A350989 Numbers k such that both k and the k-th triangular number are binary palindromes.

0, 1, 5, 9, 17, 21, 33, 65, 129, 257, 341, 513, 693, 1025, 1365, 1397, 2049, 4097, 8193, 16385, 21845, 32769, 43605, 65537, 87125, 87381, 131073, 262145, 524289, 1048577, 1398101, 2097153, 2796885, 4194305, 5592405, 5594453, 8388609, 16777217, 33554433, 67108865

Offset: 1

Author

Amiram Eldar, Jan 28 2022

Keywords

• nonn
• base

Comments

This sequence is infinite since 2^k+1 is a term for all k>1.

Examples

			5 is a term since 5 = 101_2 is a binary palindromic number and A000217(5) = 5*(5+1)/2 = 15 = 1111_2 is a triangular number and also a binary palindromic number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100

Crossrefs

The binary version of A008510.

Intersection of A006995 and A350988.

A000051 \ {3} is a subsequence.

Cf. A229236, A350987.

Programs

Mathematica

Select[Range[0, 10^6], And @@ PalindromeQ /@ IntegerDigits[{#, #*(# + 1)/2}, 2] &]

PARI

isok(k) = my(bt=binary(k*(k+1)/2), bk=binary(k)); (bt == Vecrev(bt)) && (bk==Vecrev(bk)); \\ Michel Marcus, Jan 28 2022

Python

from itertools import count, islice
def ispal(s): return s == s[::-1]
def ok(n): return ispal(bin(n)[2:]) and ispal(bin(n*(n+1)//2)[2:])
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 28 2022