A340092 -id:A340092 - 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))));

A340091 Odd numbers k such that A064989(k) is in A340151.

Original entry on oeis.org

679, 703, 1387, 1729, 1891, 2047, 2509, 2701, 2821, 3277, 3367, 5551, 7471, 7735, 8119, 8827, 9997, 10963, 11305, 12403, 13021, 13747, 13981, 14491, 14701, 15841, 16471, 17563, 19951, 21349, 21907, 21931, 22015, 23959, 24727, 25669, 26281, 27511, 28939, 29341, 31417, 32407, 38503, 39091, 39831, 39865, 40501, 41041

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Sequence A003961(A340151(i)), for i >= 1, sorted into ascending order.

By definition, this has no common terms with A340077 nor any of its subsequences like A339869 or A339880.

Antti Karttunen, Table of n, a(n) for n = 1..3290

Cf. A002997, A003961, A339869, A339880, A340077, A340151.

Cf. A340092 (Carmichael numbers in this sequence).

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A339904(n) = A000265(eulerphi(A003961(n)));
A340075(n) = { my(u=A339904(n)); u/gcd(A003961(n)-1, u); };
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
A340149(n) = A000265(A247074(A003961(n)));
isA340151(n) = ((1!=A340075(n))&&(1==A340149(n)));
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
isA340091(n) = ((n%2)&&isA340151(A064989(n)));

cp's OEIS Frontend

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

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