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A262998
Composite numbers n such that Sum_{k=1..phi(n)} k^phi(n) == phi(n) (mod n), where phi(n) = A000010(n).
1
10, 26, 34, 58, 74, 82, 106, 122, 146, 178, 194, 202, 218, 226, 274, 298, 314, 320, 346, 362, 386, 394, 458, 466, 480, 482, 514, 538, 554, 562, 586, 626, 634, 674, 698, 706, 746, 778, 794, 802, 818, 842, 866, 898, 914, 922, 1018, 1042, 1082, 1114, 1138, 1154, 1186, 1202, 1226, 1234, 1282, 1306
OFFSET
1,1
COMMENTS
The terms a(18) = 320 and a(25) = 480 are not of the form 2p, where prime p == 1 (mod 4). - Altug Alkan, Oct 07 2015
The term a(662) = 22113 is the first odd term and the third one not of the form above. - Giovanni Resta, Oct 07 2015
If n == 1 (mod 4) is in the sequence, then so is 2n. - Thomas Ordowski, Oct 07 2015
LINKS
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.
FORMULA
{2 * A002144} U {320, 480, 22113, 44226, 66339, ?}.
EXAMPLE
For a(1) = 10; phi(10) = 4, 1^4 + 2^4 + 3^4 + 4^4 = 354 == 4 (mod 10).
MAPLE
filter:= proc(n) local p;
if isprime(n) then return false fi;
p:= numtheory:-phi(n);
evalb(add(i &^ p mod n, i=1..p) mod n = p)
end proc:
select(filter, [$2..2000]); # Robert Israel, Oct 07 2015
MATHEMATICA
Select[Range[2, 3000], !PrimeQ[#] && (p= EulerPhi@ #; Mod[ Sum[ PowerMod[k, p, #], {k, p}]-p, #] == 0) &] (* Giovanni Resta, Oct 07 2015 *)
PROG
(PARI) forcomposite(n=1, 3000, if(lift(sum(k=1, eulerphi(n), Mod(k, n)^eulerphi(n))) == eulerphi(n), print1(n", "))); \\ Altug Alkan, Oct 07 2015
CROSSREFS
Cf. A007850 (see Jonathan Sondow's comment, Jan 03 2014).
Sequence in context: A339977 A157075 A369668 * A245021 A045039 A322972
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Oct 07 2015
EXTENSIONS
More terms from Altug Alkan, Oct 07 2015
STATUS
approved