A339908 -id:A339908 - cp's OEIS frontend

A339909 Carmichael numbers k for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

1729, 14676481, 84350561, 90698401, 279377281, 382536001, 413138881, 542497201, 702683101, 781347841, 851703301, 939947009, 955134181, 3480174001, 4765950001, 5255104513, 5781222721, 5985964801, 7558388641, 7816642561, 8714965001, 9237473281, 13630072501, 18189007201, 21669076801, 21863001601, 23915494401, 25477682491

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Natural numbers n that satisfy equation k * phi(n) = n - 1, for some integer k > 1, should all occur in this sequence, if they exist at all. Lehmer conjectured that there are no such numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (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 A339908.
Cf. A000010, A001222, A002322.
Cf. also A339818, A339869, A339878.

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

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA339909(n) = ((n>1)&&issquarefree(n)&&!isprime(n)&&(bigomega(eulerphi(n))A002322(n))));

A339907 Odd squarefree numbers k > 1 for which the bigomega(phi(k)) <= bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 33, 37, 41, 43, 47, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 129, 131, 137, 139, 141, 145, 149, 151, 157, 161, 163, 167, 173, 177, 179, 181, 191, 193, 197, 199, 201, 209, 211, 217, 223, 227, 229, 233, 235, 239, 241, 249, 251, 253, 257

Offset: 1

Antti Karttunen, Dec 21 2020

nonn

Terms of A003961(A019565(A339906(i))) [or equally, of A019565(2*A339906(i))], for i = 1.., sorted into ascending order.

Natural numbers n > 2 that satisfy equation k * phi(n) = n - 1 (for some integer k) all occur in this sequence. Lehmer conjectured that there are no composite solutions.

Antti Karttunen, Table of n, a(n) for n = 1..18526
D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
Wikipedia, Lehmer's totient problem.

Cf. A339906.
Cf. A065091, A339908 (subsequences).
Cf. also A339817.
Apart from initial 3, a subsequence of A339910.

isA339907(n) = ((n>1)&&(n%2)&&issquarefree(n)&&(bigomega(eulerphi(n))<=bigomega(n-1)));

A339911 Numbers k > 1 for which bigomega(k) <= bigomega(k-1)/2, where bigomega gives the number of prime factors, counted with multiplicity.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, 37, 41, 43, 47, 49, 53, 55, 57, 59, 61, 65, 67, 71, 73, 79, 82, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 157, 161, 163, 167, 169, 173, 177, 179, 181, 185, 191, 193, 197, 199, 201, 205, 209, 211, 217, 221, 223, 226, 227

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..17515; all terms <= 65537

Cf. A001222.
Subsequence of A339910.
Cf. A339908, A339912 for subsequences.

Select[Range[3, 227, 2], PrimeOmega[#] <= PrimeOmega[# - 1]/2 &] (* Michael De Vlieger, Dec 22 2020 *)

isA339911(n) = ((n>1)&&((2*bigomega(n))<=bigomega(n-1)));

A339912 Numbers k > 1 for which bigomega(k) < bigomega(k-1)/2, where bigomega gives the number of prime factors, counted with multiplicity.

Original entry on oeis.org

13, 17, 19, 29, 31, 33, 37, 41, 43, 49, 53, 61, 65, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 121, 127, 129, 131, 137, 139, 145, 149, 151, 157, 161, 163, 169, 173, 177, 181, 191, 193, 197, 199, 201, 209, 211, 217, 223, 229, 233, 239, 241, 251, 253, 257, 265, 269, 271, 277, 281, 283, 289, 293, 301, 305, 307, 311, 313

Offset: 1

Antti Karttunen, Dec 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12313; all terms <= 65537

Cf. A001222.
Subsequence of A339911 and of A339910.
Cf. also A339908.

Select[Range[3, 313, 2], PrimeOmega[#] < PrimeOmega[# - 1]/2 &] (* Michael De Vlieger, Dec 22 2020 *)
Flatten[Position[Partition[PrimeOmega[Range[400]],2,1],?(#[[2]]<#[[1]]/2&),1,Heads->False]]+1 (* _Harvey P. Dale, Jan 11 2024 *)

isA339912(n) = ((n>1)&&((2*bigomega(n))

cp's OEIS Frontend

A339909 Carmichael numbers k for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

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A339907 Odd squarefree numbers k > 1 for which the bigomega(phi(k)) <= bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

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A339911 Numbers k > 1 for which bigomega(k) <= bigomega(k-1)/2, where bigomega gives the number of prime factors, counted with multiplicity.

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A339912 Numbers k > 1 for which bigomega(k) < bigomega(k-1)/2, where bigomega gives the number of prime factors, counted with multiplicity.

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