A215486 -id:A215486 - cp's OEIS frontend

A213514 For composite n, remainder of n - 1 when divided by phi(n), where phi(n) is the totient function (A000010).

1, 1, 3, 2, 1, 3, 1, 6, 7, 5, 3, 8, 1, 7, 4, 1, 8, 3, 5, 15, 12, 1, 10, 11, 1, 14, 7, 5, 3, 20, 1, 15, 6, 9, 18, 3, 17, 14, 7, 20, 1, 11, 1, 26, 31, 16, 5, 3, 24, 21, 23, 1, 34, 3, 16, 5, 15, 26, 1, 11, 20, 1, 30, 7, 17, 18, 3, 32, 1, 22, 31, 13, 38, 19, 5, 7, 8, 1, 35, 29, 38, 15, 5, 26, 3, 44, 1, 22, 23, 10

Offset: 1

Balarka Sen, Feb 15 2013

D. Lehmer conjectured that a(k) is never 0. He proved that if such k exists, the corresponding composite n must be odd, squarefree, and divisible by at least 7 primes. Cohen and Haggis showed that such n must be larger than 10^20 and have at least 14 prime factors.

			a(1) = 1 because the first composite number is 4 and 4 - 1 = 1 mod phi(4).
a(2) = 1 because the second composite is 6 and 6 - 1 = 1 mod phi(6).
a(3) = 3 because the third composite is 8 and 8 - 1 = 3 mod phi(8).

Balarka Sen, Table of n, a(n) for n = 1..999
Eric W. Weisstein, Totient Function (MathWorld)
Wikipedia, Euler's totient function

Cf. A000010, A068494, A215486.

DeleteCases[Table[Mod[n - 1, EulerPhi[n]] - Boole[PrimeQ[n]], {n, 100}], -1] (* Alonso del Arte, Feb 15 2013 *)
Mod[#-1,EulerPhi[#]]&/@Select[Range[200],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 14 2019 *)

for(n=1,300,if(isprime(n)==0,print1((n-1)%eulerphi(n)",")))

forcomposite(n=4,100,print1((n-1)%eulerphi(n)", ")) \\ Charles R Greathouse IV, Feb 19 2013

A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

Original entry on oeis.org

6601, 11972017, 34657141, 67902031, 139952671, 258634741, 2000436751, 8801128801, 9116583841, 9462932431, 38069223721, 326170416001, 359316634951, 1860929324101, 2022188518351, 2283475947391, 2648686458601, 2697891108151, 4513362899761, 5020030521001, 5472940991761, 6163867710001, 7507903975951, 19288340548471

Offset: 1

Robert Israel and Thomas Ordowski, Jul 25 2017

nonn

Numbers k such that A215486(k) = A002322(k).
Subsequence of the Carmichael numbers (A002997).
Composite numbers k such that (k-1) == lambda(k) (mod phi(k)).
Composite numbers k such that A277127(k) == 1 (mod A000010(k)).
Problem: are there infinitely many such numbers?
Conjecture: these are numbers k such that phi(k) + lambda(k) = k - 1. Checked up to 2^64. - Amiram Eldar and Thomas Ordowski, Dec 06 2019

Amiram Eldar, Table of n, a(n) for n = 1..239 (terms below 10^22 calculated using data from Claude Goutier; terms 1..79 from Robert Israel)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A000010, A002322, A002997, A215486.

Subsequence of A264012.

# Using data files for A002997
count:= 0:
for cfile in ["carmichael-16","carmichael17","carmichael18"] do
do
    S:= readline(cfile);
    if S = 0 then break fi;
    L:= map(parse, StringTools:-Split(S));
    n:= L[1]; pm:= map(`-`,L[2..-1],1);
    phin:= convert(pm,`*`);
    lambdan:= ilcm(op(pm));
    if n-1 - lambdan mod phin = 0 then
      count:= count+1; A[count]:= n;
    fi
od:
   fclose(cfile);
od:
seq(A[i],i=1..count); # Robert Israel, Jul 26 2017

Select[Range[10^8], Divisible[# - 1, (lam = CarmichaelLambda[#])] && Mod[# - 1, EulerPhi[#]] == lam &] (* Amiram Eldar, Dec 06 2019 *)

A268597 Smallest x such that x-1 mod phi(x) = n, or 0 if no such x exists.

Original entry on oeis.org

1, 4, 9, 8, 25, 18, 15, 16, 21, 50, 35, 36, 33, 98, 39, 32, 65, 54, 51, 100, 45, 70, 95, 72, 69, 338, 63, 196, 161, 110, 87, 64, 93, 130, 75, 108, 217, 182, 99, 200, 185, 170, 123, 140, 117, 190, 215, 144, 141, 250, 235

Offset: 0

Christina Steffan, Feb 08 2016

nonn

Conjecture: a(n) > 0 for all n.

Cf. A215486.

a(n) = {my(x = 1); while ((x-1) % eulerphi(x) != n, x++); x;} \\ Michel Marcus, Feb 27 2016

cp's OEIS Frontend

A213514 For composite n, remainder of n - 1 when divided by phi(n), where phi(n) is the totient function (A000010).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A290281 Numbers k such that (k-1) mod phi(k) = lambda(k), where phi = A000010 and lambda = A002322.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A268597 Smallest x such that x-1 mod phi(x) = n, or 0 if no such x exists.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI