nov 13 2012 *)

a(n)=n%lcm(znstar(n)[2]) \\ Charles R Greathouse IV, Nov 13 2012

A263027 a(n) = A002322(n) + 1, where A002322 is Carmichael lambda.

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 7, 3, 7, 5, 11, 3, 13, 7, 5, 5, 17, 7, 19, 5, 7, 11, 23, 3, 21, 13, 19, 7, 29, 5, 31, 9, 11, 17, 13, 7, 37, 19, 13, 5, 41, 7, 43, 11, 13, 23, 47, 5, 43, 21, 17, 13, 53, 19, 21, 7, 19, 29, 59, 5, 61, 31, 7, 17, 13, 11, 67, 17, 23, 13, 71, 7

Offset: 1

Vincenzo Librandi, Oct 08 2015

nonn

The function t(k,n) = A002322(n)+k provides many prime values for k=1: for n up to 1000, for example, it returns 798 primes (with repetitions). On the other hand, for n <= 1000 and odd k from 3 to 11, t(k,n) gives 247, 387, 538, 231, 504 prime values, respectively.

Another function of this type is |A002322(n)-119|, which provides 693 prime values for n <= 1000. [Bruno Berselli, Oct 14 2015]

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A002322.
Cf. A263028: indices n for which a(n) is prime.
Cf. A263029: indices n for which a(n) is composite.
Cf. also A039649, A296076, A296077.

[2] cat [CarmichaelLambda(n)+1: n in [2..100]];

Table[CarmichaelLambda[n] + 1, {n, 1, 100}]

vector(100, n, 1 + lcm(znstar(n)[2])) \\ Altug Alkan, Oct 08 2015

Edited by Bruno Berselli, Oct 14 2015

A263029 Numbers n such that A002322(n) + 1 is not a prime, where A002322 is Carmichael lambda.

Original entry on oeis.org

25, 32, 50, 55, 75, 81, 96, 100, 110, 115, 119, 121, 128, 150, 153, 160, 162, 165, 176, 187, 200, 203, 209, 215, 220, 221, 224, 230, 235, 238, 242, 245, 253, 256, 261, 275, 287, 288, 289, 295, 297, 299, 300, 306, 319, 323, 324, 330, 335, 343, 345, 355

Offset: 1

Vincenzo Librandi, Oct 12 2015

Complement of A263028.

Antti Karttunen, Table of n, a(n) for n = 1..25794

Cf. A002322, A263027, A263028.
Positions of zeros in A296077.
Cf. also A039689.

[n: n in [2..400] | not IsPrime(CarmichaelLambda(n)+1)];

Select[Range[1, 400], ! PrimeQ[CarmichaelLambda[#] + 1] &]

for(n=1, 1e3, if(isprime((1 + lcm(znstar(n)[2]))) == 0, print1(n", "))) \\ Altug Alkan, Oct 12 2015

A296076 Least number with the same prime signature as 1 + A002322(n), where A002322 is Carmichael's lambda.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 12, 2, 2, 2, 4, 2

Offset: 1

Antti Karttunen, Dec 05 2017

nonn

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

Cf. A002322, A046523, A263027, A263028 (positions of 2's), A296077, A296078.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A296076(n) = A046523(1+lcm(znstar(n)[2]));

a(n) = A046523(A263027(n)) = A046523(1+A002322(n)).

A328935 Imprimitive Carmichael numbers: Carmichael numbers m such that if m = p_1 * p_2 * ... *p_k is the prime factorization of m then g(m) = gcd(p_1 - 1, ..., p_k - 1) > sqrt(lambda(m)), where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

294409, 399001, 488881, 512461, 1152271, 1461241, 3057601, 3828001, 4335241, 6189121, 6733693, 10267951, 14676481, 17098369, 19384289, 23382529, 50201089, 53711113, 56052361, 64377991, 68154001, 79624621, 82929001, 84350561, 96895441, 115039081, 118901521, 133800661

Offset: 1

Amiram Eldar, Oct 31 2019

nonn

Granville and Pomerance separated the Carmichael numbers into two classes, primitive and imprimitive, according to whether g(m) <= sqrt(lambda(n)) or not.
They conjectured that most Carmichael numbers are primitive and most 3-Carmichael numbers (A087788) are imprimitive.
Comment from Jeppe Stig Nielsen, Apr 21 2021: (Start)
In cases n = 1, 3, 5, 7, 8, 10, 14, 15, 19, 20, ..., there exists a primitive Carmichael number in the same "family" (Carmichael numbers that share the ratio (p_1-1):(p_2-1):...:(p_k-1) belong to the same family). However, in the remaining cases, the entire family consists of imprimitive Carmichael numbers.
There can be more than one primitive Carmichael number in a family. For example, both Carmichael numbers 5828853661 and 965507554621 are primitive, and are in the family 1:3:6:70. The first imprimitive Carmichael number in the family 1:3:6:70 is a(1639)=59610715093021. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Mathematics of Computation, Vol. 71, No. 238 (2002), pp. 883-908.

