A336237 - OEIS

%I #15 Sep 08 2022 08:46:25

%S 24,26,87,154,165,209,250,364,440,448,644,875,1240,1252,1269,1320,

%T 1434,1632,1640,1768,1996,2440,2480,2500,2656,2840,2842,4040,4400,

%U 5240,6040,6499,6544,7240,7640,8250,8360,8420,8440,8727,8832,8875,9640,10040,10344,10840

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

%C From _Marius A. Burtea_, Aug 04 2020: (Start)

%C 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).

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

%C 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)

%H Marius A. Burtea, <a href="/A336237/b336237.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%o (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)););}

%o (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

%Y Cf. A000010, A336191, A336192.

%K nonn,base

%O 1,1

%A _Michel Marcus_, Jul 13 2020