%I #26 Jul 31 2022 07:47:17

%S 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,101,111,121,131,141,

%T 151,161,171,181,191,202,212,222,232,242,252,262,272,282,292,303,313,

%U 323,333,343,353,363,373,383,393,404,414,424,434,444,454,464,474,484,494,505,515,525,535,545,555,565,575,585,595,606,616,626

%N Curious numbers base 10.

%C Palindromes in base 10 of the form a_m b_n a_m, where a,b,m,n are nonnegative integers, 0<a<=9, 0<=b<=9, a_m denotes the repdigit a..a of length m (similarly for b_n), and juxtaposition denotes concatenation.

%C By definition, a_0 b_n a_0 := b_n. Thus each repdigit is a curious number. Also, by definition, a_m b_0 a_m := a_{2m}.

%C Named after Ian Stewart's "Calculator Curiosity 1".

%D I. Stewart, Professor Stewart’s Hoard of Mathematical Treasures, Basic Books, 2010, page 7.

%H Michael S. Branicky, <a href="/A335779/b335779.txt">Table of n, a(n) for n = 1..10010</a>

%H Neelima Borade and Jacob Mayle, <a href="https://arxiv.org/abs/2006.08083">Curious squares</a>, arXiv:2006.08083 [math.NT], 2020. Defines this sequence and determines its squares.

%F a_m b_n a_m = 1/9 * (N_{a,b,m} * 10^n + M_{a,b,m}) where M_{a,b,m} := 10^m *(a - b) - a and N_{a,b,m} := 10^m * (a*10^m + b - a).

%e 3 is a curious number since 3 = 5_0 3_1 5_0 (for instance),

%e 44944 is a curious number since 44944 = 4_2 9_1 4_2,

%e 7111117 is a curious number since 7111117 = 7_1 1_5 7_1,

%e 10101 is the smallest palindrome that is not a curious number,

%e 12321 is an example of a palindrome that is not a curious number, and

%e 11121 is not a palindrome (and hence also not a curious number).

%t curQ[n_] := PalindromeQ[n] && Length @ Split @ IntegerDigits[n] < 4; Select[Range[0, 1000], curQ] (* _Amiram Eldar_, Jun 23 2020 *)

%o (Python)

%o from itertools import count, islice, product

%o def agen(): # generator of terms

%o digs, nzdigs = "0123456789", "123456789"

%o yield from map(int, digs)

%o for d in count(2):

%o yield from sorted(set(int(a*m+b*(d-2*m)+a*m) for m in range(d//2+1) for a in nzdigs for b in digs)-{0})

%o print(list(islice(agen(), 72))) # _Michael S. Branicky_, Jul 29 2022

%Y Subsequence of A002113.

%Y Supersequence of A010785.

%K nonn,base

%O 1,3

%A _Jacob Mayle_, Jun 22 2020