Cf. A002322, A002997, A087788, A258409, A335584.

aQ[n_] := Length[(f = FactorInteger[n])] > 2 && Max[f[[;; , 2]]] == 1 && Divisible[n-1, (lambda = LCM @@ (f[[;; , 1]] - 1))] && GCD @@ (f[[;; , 1]] - 1) > Sqrt[lambda]; Select[Range[4*10^6], aQ]

isA328935(m)=f=factor(m);!(issquarefree(f)&&omega(f)>2)&&return(0);p=f[,1]~;r=apply(x->x-1,p);foreach(r,x,(m-1)%x!=0&&return(0));g=gcd(r);a=r/g;g>lcm(a) \\ p, g, and a are like in Granville & Pomerance, Jeppe Stig Nielsen, Apr 21 2021

Terms m of A002997 such that A258409(m) > sqrt(A002322(m)).

A354061 Irregular table read by rows: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n, 1 <= k <= psi(n), psi = A002322.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 2, 1, 0, 5, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 1, 4, 1, 0, 1, 0, 9, 0, 1, 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15

Offset: 1

Jianing Song, May 16 2022

Given n, T(n,k) only depends on gcd(k,psi(n)).
The n-th row contains entirely 0's if and only if n == 2 (mod 4).
If n !== 2 (mod 4), T(n,psi(n)) > T(n,k) for 1 <= k < psi(n).

			Table starts
n = 1: 1;
n = 2: 0;
n = 3: 0, 1;
n = 4: 0, 1;
n = 5: 0, 1, 0, 3;
n = 6: 0, 0;
n = 7: 0, 1, 2, 1, 0, 5;
n = 8: 0, 2;
n = 9: 0, 0, 2, 0, 0, 4;
n = 10: 0, 0, 0, 0;
n = 11: 0, 1, 0, 1, 4, 1, 0, 1, 0, 9;
n = 12: 0, 1;
n = 13: 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11;
n = 14: 0, 0, 0, 0, 0, 0;
n = 15: 0, 1, 0, 3;
n = 16: 0, 0, 0, 4;
n = 17: 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15;
n = 18: 0, 0, 0, 0, 0, 0;
n = 19: 0, 1, 2, 1, 0, 5, 0, 1, 8, 1, 0, 5, 0, 1, 2, 1, 0, 17;
n = 20: 0, 1, 0, 3;
...

Jianing Song, Table of n, a(n) for n = 1..8346 (the first 200 rows)

Cf. A354058, A002322, A007431, A354060.

A354257 gives the smallest index for the nonzero terms in each row.

b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
T(n,k) = sumdiv(n, d, moebius(n/d)*b(d,k))

For odd primes p: T(p,k) = gcd(p-1,k)-1, T(p^e,k*p^(e-1)) = p^(e-2)*(p-1)*gcd(k,p-1), T(p^e,k) = 0 if k is not divisible by p^(e-1). T(2,k) = 0, T(4,k) = 1 for even k and 0 for odd k, T(2^e,k) = 2^(e-2) if k is divisible by 2^(e-2) and 0 otherwise.

T(n,psi(n)) = A007431(n). - Jianing Song, May 24 2022

cp's OEIS Frontend

A162578 Partial sums of A002322.

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A231569 Composite numbers n such that lambda(n) divides 2n-2, where lambda is the Carmichael lambda function (A002322).

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A263028 Numbers n such that A002322(n) + 1 is a prime, where A002322 is Carmichael lambda.

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A061258 a(n) = Sum_{d|n} d*psi(d), where psi(d) is reduced totient function, cf. A002322.

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A219175 a(n) = n mod lambda(n) where lambda is the Carmichael function (A002322).

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A263027 a(n) = A002322(n) + 1, where A002322 is Carmichael lambda.

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A263029 Numbers n such that A002322(n) + 1 is not a prime, where A002322 is Carmichael lambda.

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