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

1,3

Comments

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

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

The binary version of A008510.

Intersection of A006995 and A350988.

A000051 \ {3} is a subsequence.

Cf. A229236, A350987.

Sequence in context: A022341 A255651 A216877 * A268756 A095725 A005006

Adjacent sequences: A350986 A350987 A350988 * A350990 A350991 A350992

Keyword

nonn,base

Author

Amiram Eldar, Jan 28 2022

Status

approved