A354300 Numbers k such that k! and (k+1)! have the same binary weight (A000120).

0, 1, 3, 5, 7, 8, 12, 13, 15, 31, 63, 88, 127, 129, 131, 244, 255, 262, 263, 288, 300, 344, 511, 793, 914, 1012, 1023, 1045, 1116, 1196, 1538, 1549, 1565, 1652, 1817, 1931, 1989, 2047, 2067, 2096, 2459, 2548, 2862, 2918, 2961, 3372, 3478, 3540, 3588, 3673, 3707

Offset

1,3

Comments

Numbers k such that A079584(k) = A079584(k+1).

The corresponding values of A079584(k) are 1, 1, 2, 4, 6, 6, 12, 12, 18, 42, ...

This sequence is infinite as it contains A000225. - Rémy Sigrist, May 23 2022

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Example

1 is a term since A079584(1) = A079584(2) = 1.
3 is a term since A079584(3) = A079584(4) = 2.

Mathematica

s[n_] := s[n] = DigitCount[n!, 2, 1]; Select[Range[0, 4000], s[#] == s[# + 1] &]

Prog

(Python)
from itertools import count, islice
def wt(n): return bin(n).count("1")
def agen(): # generator of terms
    n, fn, fnplus, wtn, wtnplus = 0, 1, 1, 1, 1
    for n in count(0):
        if wtn == wtnplus: yield n
        fn, fnplus = fnplus, fnplus*(n+2)
        wtn, wtnplus = wtnplus, wt(fnplus)
print(list(islice(agen(), len(data)))) # Michael S. Branicky, May 23 2022
(PARI) isok(k) = hammingweight(k!) == hammingweight((k+1)!); \\ Michel Marcus, May 23 2022

Crossrefs

A354301 is a subsequence.

Cf. A000120, A000142, A000225, A079584, A353986.

Sequence in context: A276112 A228075 A309405 * A032420 A127458 A039003

Adjacent sequences: A354297 A354298 A354299 * A354301 A354302 A354303

Keyword

nonn,base

Author

Amiram Eldar, May 23 2022

Status

approved