A175381 -id:A175381 - cp's OEIS frontend

A175382 Positive integers n for which there is at least one positive integer k whose binary expansion occurs as a substring in the binary expansion of n but does not divide n.

5, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

Leroy Quet, Apr 24 2010

This is the complement of sequence A175381.

Includes all odd numbers > 3. - Robert Israel, Nov 25 2016

			14 in binary is 1110. One of the substrings of 1110 is 11, which is 3 in decimal. Since 3 does not divide 14, 14 is in this sequence.

Jeremy Gardiner (up to n=226) and Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A175381.

Select[Range@ 85, Function[n, Count[Map[IntegerDigits[#, 2] &, Complement[Range@ n, Divisors@ n]], k_ /; Length@ SequencePosition[IntegerDigits[n, 2], k] > 0] > 0]] (* Michael De Vlieger, Nov 24 2016, Version 10.1 *)

Spelling corrected by Jason G. Wurtzel, Sep 04 2010
a(12)-a(56) from Lars Blomberg, May 05 2011
a(57)-a(66) from Nathaniel Johnston, May 05 2011

A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 150, 200, 210, 220, 240, 250, 300, 330, 360, 400, 420, 440, 480, 500, 510, 520, 550, 600, 630, 660, 700, 770, 800, 840, 880

Offset: 1

Rémy Sigrist, May 01 2025

This sequence is infinite as ten times a term is also a term.
All terms are of the form A037124(k) or A037124(k) + d where k > 0 and d divides A037124(k) while having strictly less decimal digits as A037124(k).
Empirically, all terms have either one or two nonzero decimal digits.

			The number 240 is divisible by 2, 24, 240, 4 and 40, so 240 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10314
Rémy Sigrist, PARI program

Cf. A037124, A078546, A175381 (binary variant), A178157, A218978.

Select[Range[880],AllTrue[#/Select[FromDigits/@Subsequences[IntegerDigits[#]],#>0&],IntegerQ]&] (* James C. McMahon, May 13 2025 *)

is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if (n % fromdigits(d[i..j], base),
                    return (0);););););
    return (1); }

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

def ok(n):
    s = str(n)
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and all(n%v == 0 for ss in subs if (v:=int(ss)) > 0)
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, May 09 2025

cp's OEIS Frontend

A175382 Positive integers n for which there is at least one positive integer k whose binary expansion occurs as a substring in the binary expansion of n but does not divide n.

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