A093653 -id:A093653 - cp's OEIS frontend

A338516 Starts of runs of 4 consecutive numbers that are divisible by the total binary weight of their divisors (A093653).

1377595575, 4275143301, 13616091683, 13640596128, 15016388244, 15176619135, 21361749754, 23605084359, 24794290167, 28025464183, 29639590888, 30739547718, 33924433023, 35259630279, 38008366692, 38670247670, 38681191672, 40210059079, 40507412213, 49759198333, 52555068607

Offset: 1

Amiram Eldar, Oct 31 2020

Can 5 consecutive numbers be divisible by the total binary weight of their divisors? If they exist, then they are larger than 10^11.

			1377595575 is a term since the 4 consecutive numbers from 1377595575 to 1377595578 are all terms of A093705.

Subsequence of A338514 and A338515.
Cf. A000120, A093653, A093705.
Similar sequences: A141769, A330933, A334372, A338454.

divQ[n_] := Divisible[n, DivisorSum[n, DigitCount[#, 2, 1] &]]; div = divQ /@ Range[4]; Reap[Do[If[And @@ div, Sow[k - 4]]; div = Join[Rest[div], {divQ[k]}], {k, 5, 5*10^9}]][[2, 1]]
SequencePosition[Table[If[Mod[n,Total[Flatten[IntegerDigits[#,2]&/@Divisors[n]]]]==0,1,0],{n,526*10^8}],{1,1,1,1}][[;;,1]] (* The program will take a long time to run. *) (* Harvey P. Dale, May 28 2023 *)

A034690 Sum of digits of all the divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 9, 3, 19, 5, 15, 15, 22, 9, 30, 11, 15, 14, 9, 6, 33, 13, 15, 22, 29, 12, 27, 5, 27, 12, 18, 21, 46, 11, 24, 20, 27, 6, 33, 8, 21, 33, 18, 12, 52, 21, 21, 18, 26, 9, 48, 18, 48, 26, 27, 15, 42, 8, 15, 32, 37, 21, 36, 14, 36, 24, 36, 9, 69, 11, 24, 34

Offset: 1

Erich Friedman

For first occurrence of k, or 0 if k never appears, see A191000.
The only fixed points are 1 and 15. These are also the only loops of iterations of A034690: see the SeqFan thread "List the divisors...". - M. F. Hasler, Nov 08 2015
The following sequence is composed of numbers n such that the sum of digits of all divisors of n equals 15: 8, 14, 15, 20, 26, 59, 62, ... It actually depicts the positions of number 15 in this sequence: see the SeqFan thread "List the divisors...". - V.J. Pohjola, Nov 09 2015

			a(15) = 1 + 3 + 5 + (1+5) = 15. - _M. F. Hasler_, Nov 08 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Angelini et al., List the dividers [sic], sum the digits, lost messages reconstructed by _N. J. A. Sloane_, Dec 21 2024
H. Havermann et al, in reply to E. Angelini, List the dividers, sum the digits, SeqFan list, Nov. 2015. [Broken link]
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018-2020. See (22) p. 6.

Cf. A000005, A000203, A007953, A086793, A191000.

Cf. A093653 (binary equivalent)

a034690 = sum . map a007953 . a027750_row
-- Reinhard Zumkeller, Jan 20 2014

with(numtheory); read transforms; f:=proc(n) local t1, t2, i; t1:=divisors(n); t2:=0; for i from 1 to nops(t1) do t2:=t2+digsum(t1[i]); od: t2; end;
# Alternative:
sd:= proc(n) option remember; local k; k:= n mod 10; k + procname((n-k)/10) end proc:
for n from 0 to 9 do sd(n):= n od:
a:= n -> add(sd(d), d=numtheory:-divisors(n)):
map(a, [$1..100]); # Robert Israel, Nov 17 2015

Table[Plus @@ Flatten@ IntegerDigits@ Divisors@n, {n, 75}] (* Robert G. Wilson v, Sep 30 2006 *)

vector(100, n, sumdiv(n, d, sumdigits(d))) \\ Michel Marcus, Jun 28 2015

A034690(n)=sumdiv(n,d,sumdigits(d)) \\ For use in other sequences. - M. F. Hasler, Nov 08 2015

from sympy import divisors
def sd(n): return sum(map(int, str(n)))
def a(n): return sum(sd(d) for d in divisors(n))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Oct 06 2021

A192895 A000120-deficiency of n.

