|
%I #10 Oct 28 2021 01:59:42
%S 13,17,18,19,26,31,37,39,48,49,56,62,65,71,73,79,81,84,91,93,94,97,103
%N Integers for which there exists a self-repetition that is a term of A338166.
%H Daniel Tsai, <a href="https://arxiv.org/abs/2010.03151">A recurring pattern in natural numbers of a certain property</a>, arXiv:2010.03151 [math.NT], 2020.
%H Daniel Tsai, <a href="http://math.colgate.edu/~integers/v32/v32.mail.html">A recurring pattern in natural numbers of a certain property</a>, Integers (2021) Vol. 21, Article #A32.
%e 18 is a term since 1818 is a term of A338166.
%e 48 is a term since 484848 is a term of A338166.
%e List of terms with their minimum number of repetitions : [13, 15], [17, 280], [18, 2], [19, 819], [26, 15], [31, 15], [37, 12], [39, 15], [48, 3], [49, 3243], [56, 3], [62, 15], [65, 3], [71, 280], [73, 12], [79, 624], [81, 2], [84, 3], [91, 819], [93, 15], [94, 3243], [97, 624], [103, 10234].
%o (PARI) f(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2])); \\ A338038
%o period(vp,n) = {my(p = 1, pten = 10^#Str(n)); for (i=1, #vp, if ((vp[i] != 2) && (vp[i] != 5), p = lcm(p, znorder(Mod(pten, vp[i]))); p = lcm(p, znorder(Mod(pten, vp[i]^2))););); p;}
%o isok(n) = {my(r = fromdigits(Vecrev(digits(n)))); my(vp = setunion(factor(n)[,1]~, factor(r)[,1]~)); my(nbmax = period(vp, n)); if (nbmax == 1, nbmax = 2); my(krep=1); my(pten = 10^#Str(n)); for (k=2, nbmax, krep = pten*krep+1; my(q=1); for (i=1, #vp, my(va = valuation(krep, vp[i])); q *= vp[i]^va;); if (f(n*q) == f(r*q), return(k);););}
%o ispal(n) = my(d=Vecrev(digits(n))); n == fromdigits(d);
%o lista(nn) = {for (n=1, nn, if ((n % 10) && !ispal(n), if (isok(n), print1(n, ", "));););}
%Y Cf. A338038, A338039, A338166.
%K nonn,base,more
%O 1,1
%A _Michel Marcus_, Oct 23 2020
|
