A336191 Numbers k of the form k = ab (the decimal concatenation of a and b) such that phi(ab) = a*b + 1.
57, 195, 319, 5595, 11709, 77097, 114765, 1313667, 1348559, 4752465, 10219099, 11031119, 185573199, 2918945715, 3165616929, 12233666703, 16996664613, 18052811909, 20650199699, 38081370319, 58943659521, 195823876095, 236323024041, 242687655369, 342764528277, 677924155713
Offset: 1
Examples
phi(57) = 5 * 7 + 1 phi(195) = 1 * 95 + 1 = 19 * 5 + 1 phi(319) = 31 * 9 + 1 phi(5595) = 5 * 595 + 1 phi(11709) = 11 * 709 + 1 phi(77097) = 7 * 7097 + 1 phi(114765) = 11 * 4765 + 1 phi(1313667) = 1313 * 667 + 1 phi(1348559) = 134855 * 9 + 1 phi(4752465) = 47 * 52465 + 1 phi(10219099) = 1021 * 9099 + 1 phi(11031119) = 1103111 * 9 + 1 phi(185573199) = 185 * 573199 + 1
Programs
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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] - 1]]; Select[Range[10, 10^5], seqQ] (* Amiram Eldar, Jul 11 2020 *) -
PARI
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
Extensions
Missing terms 1348559 & 4752465, and a(12) from Amiram Eldar, Jul 11 2020
More terms from Giovanni Resta, Jul 13 2020
Comments