cp's OEIS Frontend

A143869 An integer k is called regular (mod n) if there is an integer x such that k^2 x == k (mod n). Then these numbers are the sum of regular integers k (mod n) such that 1 <= k <= n for n=1,2,... .

%I A143869 #32 May 30 2025 03:14:06
%S A143869 1,3,6,8,15,21,28,24,36,55,66,60,91,105,120,80,153,135,190,160,231,
%T A143869 253,276,192,275,351,270,308,435,465,496,288,561,595,630,396,703,741,
%U A143869 780,520,861,903,946,748,810,1081,1128,672,1078,1075,1326,1040,1431,1053
%N A143869 An integer k is called regular (mod n) if there is an integer x such that k^2 x == k (mod n). Then these numbers are the sum of regular integers k (mod n) such that 1 <= k <= n for n=1,2,... .
%H A143869 Amiram Eldar, <a href="/A143869/b143869.txt">Table of n, a(n) for n = 1..10000</a>
%H A143869 Osama Alkam and Emad Abu Osba, <a href="http://journals.tubitak.gov.tr/math/issues/mat-08-32-1/mat-32-1-4-0610-4.pdf">On the regular elements in Zn</a>, Turk J Math, 32 (2008), 31-39.
%H A143869 B. Apostol and L. Petrescu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Apostol/apostol3.html">Extremal Orders of Certain Functions Associated with Regular Integers (mod n)</a>, Journal of Integer Sequences, 2013, # 13.7.5.
%H A143869 L. Tóth, <a href="https://ac.inf.elte.hu/Vol_029_2008/doi/263_29.pdf">Regular integers modulo n</a>, Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275.
%F A143869 a(n) = n*(A055653(n)+1)/2.
%o A143869 (PARI) isregu(k, n) = {g = gcd(k, n); if ((n % g == 0) && (gcd(g, n/g) == 1), return(k), return(0));}
%o A143869 a(n) = sum(k=1, n, isregu(k, n)) \\ _Michel Marcus_, May 25 2013
%Y A143869 Cf. A055653, A176345.
%K A143869 nonn
%O A143869 1,2
%A A143869 _Laszlo Toth_, Sep 04 2008
%E A143869 Extended by _R. J. Mathar_, Sep 05 2008