cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 22 2020

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Antti Karttunen, Dec 20 2020

Keywords

Links

Crossrefs

Intersection of A002997 and A339817 (see comments in latter).
Cf. also A339869, A339878, A339909.

Programs

  • Mathematica
    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 *)
  • PARI
    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))));

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 22 2020

Keywords

Comments

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.

Links

Crossrefs

Intersection of A002997 and A339908.
Cf. A000010, A001222, A002322.
Cf. also A339818, A339869, A339878.

Programs

  • Mathematica
    carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; Select[carmichaels, PrimeOmega[EulerPhi[#]] < PrimeOmega[# - 1] &] (* Amiram Eldar, Dec 26 2020 *)
  • PARI
    A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA339909(n) = ((n>1)&&issquarefree(n)&&!isprime(n)&&(bigomega(eulerphi(n))A002322(n))));

A338998 Composite numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1.

Original entry on oeis.org

1729, 12801, 5079361, 34479361, 3069196417, 23915494401
Offset: 1

Views

Author

Tomohiro Yamada, Nov 18 2020

Keywords

Comments

All terms of this sequence are terms of A337316 and all Lehmer numbers (if there are any) are contained in this sequence.
Terms 1729 and 3069196417 and several others are also Carmichael numbers (A002997), they are given in A339878.
The sequence also includes: 1334063001601, 6767608320001, 33812972024833, 380655711289345, 1584348087168001, 1602991137369601, 6166793784729601, 1531757211193440001. - Daniel Suteu, Nov 24 2020
Apparently, a(n) == 1 (mod 64). - Hugo Pfoertner, Dec 08 2020
The "Lehmer numbers" above refers to composite 1-Lehmer numbers, that is, numbers n that would satisfy the equation y * phi(n) = n-1, for some integer y > 1. Lehmer conjectured that no such numbers exist. See the assorted Web-links. - Antti Karttunen, Dec 26 2020

Examples

			phi(1729) = 1296 divides 3 * 1728.

Links

Crossrefs

Subsequence of A173703 (2-Lehmer numbers).
Cf. A337316 (with "squarefree divisor" instead of "prime factor").
Cf. A000010 (phi), A238574 (k-Lehmer numbers for some k), A339878 (Carmichael numbers in this sequence).

Programs

  • PARI
    is(n)={my(s=denominator((n-1)/eulerphi(n))); !isprime(n) && isprime(s) && ((n-1)%s==0) && n>1}
{ forcomposite(n=1, 2^32, if(is(n), print1(n, ", "))) }

Extensions

a(5) from Amiram Eldar, Nov 18 2020
a(6) from Daniel Suteu, confirmed by Max Alekseyev, Sep 29 2023
Showing 1-4 of 4 results.

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