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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 23:38 EST 2021. Contains 349558 sequences. (Running on oeis4.)