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A352978
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Lesser term of pairs of numbers of the form (x^y, y^x) whose numbers of digits are repdigits.
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1
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8, 25, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 1000, 1156, 1225, 1296, 1331, 1728, 2197, 2401, 2744, 3375, 4096, 4900, 4913, 5041, 5184, 5329, 5832, 6561, 7776, 9261, 10000
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OFFSET
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1,1
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COMMENTS
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Such pairs are called amicable constant word self powers in the Cobeli paper.
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LINKS
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Cristian Cobeli, DOI^2, arXiv:1911.09003 [math.HO], 2019.
Cristian Cobeli, DOI^2, Romanian Journal Of Pure And Applied Mathematics, Tome LXVI, No. 3-4, 2021.
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EXAMPLE
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8=2^3 and 9=3^2 is such a pair because their decimal lengths are both 1, which is a repdigit; so 8 is a term.
81=2^9 and 512=9^2 is such a pair because their decimal lengths are 2 and 3, which are repdigits; so 81 is a term.
368^4=18339659776 and 4^368 is such a pair because their decimal lengths are 11 and 222, which are repdigits; so 18339659776 is a term. See Cobeli paper.
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MATHEMATICA
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repQ[n_] := Length[Union[IntegerDigits[IntegerLength[n]]]] == 1; q[n_] := n > 1 && repQ[n] && Module[{f = FactorInteger[n], e, g, d, x, y, ans = False}, e = f[[;; , 2]]; g = GCD @@ e; If[g > 1, Do[x = Surd[n, y]; If[y^x > n && repQ[y^x], ans = True; Break[]], {y, Rest @ Divisors[g]}]]; ans]; Select[Range[10000], q] (* Amiram Eldar, Apr 24 2022 *)
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PROG
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(PARI) iscw(n) = (#Set(digits(#Str(n))) == 1);
isokd(na, r, k) = fordiv(k, d, if (d < k, my(nb = (k/d)^(r^d)); if ((na < nb) && iscw(nb), return(1)); ); );
isok(na) = if (iscw(na), my(k, r); if (k=ispower(na, , &r), if (isokd(na, r, k), return(1)); ));
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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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