A273292 Numbers n such that n = Sum_{i=1..j} (phi(n) mod d(i)), where phi(n) is the Euler totient function of n and d(i) are the divisors of n.

80, 400, 693, 2000, 3290, 7030, 10000, 24150, 50000, 191961, 250000, 610718, 1214425, 1250000, 2194778, 6250000, 31250000, 75369362, 156250000, 234392726, 572760397, 588270806, 590434574, 595208594, 781250000, 1547001099, 2889682738, 3906250000, 7627258546, 10614420142

Offset

1,1

Comments

This sequence is infinite, since it contains all the numbers of the form 2^4*5^k, for k>0. The key fact is that phi(2^4*5^k) = 2^5*5^(k-1), so only 5 moduli do not vanish. - Giovanni Resta, May 25 2016

Links

Table of n, a(n) for n=1..30.

Example

Divisors of 693 are 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693 and its Euler totient function is phi(693) = 360. Then 360 mod 1 = 0, 360 mod 3 = 0, 360 mod 7 = 3, 360 mod 9 = 0, 360 mod 11 = 8, 360 mod 21 = 3, 360 mod 33 = 30, 360 mod 63 = 45, 360 mod 77 = 52, 360 mod 99 = 63, 360 mod 231 = 129, 360 mod 693 = 360 and 0 + 0 + 3 + 0 + 8 + 3 + 30 + 45 + 52 + 63 + 129 + 360 = 693.

Maple

with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do a:=divisors(n);
if add(phi(n) mod a[k], k=1..nops(a))=n then print(n); fi; od; end: P(10^9);

Mathematica

Select[Range[10^5], # == Plus @@ Mod[EulerPhi@ #, Divisors@ #] &] (* Giovanni Resta, May 25 2016 *)

Crossrefs

Cf. A000010.

Sequence in context: A233951 A045666 A045657 * A234333 A234326 A085774

Adjacent sequences: A273289 A273290 A273291 * A273293 A273294 A273295

Keyword

nonn

Author

Paolo P. Lava, May 19 2016

Extensions

a(17)-a(30) from Giovanni Resta, May 25 2016

Status

approved