A382441 -id:A382441 - cp's OEIS frontend

A382442 Lexicographically earliest sequence of positive integers such that for any n > 1, a(n) does not divide any of the positive numbers whose binary expansion appears as a contiguous subword in the concatenation of the previous terms.

1, 2, 4, 7, 8, 16, 18, 27, 32, 42, 54, 64, 84, 126, 128, 133, 172, 238, 256, 276, 379, 381, 444, 512, 524, 582, 621, 765, 948, 1024, 1048, 1179, 1241, 1449, 1496, 1557, 1861, 1896, 1982, 2048, 2132, 2155, 2227, 2386, 2667, 2900, 3013, 3058, 3236, 3444, 3613

Offset: 1

Rémy Sigrist, Mar 26 2025

This sequence contains all powers of 2.

			a(1) = 1.
a(2) must not divide 1 ("1" in binary); we can take a(2) = 2.
a(3) must not divide 1, 2, 3 or 6 ("1", "10", "11", "110" in binary); we can take a(3) = 4.

Rémy Sigrist, C++ program

Cf. A382441 (decimal variant).

from itertools import count, islice
def agen(): # generator of terms
    an, s, d = 1, "1", {1}
    while True:
        yield an
        an = next(k for k in count(an+1) if not any(di%k == 0 for di in d))
        for di in bin(an)[2:]:
            s += di
            d |= set(si for i in range(len(s)) if (si:=int(s[i:], 2)) > an)
print(list(islice(agen(), 51))) # Michael S. Branicky, Mar 26 2025

A382445 Lexicographically least increasing sequence of distinct positive integers such that for any n > 1, a(n) does not divide the concatenation of the earlier terms.

Original entry on oeis.org

1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Rémy Sigrist, Mar 25 2025

			a(1) = 1.
a(2) must not divide 1; we can take a(2) = 2.
a(3) must not divide 12; we can take a(3) = 5.

Rémy Sigrist, PARI program

Cf. A007978, A096098, A382441.

PARI
```
\\ See Links section.
```

from itertools import count, islice
def agen(): # generator of terms
    an = t = 1
    while True:
        yield an
        an = next(k for k in count(an+1) if t%k != 0)
        t = t*10**len(str(an)) + an
print(list(islice(agen(), 54))) # Michael S. Branicky, Mar 26 2025

cp's OEIS Frontend

A382442 Lexicographically earliest sequence of positive integers such that for any n > 1, a(n) does not divide any of the positive numbers whose binary expansion appears as a contiguous subword in the concatenation of the previous terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python