A377164 a(n) is the least exponent k such that the binary representation of n occurs at exactly n positions within the binary representation of n^k, or -1 if this is not possible.

1, -1, 10, -1, 11, 30, 9, -1, 36, 51, 38, 80, 44, 70, 41, -1, 96, 97, 118, 203, 138, 123, 104, 321, 152, 202

Offset

1,3

Comments

After a(27), which seems to be -1 (proof or counterexample needed), the sequence continues 258, 158, 185, 137, -1, 372, 374, 336, 505, 350, 435, 404, 876, 435, 365, 385, 657, 405, 540, 396, 1298, 465, 659, 488, 647, 482, 694, 557, 1049, 549, 725, 581, 861, 558, 638, 636, -1, ... .

Links

Table of n, a(n) for n=1..26.

Example

Overlapping occurrences are counted.
a(3) = 10: 3 = 11_2, 3^10 = 59049 = 1110011010101001_2,
3 occurrences of 11_2:              ^^   ^
a(5) = 11: 5 = 101_2, 5^11 = 48828125 = 10111010010000111011011101_2,
5 occurrences of 101_2:                 ^   ^           ^  ^   ^
a(7) = 9: 7 = 111_2, 7^9 = 40353607 = 10011001111011111101000111
7 occurrences of 111_2:                      ^^   ^^^^       ^

Prog

(PARI) findstr(m, n) = my(dm=digits(m, 2), dn=digits(n, 2)); sum(j=1, #dm-#dn+1, dn==dm[j..j+#dn-1]);
a377164(n) = if(n>1 && n==2^valuation(n, 2), -1, for(x=1, oo, if(findstr(n^x, n)==n, return(x))))

Crossrefs

Cf. A377163.

Sequence in context: A370401 A172171 A327723 * A164899 A164844 A287015

Adjacent sequences: A377161 A377162 A377163 * A377165 A377166 A377167

Keyword

base,sign,more

Author

Hugo Pfoertner, Nov 01 2024

Status

approved