Original entry on oeis.org

-1, 0, -1, 1, -1, 2, -2, 2, 1, 2, -2, 5, -2, 2, 1, 3, -1, 6, -2, 5, 3, 2, -3, 8, 0, 2, 1, 6, -3, 10, -4, 4, 4, 2, 3, 11, -2, 2, 2, 8, -2, 12, -3, 6, 7, 2, -4, 11, 1, 6, 1, 6, -3, 10, 1, 10, 2, 2, -4, 19, -4, 2, 5, 5, 4, 12, -2, 5, 4, 12, -3, 16, -2, 2, 8, 6

Offset: 1

Reinhard Zumkeller, Jul 12 2011

sign

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

Cf. A000120, A033879, A093653, A175522, A175524, A175526, A292257.

Cf. A257691 (positions where a(n) <= 0), A294905 (and its char.fun).

a192895 n =
   sum (map a000120 $ filter ((== 0) . (mod n)) [1..n-1]) - a000120 n
a192895_list = map a192895 [1..]

a[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[# == n]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015, adapted from PARI *)

a(n)=sumdiv(n,d,hammingweight(d)*(-1)^(d==n)) \\ Charles R Greathouse IV, Feb 07 2013

from sympy import divisors
def A192895(n): return sum((d.bit_count() if dChai Wah Wu, Jul 25 2023

a(n) = Sum(A000120(d): 1 <= d < n and n mod d = 0) - A000120(n); see A175522 for motivation and more information;
a(A175524(n)) < 0; a(A175522(n)) = 0; a(A175526(n)) > 0.
a(n) = A292257(n) - A000120(n). - Antti Karttunen, Nov 10 2017

A292257 a(n) is the total number of 1's in binary expansion of all proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 4, 1, 7, 1, 5, 5, 4, 1, 8, 1, 7, 6, 5, 1, 10, 3, 5, 5, 9, 1, 14, 1, 5, 6, 4, 6, 13, 1, 5, 6, 10, 1, 15, 1, 9, 11, 6, 1, 13, 4, 9, 5, 9, 1, 14, 6, 13, 6, 6, 1, 23, 1, 7, 11, 6, 6, 14, 1, 7, 7, 15, 1, 18, 1, 5, 12, 9, 7, 16, 1, 13, 9, 5, 1, 24, 5, 6, 7, 13, 1, 26, 7, 11, 8, 7, 6, 16, 1, 11, 10, 15, 1, 14, 1, 13, 18

Offset: 1

Antti Karttunen, Oct 04 2017

If a(n) == A000120(n), then n is in A175522, if a(n) < A000120(n), then n is in A175524, and if a(n) > A000120(n), then n is in A175526.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n.

Cf. A000120, A093653, A175522, A175524, A175526, A192895, A290090, A293214.

a[n_] := DivisorSum[n, DigitCount[#, 2, 1] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
Table[Total[Flatten[IntegerDigits[#,2]&/@Most[Divisors[n]]]],{n,120}] (* Harvey P. Dale, Oct 11 2024 *)

