A375387 - OEIS

%I #43 Sep 03 2024 01:36:31

%S -1,130,41,123,16,170,-1,55,155,39,274,239,96,187,494,2925,685,1784,

%T 1389,859,599,1779,1978,989,6597,5887,6968,8499,5989,17969,29859,

%U 17899,28898,435897,38989,2089469,1788960,498847,2886278,487878,919996,4098689,898794,1896967

%N a(n) is the least number k whose sum of digits in base 10 is n and that is palindromic in base n, or -1 if no such number exists.

%C A positive integer that is a multiple of 3 ends with 0 in base 3, so it cannot be a palindrome in base 3.

%C A positive integer that is a multiple of 9 ends with 0 in base 9, so it cannot be a palindrome in base 9.

%C From _Michael S. Branicky_, Aug 15 2024: (Start)

%C Regarding a(2): To be a palindrome in base 2, it must end with 1, hence odd. To be odd and have digit sum 2 in base 10, it must be of the form t_d = 10^(d-1) + 1, d > 1 (a d-digit base-10 number).  t_d is not divisible by 3, and base-2 palindromes with even length (i.e., number of binary digits) are divisible by 3, so, if a(2) exists, it must be a base-2 palindrome with odd length.

%C Computer search shows no such terms with d <= 10^6, so a(2), if it exists, has > 10^6 decimal digits. (End)

%H Michael S. Branicky, <a href="/A375387/b375387.txt">Table of n, a(n) for n = 3..149</a>

%H Michael S. Branicky, <a href="/A375387/a375387.py.txt">Python programs for OEIS A375387</a>

%H Jean-Marc Rebert, <a href="/A375387/a375387.txt">a375387_1e10</a>

%e a(5) = 41, because 4 + 1 = 5 and 41 = 131_5, and no lesser number has this property.

%e First terms are:

%e   130 = 2002_4

%e   41  = 131_5

%e   123 = 3323_6

%e   16  = 22_7

%e   170 = 252_8

%o (PARI) isok(k, n) = if (sumdigits(k)==n, my(d=digits(k, n)); d==Vecrev(d));

%o a(n) = if ((n==3) || (n==9), return((-1))); my(k=1); while (!isok(k,n), k++); k; \\ _Michel Marcus_, Aug 13 2024

%o (Python) # see Links for faster variants

%o from itertools import count

%o from sympy.ntheory import is_palindromic

%o def a(n):

%o     if n in {3, 9}: return -1

%o     return next(k for k in count(10**(n//9)-1) if sum(map(int, str(k)))==n and is_palindromic(k, n))

%o print([a(n) for n in range(3, 47)]) # _Michael S. Branicky_, Aug 13 2024

%Y Cf. A002113, A007953, A051885, A065531, A107129, A228915, A253594.

%K sign,base

%O 3,2

%A _Jean-Marc Rebert_, Aug 13 2024