|
%I #41 Jun 23 2022 17:46:32
%S 1,2,4,8,272,25952,2112,4224,8448,44544,2532352,6547456,405504,
%T 236101632,8634368,487030784,677707776,61486268416,25135153152,
%U 4285005824,2912594952192,63115033051136,2114151514112,65892155129856,29142024192,827163020361728,42643629092634624
%N a(n) is the smallest positive palindromic number whose binary expansion ends in exactly n zeros, or 0 if no such number exists.
%C Equivalently, a(n) is the smallest base-10 palindrome of the form (2k+1)*2^n.
%C Conjecture: a(n) > 0 for all n >= 0.
%H Chai Wah Wu, <a href="/A302864/b302864.txt">Table of n, a(n) for n = 0..40</a>
%F If A082613(n)/A082614(n) = 2^n (see conjecture at A082613) and a(n) > 0, then a(n) >= A082613(n). - _Altug Alkan_, Apr 15 2018
%e n a(n) binary expansion of a(n)
%e == ============= ==========================================
%e 0 1 1
%e 1 2 10
%e 2 4 100
%e 3 8 1000
%e 4 272 100010000
%e 5 25952 110010101100000
%e 6 2112 100001000000
%e 7 4224 1000010000000
%e 8 8448 10000100000000
%e 9 44544 1010111000000000
%e 10 2532352 1001101010010000000000
%e 11 6547456 11000111110100000000000
%e 12 405504 1100011000000000000
%e 13 236101632 1110000100101010000000000000
%e 14 8634368 100000111100000000000000
%e 15 487030784 11101000001111000000000000000
%e 16 677707776 101000011001010000000000000000
%e 17 61486268416 111001010000110111100000000000000000
%e 18 25135153152 10111011010001011000000000000000000
%e 19 4285005824 11111111011010000000000000000000
%e 20 2912594952192 101010011000100100001100000000000000000000
%o (PARI) isp(n) = n=digits(n); for(i=1, #n\2, if(n[i]!=n[#n+1-i], return(0))); 1;
%o a(n) = forstep(k=1, oo, 2, if(isp(p=k*2^n), return(p))); \\ _Altug Alkan_, Apr 15 2018, after _Charles R Greathouse IV_ at A002113
%o (Python)
%o from itertools import count
%o def A302864(n):
%o for k in count(0,2):
%o if (s:=str(k+1<<n))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1]:
%o return k+1<<n # _Chai Wah Wu_, Jun 23 2022
%Y Cf. A002113, A082613, A082614.
%K nonn,base
%O 0,2
%A _Jon E. Schoenfield_, Apr 14 2018
%E a(26) from _Chai Wah Wu_, Mar 06 2019
| 