A093640 -id:A093640 - cp's OEIS frontend

A355634 Irregular triangle T(n, k), n > 0, k = 1..A093640(n), read by rows; the n-th row contains in ascending order the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 25

Offset: 1

Rémy Sigrist, Jul 11 2022

			Triangle T(n, k) begins:
     1: [1]
     2: [1, 2]
     3: [1, 3]
     4: [1, 2, 4]
     5: [1, 5]
     6: [1, 2, 3, 6]
     7: [1, 7]
     8: [1, 2, 4, 8]
     9: [1, 9]
    10: [1, 2, 5, 10]
    11: [1, 11]
    12: [1, 2, 3, 4, 6, 12]
    13: [1, 13]
    14: [1, 2, 7, 14]
    15: [1, 3, 15]
    16: [1, 2, 4, 8, 16]

Paolo Xausa, Table of n, a(n) for n = 1..10019 (rows 1..2000 of the triangle, flattened).
Index entries for sequences related to binary expansion of n
Index entries for sequences related to divisors

Cf. A027750, A093640 (row lengths), A355632 (decimal analog), A355633 (row sums).

Table[Select[Divisors[n], StringContainsQ[IntegerString[n, 2], IntegerString[#, 2]] &], {n, 50}] (* Paolo Xausa, Jul 23 2024 *)

row(n, base=2) = { my (d=digits(n, base), s=setbinop((i, j) -> fromdigits(d[i..j], base), [1..#d]), v=0); select(v -> v && n%v==0, s) }

from sympy import divisors
def row(n):
    s = bin(n)[2:]
    return sorted(d for d in divisors(n, generator=True) if bin(d)[2:] in s)
def table(r): return [i for n in range(1, r+1) for i in row(n)]
print(table(25)) # Michael S. Branicky, Jul 11 2022

T(n, 1) = 1.
T(n, A093640(n)) = n.
Sum_{k = 1..A093640(n)} T(n, k) = A355633(n).

A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112

Offset: 1

Reinhard Zumkeller, Apr 07 2004

a(n) is either 1, prime, or of form 2a(m), m

1 and Heinz numbers of hook integer partitions. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). A hook is a partition of the form (n,1,1,...,1). - Gus Wiseman, Sep 15 2018

Numbers whose odd part is noncomposite. - Peter Munn, Aug 06 2020

			55 is not a member, as 5*11 is not of the form 2^i * prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A093640(a(n)) = A000005(a(n)); A000040 and A000079 are subsequences.
A105440 is a subsequence, see also A105442. - Reinhard Zumkeller, Apr 09 2005
Cf. A078822, A007088.
Complement of A105441; A001221(a(n))<=2; A005087(a(n))<=1; A087436(a(n))<=1.
See also A105442.
Union of A038550 and A000079, see also A008578.
Cf. A082733, A153452, A296188, A296561, A300121, A304438, A305940, A317554.
Cf. A000265 (odd part), A008578 (noncomposite).

a093641 n = a093641_list !! (n-1)
a093641_list = filter ((<= 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n],{2,}],{}|{{,1}}];
Select[Range[100],hookQ] (* Gus Wiseman, Sep 15 2018 *)

upTo(lim)=my(v=List([1])); for(e=0, log(lim)\log(2), forprime(p=2, lim>>e, listput(v,p<Charles R Greathouse IV, Aug 21 2011

isok(m) = my(k=m/2^valuation(m,2)); (k == 1) || isprime(k); \\ Michel Marcus, Mar 16 2023

from sympy import primepi
def A093641(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum(primepi(x>>i) for i in range(x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Feb 02 2025

A001227(a(n)) <= 2. - Reinhard Zumkeller, May 01 2012

Number A(x) of a(n) not exceeding x equals 1 + pi(x) + pi(x/2) + pi(x/4) + ..., where pi(x) is the number of primes <= x. If x goes to infinity, A(x)~2*x/log(x) and a(n)~n*log(n)/2 (n-->infinity). - Vladimir Shevelev, Feb 06 2014

A093642 Numbers not containing all divisors in their binary representation.

Original entry on oeis.org

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141

Offset: 1

Reinhard Zumkeller, Apr 07 2004

			55 is not a member, as the binary representations of 5 ("101") and 11 ("1011") both appear in the binary representation of 55 ("110111").