PARI
```
A292257(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA000120(d).
a(n) = A093653(n) - A000120(n).
a(n) = A192895(n) + A000120(n).
a(n) = A001222(A293214(n)).
A000035(a(n)) = A000035(A290090(n)). [Parity-wise equivalent with A290090.]

A300837 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

			For n=12, its divisors are 1, 2, 3, 4, 6 and 12. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101, 1001 and 10101. Total number of 1's present is 10 (ten), thus a(12) = 10.

Antti Karttunen, Table of n, a(n) for n = 1..10946

Cf. A000045, A007895, A014417, A072649, A300835, A300836.

Cf. also A005086, A093653.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); };
A300837(n) = sumdiv(n,d,A007895(d));

a(n) = Sum_{d|n} A007895(d).
a(n) = A300836(n) + A007895(n).
For all n >=1, a(n) >= A005086(n).

A093705 Numbers that are divisible by the total number of 1's in the binary expansions of all their divisors.

Original entry on oeis.org

1, 2, 3, 6, 8, 24, 27, 45, 49, 54, 55, 77, 90, 98, 108, 110, 128, 154, 180, 189, 209, 216, 299, 324, 360, 378, 384, 392, 418, 425, 440, 448, 598, 616, 689, 765, 783, 850, 855, 864, 880, 891, 896, 931, 972, 1023, 1040, 1056, 1160, 1188, 1200, 1209, 1215, 1378

Offset: 1

Jason Earls, May 17 2004

Numbers of the form 2^(2^k-1) (A058891) are terms of this sequence since A093653(2^(2^k-1)) = 2^k. - Amiram Eldar, Oct 31 2020

			a(5) = 8 because the divisors of 8 in binary are: 1, 10, 100, 1000, with four 1's and 8/4 = 2.

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

Cf. A093653.

A058891 is a subsequence.

f:=func< n|&+[&+Intseq(d,2):d in Divisors(n)]>; [k:k in [1..1500]| k mod f(k) eq 0]; // Marius A. Burtea, Dec 16 2019

Select[Range[1500], Divisible[#, Plus @@ DigitCount[Divisors[#], 2, 1]] &] (* Amiram Eldar, Dec 16 2019 *)

A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 13, 14, 15, 5, 16, 11, 17, 18, 19, 20, 21, 22, 19, 23, 24, 20, 25, 26, 27, 28, 10, 29, 30, 31, 19, 32, 33, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 23, 36, 35, 43, 44, 45, 32, 38, 46, 47, 39, 48, 49, 50, 51, 52, 11, 17, 53, 54, 20, 55, 31, 56, 57, 36, 58, 59, 39, 60, 61, 56, 35, 62, 63, 64, 65, 66, 35, 67

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A286622, A305794.

Cf. also A304105, A305793, A305815.

\\ Needs also code from A286622:
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A305794(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A286622(d)-1))); (m); };
v305795 = rgs_transform(vector(up_to, n, A305794(n)));
A305795(n) = v305795[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A007814(i) = A007814(j).
  a(i) = a(j) => A093653(i) = A093653(j).
  a(i) = a(j) => A154402(i) = A154402(j).
  a(i) = a(j) => A305436(i) = A305436(j).

A093687 Numbers k such that the total number of 1's in the binary expansion of all the divisors of k sets a new record.

Original entry on oeis.org

1, 2, 3, 6, 12, 18, 24, 30, 60, 90, 120, 180, 252, 360, 420, 504, 540, 630, 756, 840, 1080, 1260, 1980, 2520, 3780, 5040, 6300, 7560, 12600, 13860, 15120, 21420, 25200, 27720, 32760, 41580, 42840, 49140, 55440, 65520, 81900, 83160, 98280, 128520, 138600

Offset: 1

Jason Earls, May 16 2004

Amiram Eldar, Table of n, a(n) for n = 1..120

Cf. A093653.

a[n_] := Plus @@ DigitCount[Divisors[n], 2, 1]; am = -1; c = 0; seq={}; Do[a1 = a[n]; If[a1 > am, am = a1; c++; AppendTo[seq, n]], {n, 1, 10^4}]; seq (* Amiram Eldar, Dec 17 2019 *)

# uses imports and definitions in A093653
from itertools import count, islice
def f(n): return A093653(n)
def agen(r=0): yield from ((m, r:=fm)[0] for m in count(1) if (fm:=f(m))>r)
print(list(islice(agen(), 45))) # Michael S. Branicky, Feb 15 2023

A182627 Total number of digits in binary expansion of all divisors of n.

Original entry on oeis.org

1, 3, 3, 6, 4, 8, 4, 10, 7, 10, 5, 15, 5, 10, 10, 15, 6, 17, 6, 18, 11, 12, 6, 24, 9, 12, 12, 18, 6, 24, 6, 21, 13, 14, 13, 30, 7, 14, 13, 28, 7, 26, 7, 21, 20, 14, 7, 35, 10, 21, 14, 21, 7, 28, 14, 28, 14, 14, 7, 42, 7, 14, 21, 28, 15, 30, 8, 24, 15, 30, 8

Offset: 1

Omar E. Pol, Nov 23 2010

Also, total number of digits in row n of triangle A182620.
Also, number of digits of A182621(n).
