A336237 Numbers of the form ab such that phi(ab) = a*b where ab is the concatenation of a and b.

24, 26, 87, 154, 165, 209, 250, 364, 440, 448, 644, 875, 1240, 1252, 1269, 1320, 1434, 1632, 1640, 1768, 1996, 2440, 2480, 2500, 2656, 2840, 2842, 4040, 4400, 5240, 6040, 6499, 6544, 7240, 7640, 8250, 8360, 8420, 8440, 8727, 8832, 8875, 9640, 10040, 10344, 10840

Offset: 1

Author

Michel Marcus, Jul 13 2020

Keywords

• nonn
• base

Comments

From Marius A. Burtea, Aug 04 2020: (Start)

The sequence is infinite because it contains the family of terms 25 * 10^k, k >= 1. Indeed, phi(25 * 10^k) = phi(2^k * 5^(k + 2)) = 2^(k-1) * 4 * 5^(k + 1) = 2 * (5 * 10^k).

More generally, if 10 * m is a term then 10^k * m, k >= 1, is a term.

For example, for k = 2, let 10 * m = a_b and phi(10 * m) = a * b. If m = 2^u * 5^v * s, with u, v >= 0 and gcd(10, s) = 1, then phi(100 * m) = phi(100 * 2^u * 5^v * s) = phi(2^(u + 2) * 5^(v + 2) * s) = 2^(u + 1) * 5^(v + 1) * 4 * phi(s) = 10 * 2^u * 5^v * 4 * phi(s) = 10 * phi(2^(u + 1) * 5^(v + 1) * s) = 10 * phi(10 * m) = 10 * a_b = a_(10*b). (End)

Examples

			2*4=8 and phi(24)=8 so 24 is a term.
		

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A336191, A336192.

Programs

Magma

a:=[]; for n in [1..11000] do s:=#Intseq(n); if exists(c){i:i in [1..s-1]| ((n mod 10^i)*(n div 10^i)) eq EulerPhi(n)} then Append(~a,n); end if; end for; a; // Marius A. Burtea, Aug 04 2020

Mathematica

seqQ[n_] := Module[{d = IntegerDigits[n]}, MemberQ[Times @@@ Table[FromDigits /@ {Take[d, k], Take[d, -Length[d] + k]}, {k, 1, Length[d] - 1}], EulerPhi[n]]]; Select[Range[10, 10^4], seqQ] (* Amiram Eldar, Jul 13 2020 *)

PARI

isok(m) = {my(tm=eulerphi(m), d=digits(m)); for (i=1, #d-1, if (fromdigits(vector(i, k, d[k]))*fromdigits(vector(#d-i, k, d[i+k])) == tm, return(1)););}