A335779 Curious numbers base 10.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 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

Offset

1,3

Comments

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.

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}.

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

References

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

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10010

Neelima Borade and Jacob Mayle, Curious squares, arXiv:2006.08083 [math.NT], 2020. Defines this sequence and determines its squares.

Formula

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).

Example

3 is a curious number since 3 = 5_0 3_1 5_0 (for instance),
44944 is a curious number since 44944 = 4_2 9_1 4_2,
7111117 is a curious number since 7111117 = 7_1 1_5 7_1,
10101 is the smallest palindrome that is not a curious number,
12321 is an example of a palindrome that is not a curious number, and
11121 is not a palindrome (and hence also not a curious number).

Mathematica

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

Prog

(Python)
from itertools import count, islice, product
def agen(): # generator of terms
    digs, nzdigs = "0123456789", "123456789"
    yield from map(int, digs)
    for d in count(2):
        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})
print(list(islice(agen(), 72))) # Michael S. Branicky, Jul 29 2022

Crossrefs

Subsequence of A002113.

Supersequence of A010785.

Sequence in context: A147882 A002113 A227858 * A240601 A324988 A276354

Adjacent sequences: A335776 A335777 A335778 * A335780 A335781 A335782

Keyword

nonn,base

Author

Jacob Mayle, Jun 22 2020

Status

approved