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A123345.
Subsequence of A105441. - Reinhard Zumkeller, Apr 09 2005
Cf. A000005, A078822, A007088, A027750, A093640.

import Data.List (unfoldr, isInfixOf)
a093642 n = a093642_list !! (n-1)
a093642_list = filter
  (\x -> not $ all (`isInfixOf` b x) $ map b $ a027750_row x) [1..] where
  b = unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Oct 27 2012

q[n_] := !AllTrue[Divisors[n], StringContainsQ[IntegerString[n, 2], IntegerString[#, 2]] &]; Select[Range[150], q] (* Amiram Eldar, Jun 05 2022 *)

from sympy import divisors
def ok(n):
    b = bin(n)[2:]
    return not all(bin(d)[2:] in b for d in divisors(n, generator=True))
print([k for k in range(119) if ok(k)]) # Michael S. Branicky, Jun 05 2022

A093640(a(n)) < A000005(a(n)).

A355633 a(n) is the sum of the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 10, 18, 12, 28, 14, 24, 19, 31, 18, 30, 20, 42, 22, 36, 24, 60, 26, 42, 31, 56, 30, 57, 32, 63, 34, 54, 36, 70, 38, 60, 43, 90, 42, 66, 44, 84, 54, 72, 48, 124, 50, 78, 55, 98, 54, 93, 72, 120, 61, 90, 60, 133, 62, 96, 74, 127, 66

Offset: 1

Rémy Sigrist, Jul 11 2022

			For n = 84:
- the binary expansion of 84 is "1010100",
- we have the following divisors:
  d   bin(d)   in bin(84)?
  --  -------  -----------
   1        1  Yes
   2       10  Yes
   3       11  No
   4      100  Yes
   6      110  No
   7      111  No
  12     1100  No
  14     1110  No
  21    10101  Yes
  28    11100  No
  42   101010  Yes
  84  1010100  Yes
- so a(84) = 1 + 2 + 4 + 21 + 42 + 84 = 154.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 100000 terms (the color is function of the 2-adic valuation of n)
Index entries for sequences related to binary expansion of n
Index entries for sequences related to divisors

Cf. A000203, A027750, A093640, A355620 (decimal analog), A355634.

a[n_] := DivisorSum[n, # &, StringContainsQ @@ IntegerString[{n, #}, 2] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

a(n, base=2) = { my (d=digits(n, base), s=setbinop((i, j) -> fromdigits(d[i..j], base), [1..#d]), v=0); for (k=1, #s, if (s[k] && n%s[k]==0, v+=s[k])); return (v) }

from sympy import divisors
def a(n):
    s = bin(n)[2:]
    return sum(d for d in divisors(n, generator=True) if bin(d)[2:] in s)
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Jul 11 2022

a(n) <= A000203(n).

a(2^n) = 2^(n+1) - 1 for any n >= 0.

A320538 Assuming the truth of the Collatz conjecture, a(n) is the number of divisors of n appearing in the Collatz trajectory of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 3, 5, 2, 4, 2, 6, 2, 4, 2, 8, 3, 4, 2, 6, 2, 6, 2, 6, 3, 4, 3, 6, 2, 4, 3, 8, 2, 4, 2, 6, 3, 4, 2, 10, 3, 6, 3, 6, 2, 4, 3, 8, 2, 4, 2, 9, 2, 4, 2, 7, 4, 6, 2, 6, 2, 6, 2, 8, 2, 4, 2, 6, 3, 6, 2, 10, 2, 4, 2, 6, 2, 4, 2

Offset: 1

Michel Lagneau, Oct 15 2018

nonn

a(p) = 2 for p prime.
a((2^2k - 1)/3) = 2, k = 1, 2, ...
We observe that a(n) differs from A093640(n) for n = 25, 27, 33, 35, 45, 49, 50, 54, 55, 57, 63, 65, 66, 70, 75, 77, 85, ...
7 occurs only eighteen times among the first 65537 terms. - Antti Karttunen, May 18 2019

			a(6) = 4 because the Collatz trajectory 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 contains 4 divisors of 6: 1, 2, 3 and 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A027750, A070165, A093640, A207674, A207675.

lst={}; coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Do[AppendTo[lst,Length[Intersection[Divisors[n],coll[n]]]],{n,1,100}]; lst

f(n) = if(n%2, 3*n+1, n/2);
a(n) = {my(kn = n, nb = 1); while (n != 1, n = f(n); if ((kn % n) == 0, nb++);); nb;}

Showing 1-5 of 5 results.

cp's OEIS Frontend

A355634 Irregular triangle T(n, k), n > 0, k = 1..A093640(n), read by rows; the n-th row contains in ascending order the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

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A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.

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A093642 Numbers not containing all divisors in their binary representation.

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A355633 a(n) is the sum of the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

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A320538 Assuming the truth of the Collatz conjecture, a(n) is the number of divisors of n appearing in the Collatz trajectory of n.

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