Rows sums of triangle A182628.
From Davide Rotondo, Apr 20 2022: (Start)
Can be constructed by writing the sequence of natural numbers with 1 one, 2 twos, 4 threes, 8 fours, ..., where 1,2,4,8,... are consecutive powers of 2; then the same sequence spaced by a zero, then the same sequence spaced by two zeros, and so on. Finally add the values of the columns.
     1  2  2  3  3  3  3  4  4  4  4  4  4  4  4  5 ...
     0  1  0  2  0  2  0  3  0  3  0  3  0  3  0  4 ...
     0  0  1  0  0  2  0  0  2  0  0  3  0  0  3  0 ...
     0  0  0  1  0  0  0  2  0  0  0  2  0  0  0  3 ...
     0  0  0  0  1  0  0  0  0  2  0  0  0  0  2  0 ...
     0  0  0  0  0  1  0  0  0  0  0  2  0  0  0  0 ...
     0  0  0  0  0  0  1  0  0  0  0  0  0  2  0  0 ...
     0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  2 ...
     0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0 ...
     0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1 ...
     ...
     ----------------------------------------------
Tot. 1  3  3  6  4  8  4 10  7 10  5 15  5 10 10 15 ...  (End)

			The divisors of 12 are 1, 2, 3, 4, 6, 12. These divisors written in base 2 are 1, 10, 11, 100, 110, 1100. Then a(12)=15 because 1+2+2+3+3+4 = 15.

Jaroslav Krizek, Table of n, a(n) for n = 1..500

Cf. A093653, A135539, A182620, A182621, A182628.
Cf. A093653 (number of 1's in binary expansion of all divisors of n).
Cf. A226590 (number of 0's in binary expansion of all divisors of n).

Table[Total[IntegerLength[Divisors[n],2]],{n,60}] (* Harvey P. Dale, Jan 26 2012 *)

a(n) = sumdiv(n, d, 1+logint(d, 2)); \\ Michel Marcus, Dec 11 2020

from sympy import divisors
def a(n): return sum(d.bit_length() for d in divisors(n))
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Apr 21 2022

a(n) = A093653(n) + A226590(n). - Jaroslav Krizek, Sep 01 2013
a(n) = tau(n) + Sum_{d|n} floor(log_2(d)). - Ridouane Oudra, Dec 11 2020
a(n) = Sum_{i=0..floor(log_2(n))} A135539(n,2^i). - Ridouane Oudra, Sep 19 2022

A339549 a(n) is the product of the binary weights (A000120) of the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 3, 1, 4, 4, 3, 8, 3, 9, 16, 1, 2, 16, 3, 8, 18, 9, 4, 16, 6, 9, 16, 27, 4, 256, 5, 1, 12, 4, 18, 64, 3, 9, 24, 16, 3, 324, 4, 27, 128, 16, 5, 32, 9, 36, 16, 27, 4, 256, 30, 81, 24, 16, 5, 4096, 5, 25, 216, 1, 12, 144, 3, 8, 24, 324, 4, 256, 3

Offset: 1

Amiram Eldar, Dec 08 2020

Analogous to A093653 with product instead of sum.

			a(6) = 4 since the divisors of 6 are {1, 2, 3, 6}, and in binary representation {1, 10, 11, 110}. The number of 1's are {1, 1, 2, 2} and their product is 1*1*2*2 = 4.

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

Cf. A000079, A000120, A093653.

a[n_] := Times @@ (DigitCount[#, 2, 1] & /@ Divisors[n]); Array[a, 100]

a(n) = vecprod(apply(hammingweight, divisors(n))); \\ Michel Marcus, Dec 08 2020

a(n) = Product_{d|n} A000120(d).

a(n) = 1 if and only if n is a power of 2 (A000079).

cp's OEIS Frontend

A338516 Starts of runs of 4 consecutive numbers that are divisible by the total binary weight of their divisors (A093653).

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Mathematica

A034690 Sum of digits of all the divisors of n.

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Maple

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A192895 A000120-deficiency of n.

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A292257 a(n) is the total number of 1's in binary expansion of all proper divisors of n.

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A300837 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all divisors of n.

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A093705 Numbers that are divisible by the total number of 1's in the binary expansions of all their divisors.

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Magma

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A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

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A093687 Numbers k such that the total number of 1's in the binary expansion of all the divisors of k sets a new record.

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