A392102 Squares which are the concatenation of two or more powers of 2.

81, 121, 144, 324, 441, 484, 841, 1444, 1681, 2116, 8281, 8464, 11664, 11881, 12321, 14161, 14641, 14884, 16641, 28224, 41616, 43264, 48841, 114244, 116281, 128164, 128881, 142884, 163216, 166464, 221841, 228484, 232324, 412164, 484416, 646416, 824464, 848241, 1024144

Offset

1,1

Comments

No concatenated part can begin with a 0, but 0's are permitted within a part; i.e., within a power of 2.

Sequence is infinite since it contains A272884, which is infinite as shown there. - Michael S. Branicky, Dec 31 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Formula

a(n) = A392101(n)^2.

Example

324 is a term because it is 18^2 and it can be split up into powers of 2: 32|4.
12321 is a term because it is 111^2 and it can be split up into powers of 2: 1|2|32|1.
2401 is not a term because although 49^2 = 2401, it cannot be split into powers of 2 without using leading zeros (2|4|01), which is disallowed.
However, 1024144 is a term because it is not split into 1|02|4|1|4|4, but 1024|1|4|4.
Term 128164 is the first that can be split in 2 different ways: 128|16|4, 128|1|64.

Prog

(Python)
pow2strings = {str(2**i) for i in range(4000)}
def split(s, has_split):
    global pow2strings
    if s in pow2strings: return has_split
    return any(s[:i] in pow2strings and split(s[i:], True) for i in range(1, len(s)))
def ok(n): return split(str(n), False)
print([k**2 for k in range(1, 1013) if ok(k**2)]) # Michael S. Branicky, Dec 31 2025

Crossrefs

Cf. A392101.

Supersequence of A272884, minus {1, 4}.

Subsequence of the intersection of A381259 and A000290.

Sequence in context: A039546 A223020 A357168 * A253261 A280587 A273372

Adjacent sequences: A392099 A392100 A392101 * A392103 A392104 A392105

Keyword

nonn,base

Author

Rhys Feltman, Dec 30 2025

Extensions

Terms corrected and more terms from Michael S. Branicky, Dec 31 2025

Status

approved