A339909 -id:A339909 - cp's OEIS frontend

A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.

561, 1105, 2465, 6601, 8911, 10585, 46657, 62745, 162401, 410041, 449065, 5148001, 5632705, 6313681, 6840001, 7207201, 11119105, 11921001, 19683001, 21584305, 26719701, 41298985, 55462177, 64774081, 67371265, 79411201, 83966401, 87318001, 99861985, 100427041, 172290241, 189941761, 484662529, 790623289, 809883361

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Lehmer conjectured that the equation k * phi(n) = n - 1 (with k integer) has no solutions for any composite n (i.e., when k > 1). If this sequence has no common terms with A339818, then the conjecture certainly holds.

Amiram Eldar, Table of n, a(n) for n = 1..1150 (terms below 10^22 calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 and A339880.
Complement of A340092 in A002997.
Cf. A000010, A000265, A002322, A053575.
Cf. also A339818, A339878, A339909.

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; oddPart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := Divisible[n - 1, oddPart[EulerPhi[n]]]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

A000265(n) = (n>>valuation(n, 2));
A002322(n) = lcm(znstar(n)[2]);
isA339869(n) = ((n>1)&&!isprime(n)&&(!((n-1)%A002322(n)))&&!((n-1)%A000265(eulerphi(n))));

A339818 Carmichael numbers k for which the 2-adic valuation of phi(k) does not exceed the 2-adic valuation of k-1.

Original entry on oeis.org

1729, 15841, 3057601, 3828001, 5310721, 8355841, 8830801, 9439201, 14676481, 15829633, 17236801, 40280065, 78091201, 83099521, 84350561, 92625121, 94536001, 104852881, 118901521, 129762001, 157731841, 163954561, 180115489, 193708801, 214852609, 221884001, 279377281, 382304161, 382536001, 438359041, 481239361, 511338241

Offset: 1

Antti Karttunen, Dec 20 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..22784 (terms below 2^64)

Intersection of A002997 and A339817 (see comments in latter).

Cf. also A339869, A339878, A339909.

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; q[n_] := IntegerExponent[EulerPhi[n], 2] <= IntegerExponent[n - 1, 2]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA339818(n) = ((n>1)&&issquarefree(n)&&!isprime(n)&&(valuation(eulerphi(n),2)<=valuation(n-1,2))&&(0==((n-1)%A002322(n))));

A339908 Odd squarefree numbers k > 1 for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

Original entry on oeis.org

33, 65, 129, 141, 145, 161, 177, 201, 217, 249, 253, 321, 385, 393, 417, 501, 537, 649, 681, 705, 721, 737, 849, 865, 897, 913, 973, 993, 1041, 1057, 1081, 1101, 1121, 1135, 1149, 1169, 1177, 1281, 1329, 1345, 1401, 1441, 1457, 1473, 1497, 1509, 1537, 1561, 1569, 1585, 1633, 1689, 1729, 1761, 1793, 1821, 1837, 1841

Offset: 1

Antti Karttunen, Dec 21 2020

nonn

All terms can be found in A339911. Also, all nonmultiples of 3 certainly occur in A339912 also.

The first term of the form 4u+3 is 1135.

Antti Karttunen, Table of n, a(n) for n = 1..26693; all terms < 2^20

Cf. A000010, A001222.
Subsequence of A056911 and of A339907, A339911.
Cf. also A339912.
Cf. A339909 (a subsequence).

isA339908(n) = ((n>1)&&(n%2)&&issquarefree(n)&&(bigomega(eulerphi(n))

A339878 Carmichael numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1.

Original entry on oeis.org

1729, 3069196417, 23915494401, 1334063001601, 6767608320001, 33812972024833, 1584348087168001, 1602991137369601, 6166793784729601, 1531757211193440001, 84388996672599528001

Offset: 1

Antti Karttunen (after Thomas Ordowski's and Amiram Eldar's SeqFan-posting), Dec 26 2020

The first ten terms are all in A339818, none is in A339869, and all except a(2) and a(6) are in A339909.

Also, for all ten, a(n) == 1 (mod 64). (Cf. a similar comment in A338998).

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Thomas Ordowski and Amiram Eldar, A new look at the Lehmer's totient problem, SeqFan, February 10 2019.

Intersection of A002997 and A338998.

Cf. also A339818, A339869, A339909.

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; q[n_] := Module[{p = FactorInteger[n - 1][[;; , 1]], phi = EulerPhi[n]}, AnyTrue[(n - 1)*p, Divisible[#, phi] &]]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

a(10) from Amiram Eldar, Dec 26 2020

a(11) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 21 2024

cp's OEIS Frontend

A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.

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A339818 Carmichael numbers k for which the 2-adic valuation of phi(k) does not exceed the 2-adic valuation of k-1.

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A339908 Odd squarefree numbers k > 1 for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

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A339878 Carmichael numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1.

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