A231569 -id:A231569 - cp's OEIS frontend

A231570 Composite numbers n such that lambda(n) divides 3n-3, where lambda is the Carmichael lambda function (A002322).

9, 21, 45, 63, 65, 105, 117, 133, 231, 273, 341, 481, 561, 585, 645, 651, 1001, 1105, 1281, 1365, 1541, 1729, 2465, 2821, 3201, 3605, 4033, 4371, 4641, 4921, 5461, 5565, 6305, 6533, 6601, 7107, 7161, 8321, 8911, 10585, 11041, 12545, 13333, 13833, 14981

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

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, A231571, A231572, A231573.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[3 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (3*n-3)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

A231572 Composite numbers n such that lambda(n) divides 5n-5, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

25, 33, 165, 217, 325, 385, 561, 793, 825, 1025, 1045, 1065, 1105, 1353, 1525, 1705, 1729, 2465, 2665, 2821, 3565, 4123, 4681, 5005, 5185, 5425, 6601, 6697, 8029, 8569, 8911, 9073, 10585, 11005, 12025, 12505, 12801, 13237, 13741, 14707, 14725, 14905, 15457

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.

Charles R Greathouse IV, 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. A231569-A231575, A002322.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[5 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (5*n-5)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

A231573 Composite numbers n such that lambda(n) divides 6n-6, where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

4, 6, 8, 9, 12, 14, 15, 18, 21, 24, 28, 35, 36, 39, 42, 45, 56, 63, 65, 66, 72, 76, 84, 91, 105, 117, 126, 133, 153, 168, 186, 195, 231, 247, 252, 259, 273, 276, 315, 341, 344, 396, 435, 455, 481, 504, 532, 561, 585, 616, 645, 651, 671, 703, 804, 819, 861

Offset: 1

José María Grau Ribas, Nov 11 2013

nonn

Contains the Carmichael numbers, A231569 and A231570.

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

Charles R Greathouse IV, 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. A231569-A231575, A002322.

Select [1 + Range[100000], ! PrimeQ[#] && IntegerQ[6 (# -1)/ CarmichaelLambda[#]] &]

is(n)=!isprime(n) && (6*n-6)%lcm(znstar(n)[2])==0 && n>1 \\ Charles R Greathouse IV, Nov 13 2013

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

A318555 "Strong impostors" not divisible by 4: Those numbers s !== 0 (mod 4) such that lambda(s) | 2(s-1), where lambda is the Carmichael function (A002322).

Original entry on oeis.org

6, 15, 66, 91, 435, 561, 703, 946, 1105, 1729, 1891, 2465, 2701, 2821, 2926, 3367, 5551, 6601, 8646, 8695, 8911, 10585, 11305, 12403, 13981, 15051, 15841, 16471, 18721, 23001, 26335, 29341, 30889, 38503, 39865, 41041, 46657, 49141, 52633, 53131, 62745, 63973, 68101, 75361, 76627, 76798, 79003, 88561, 88831, 91001

Offset: 1

Barry Fagin, Aug 28 2018

nonn

Strong impostors not == 0 (mod 4) have the property that, even though they are composite, when paired with any odd prime r such that (s,r) = 1, they produce valid RSA key pairs. More specifically, if n=sr, all a in Z_n will be correctly encrypted and decrypted for any (e,d) key pair such that ed == 1 mod (s-1)(r-1). They include the Carmichael numbers and are squarefree. The set of their odd prime factors is always normal: If p_i and p_j are odd prime factors, no p_i == 1 mod p_j.

Amiram Eldar, Table of n, a(n) for n = 1..8000 (terms 1..2773 from Barry Fagin)
D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's Conjecture on Primality, Amer. Math. Monthly 103, No. 1, 40-50 (1996).
B. S. Fagin, Composite Numbers That Give Valid RSA Key Pairs For Any Coprime p, Information, 9, 216; doi:10.3390/info9090216.
J. M. Grau and Antonio Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.

Cf. A002997 (Carmichael numbers), A005117 (squarefree numbers).

Subsequence of A231569.

Reap[For[s = 1, s < 10^5, s++, If[!Divisible[s, 4] && CompositeQ[s], If[ Divisible[2(s-1), CarmichaelLambda[s]], Print[s]; Sow[s]]]]][[2, 1]] (* Jean-François Alcover, Feb 18 2019 *)

isok(s) = s>1 && s%4>0 && !isprime(s) && (2*s-2)%lcm(znstar(s)[2])==0; \\ Jinyuan Wang, Mar 01 2020

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A231570 Composite numbers n such that lambda(n) divides 3n-3, where lambda is the Carmichael lambda function (A002322).

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A231572 Composite numbers n such that lambda(n) divides 5n-5, where lambda is the Carmichael lambda function (A002322).

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A231573 Composite numbers n such that lambda(n) divides 6n-6, where lambda is 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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A318555 "Strong impostors" not divisible by 4: Those numbers s !== 0 (mod 4) such that lambda(s) | 2(s-1), where lambda is the Carmichael function (A002322).

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