A336191 -id:A336191 - cp's OEIS frontend

A336192 Numbers of the form ab such that phi(ab) = a*b - 1 where ab is the concatenation of a and b.

1385, 1397, 15663, 19835, 37037, 238903, 719719, 1983035, 4337785, 5946445, 8099989, 15276063, 64438507, 97919791, 238639687, 325776657, 1926629941, 3228792383, 4387457627, 4652069941, 9801019901, 44898935609, 68135795923, 115563539473, 129898064149, 390084197561

Offset: 1

M. Farrokhi D. G., Jul 11 2020

Is the sequence infinite?

If phi(ab) = a*b - 1 then ab is a composite number.

			phi(1385) = 13 * 85 - 1
phi(1397) = 13 * 97 - 1
phi(15663) = 15 * 663 - 1
phi(19835) = 19 * 835 - 1
phi(37037) = 3703 * 7 - 1
phi(238903) = 23 * 8903 - 1
phi(719719) = 719 * 719 - 1
phi(1983035) = 19 * 83035 - 1
phi(4337785) = 4337 * 785 - 1
phi(5946445) = 5 * 946445 - 1
phi(8099989) = 809 * 9989 - 1
phi(15276063) = 1527 * 6063 - 1
phi(64438507) = 6443 * 8507 - 1
phi(97919791) = 9791 * 9791 - 1
phi(238639687) = 23 * 8639687 - 1
phi(325776657) = 32577 * 6657 - 1

Cf. A000010, A039649, A336191.

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] + 1]]; Select[Range[10, 10^5], seqQ] (* Amiram Eldar, Jul 11 2020 *)

isok(m) = {my(tm=eulerphi(m)+1, 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)););} \\ Michel Marcus, Jul 11 2020

Missing terms 1983035 & 5946445 from Amiram Eldar, Jul 11 2020

More terms from Giovanni Resta, Jul 13 2020

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

Original entry on oeis.org

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

Michel Marcus, Jul 13 2020

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)

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

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

Cf. A000010, A336191, A336192.

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

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 *)

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)););}

cp's OEIS Frontend

A336192 Numbers of the form ab such that phi(ab) = a*b - 1 where ab is the concatenation of a and b.

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