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A213514
For composite n, remainder of n - 1 when divided by phi(n), where phi(n) is the totient function (A000010).
1
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
1,3
COMMENTS
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.
LINKS
Eric W. Weisstein, Totient Function (MathWorld)
EXAMPLE
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).
MATHEMATICA
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 *)
PROG
(PARI) for(n=1, 300, if(isprime(n)==0, print1((n-1)%eulerphi(n)", ")))
(PARI) forcomposite(n=4, 100, print1((n-1)%eulerphi(n)", ")) \\ Charles R Greathouse IV, Feb 19 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Balarka Sen, Feb 15 2013
STATUS
approved