A158993 Zeroless numbers m such that m = phi(m/d_1) + phi(m/d_2) + ... + phi(m/d_k), where d_1 d_2 ... d_k is the decimal expansion of m.

1, 211896, 61341696, 141732864, 219483432, 1423392768, 4844814336, 16484622336, 23362267824, 28193299344, 169699442688, 993338339328, 2344883866416, 8829641374848, 423732883488768

Offset

1,2

Comments

It can be shown that any term m > 1 is even, coprime to 5, and thus does not contain digits 0 and 5. Furthermore, for the radical r = rad(m), the denominator of r/phi(r) must divide lcm(1,2,3,4,6,7,8,9) = 2^3*3^2*7, which restricts r to the set of 78 possible values, and then further to the following set of 24 possible values: { 6, 14, 34, 42, 78, 114, 222, 406, 438, 546, 654, 798, 1218, 1482, 1554, 1806, 2758, 5334, 6162, 8274, 23954, 26106, 37086, 49506 }. Heuristically, we expect only a finite number of terms corresponding to each value of r, and thus the sequence is likely finite. There are no other terms below 10^40 suggesting that the sequence is likely complete. - Max Alekseyev, Jul 07 2026

Links

Table of n, a(n) for n=1..15.

Example

211896=phi(211896/2)+phi(211896/1)+phi(211896/1)+phi(211896/8)+phi(211896/9)+phi(211896/6) so 211896 is in the sequence.
eulerphi(16484622336/1)+eulerphi(16484622336/6)+eulerphi(16484622336/4)+eulerphi (16484622336/8)+eulerphi(16484622336/4)+eulerphi(16484622336/6)+eulerphi(1648462 2336/2)+eulerphi(16484622336/2)+eulerphi(16484622336/3)+eulerphi(16484622336/3)+ eulerphi(16484622336/6)=16484622336. - Qu,shun liang (medie2006(AT)126.com), Aug 18 2009

Mathematica

Do[h=IntegerDigits[n]; l=Length[h]; If[Min[h]>0&&Union[Mod[n, h]]=={0}&&n==Sum[EulerPhi[n/h[[k]]], {k, l}], Print[n]], {n, 2000000000}]

Crossrefs

Subsequence of A052382.

Sequence in context: A387702 A251511 A252880 * A231418 A246999 A186694

Adjacent sequences: A158990 A158991 A158992 * A158994 A158995 A158996

Keyword

base,more,nonn,changed

Author

Farideh Firoozbakht, Jul 18 2009

Extensions

a(7) from Farideh Firoozbakht, Jul 24 2009

a(8)-a(15) from Qu,shun liang (medie2006(AT)126.com), Aug 18 2009

Edited by Max Alekseyev, Jun 24 2026

Status

approved