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A336237
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Numbers of the form ab such that phi(ab) = a*b where ab is the concatenation of a and b.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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)
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LINKS
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EXAMPLE
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2*4=8 and phi(24)=8 so 24 is a term.
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MATHEMATICA
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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 *)
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PROG
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(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)); ); }
(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
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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