A386220 a(n) = d + 2*2^e where n = d*2^e with d = A000265(n) odd.

3, 5, 5, 9, 7, 7, 9, 17, 11, 9, 13, 11, 15, 11, 17, 33, 19, 13, 21, 13, 23, 15, 25, 19, 27, 17, 29, 15, 31, 19, 33, 65, 35, 21, 37, 17, 39, 23, 41, 21, 43, 25, 45, 19, 47, 27, 49, 35, 51, 29, 53, 21, 55, 31, 57, 23, 59, 33, 61, 23, 63, 35, 65, 129, 67, 37, 69, 25, 71, 39

Offset

1,1

Comments

a(n) is odd, by construction, and can be defined as a(n) = n/2^e + 2*2^e for the largest e >= 0 which makes a(n) an integer (an odd integer), that being e = A007814(n).

Every positive odd number k appears exactly M times in this sequence, where M = floor(log_2(k)), being at n = (k - 2*2^e)*2^e for 0 <= e < M, which is n = A386386(i) + 1 for terms there of odd k >= 3 and m = e + 1 (consecutive terms).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000265(n) + A171977(n).

Mathematica

OddPart[n_]:=n/2^IntegerExponent[n, 2]; a[n_]:=OddPart[n]+2n/OddPart[n]; Array[a, 70] (* Stefano Spezia, Aug 20 2025 *)

Prog

(PARI) a(n) = my(o=n >> valuation(n, 2)); o + 2*n/o; \\ Michel Marcus, Aug 20 2025

Crossrefs

Cf. A000265 (odd part), A007814 (2-adic valuation), A171977, A386386.

Sequence in context: A229428 A388898 A348374 * A029639 A087349 A212631

Adjacent sequences: A386217 A386218 A386219 * A386221 A386222 A386223

Keyword

nonn,look

Author

Thomas Ordowski, Aug 20 2025

Extensions

More terms from Michel Marcus, Aug 20 2025

Status

approved