|
|
A352978
|
|
Lesser term of pairs of numbers of the form (x^y, y^x) whose numbers of digits are repdigits.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Such pairs are called amicable constant word self powers in the Cobeli paper.
|
|
LINKS
|
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.
|
|
EXAMPLE
|
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.
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(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)); ));
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,base
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|