A343903 - OEIS

%I #12 Dec 06 2024 11:28:05

%S 1,2,6,4,11,8,17,16,10,13,16,15,44,21,24,23,43,20,42,22,67,28,31,30,

%T 33,63,57,60,36,10,100,44,48,52,91,83,162,80,98,59,104,109,96,71,111,

%U 49,52,74,54,57,61,64,65,61,119,71,190,72,67,76,275,147,219,79

%N a(n) = Sum{i=1..A343901(n)} n (mod A343902(n, i)^2).

%C A theorem of Pomerance says that if a(n) = 0, then A000010(x) = A000010(y) has only the solution y = x (cf. Pomerance, 1974).

%H Carl Pomerance, <a href="https://doi.org/10.1090/S0002-9939-1974-0340161-0">On Carmichael's conjecture</a>, Proceedings of the American Mathematical Society 43 (1974), 297-298.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Carmichael%27s_totient_function_conjecture">Carmichael's totient function conjecture</a>.

%t a[n_] := DivisorSum[EulerPhi[n], Mod[n, (#+1)^2] &, PrimeQ[# + 1] &]; Array[a, 100] (* _Amiram Eldar_, Dec 06 2024 *)

%o (PARI) row_a343902(n) = my(e=eulerphi(n), v=[]); forprime(p=1, e+1, if(e%(p-1)==0, v=concat(v, [p]))); v \\ corrected by _Amiram Eldar_, Dec 06 2024

%o a(n) = my(v=row_a343902(n)); sum(i=1, #v, n%v[i]^2)

%o (PARI) a(n) = sumdiv(eulerphi(n), d, isprime(d+1) * n % (d+1)^2); \\ _Amiram Eldar_, Dec 06 2024

%Y Cf. A000010, A343901, A343902.

%K nonn

%O 1,2

%A _Felix Fröhlich_, May 03 2021

%E Offset changed to 1 and data corrected by _Amiram Eldar_, Dec 06 2024