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Search: a292257 -id:a292257
Displaying 1-10 of 15 results found. page 1 2
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A294905 Characteristic function for A000120-nonabundant numbers: a(n) = 1 if A292257(n) <= A000120(n), and 0 otherwise. +20
7
1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions.
Index entries for sequences related to binary expansion of n.
FORMULA
a(n) = 1 if A192895(n) <= 0, and 0 otherwise.
a(n) = [A292257(n) <= A000120(n)].
EXAMPLE
For n=25, its proper divisors are 1 and 5, in binary "1" and "101", so the total number of 1's in them is 3, while 25 in binary is "11001", with binary weight 3, thus as A292257(25) <= A000120(25), a(25) = 1.
For n=55, its proper divisors are 1, 5 and 11, in binary "1", "101" and "1011", so the total number of 1's in them is 6, while 55 in binary is "110111", with binary weight 5, thus as A292257(55) > A000120(55), a(55) = 0.
MATHEMATICA
a[n_] := If[DivisorSum[n, DigitCount[#, 2, 1] &] > 2 * DigitCount[n, 2, 1], 0, 1]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI)
A292257(n) = sumdiv(n, d, (d<n)*hammingweight(d));
A294905(n) = (A292257(n) <= hammingweight(n));
CROSSREFS
Cf. A000120, A192895, A292257.
Cf. A175526 (positions of zeros), A257691 (of ones).
Cf. A294901, A294902, A294903, A294904.
After n=1, differs from A010051 for the next time at n=25, and from both A283991 and A257000 at n=55.
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 10 2017
STATUS
approved
A093653 Total number of 1's in binary expansion of all divisors of n. +10
34
1, 2, 3, 3, 3, 6, 4, 4, 5, 6, 4, 9, 4, 8, 9, 5, 3, 10, 4, 9, 9, 8, 5, 12, 6, 8, 9, 12, 5, 18, 6, 6, 8, 6, 9, 15, 4, 8, 10, 12, 4, 18, 5, 12, 15, 10, 6, 15, 7, 12, 9, 12, 5, 18, 11, 16, 10, 10, 6, 27, 6, 12, 17, 7, 8, 16, 4, 9, 10, 18, 5, 20, 4, 8, 16, 12, 11, 20, 6, 15, 12, 8, 5, 27, 9, 10, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 500 terms from Jaroslav Krizek)
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018. See (22) p. 6.
Index entries for sequences related to binary expansion of n
FORMULA
a(n) = Sum_{k = 0..n} if(mod(n, k) = 0, A000120(k), 0). - Paul Barry, Jan 14 2005
a(n) = A182627(n) - A226590(n). - Jaroslav Krizek, Sep 01 2013
a(n) = A292257(n) + A000120(n). - Antti Karttunen, Dec 14 2017
From Bernard Schott, May 16 2022: (Start)
If prime p = A000043(n), then a(2^p-1) = a(A000668(n)) = p+1 = A050475(n).
a(2^n) = n+1 (End)
EXAMPLE
a(8) = 4 because the divisors of 8 are [1, 2, 4, 8] and in binary: 1, 10, 100, 1000, so four 1's.
MAPLE
a:= n-> add(add(i, i=Bits[Split](d)), d=numtheory[divisors](n)):
seq(a(n), n=1..100); # Alois P. Heinz, May 17 2022
MATHEMATICA
Table[Plus@@DigitCount[Divisors[n], 2, 1], {n, 75}] (* Alonso del Arte, Sep 01 2013 *)
PROG
(PARI) A093653(n) = sumdiv(n, d, hammingweight(d)); \\ Antti Karttunen, Dec 14 2017
(PARI) a(n) = {my(v = valuation(n, 2), n = (n>>v)); sumdiv(n, d, hammingweight(d)) * (v + 1)} \\ David A. Corneth, Feb 15 2023
(Python)
from sympy import divisors
def a(n): return sum(bin(d).count("1") for d in divisors(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Apr 20 2022
(Python)
from sympy import divisors
def A093653(n): return sum(d.bit_count() for d in divisors(n, generator=True))
print([A093653(n) for n in range(1, 88)]) # Michael S. Branicky, Feb 15 2023
CROSSREFS
Cf. A000120, A093687, A192895, A292257.
Cf. A226590 (number of 0's in binary expansion of all divisors of n).
Cf. A182627 (number of digits in binary expansion of all divisors of n).
Cf. A034690 (a decimal equivalent).
KEYWORD
base,easy,nonn
AUTHOR
Jason Earls, May 16 2004
STATUS
approved
A293214 a(n) = Product_{d|n, d<n} A019565(d). +10
25
1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to binary expansion of n
FORMULA
a(n) = Product_{d|n, d<n} A019565(d).
a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018
Other identities.
For n >= 0, a(2^n) = A002110(n).
For n >= 1:
A048675(a(n)) = A001065(n).
A001222(a(n)) = A292257(n).
A007814(a(n)) = A091954(n).
A087207(a(n)) = A218403(n).
A248663(a(n)) = A227320(n).
PROG
(PARI)
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(d))); m; };
CROSSREFS
Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).
Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Oct 03 2017
STATUS
approved
A192895 A000120-deficiency of n. +10
18
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
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
MATHEMATICA
a[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[# == n]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015, adapted from PARI *)
PROG
(Haskell)
a192895 n =
sum (map a000120 $ filter ((== 0) . (mod n)) [1..n-1]) - a000120 n
a192895_list = map a192895 [1..]
(PARI) a(n)=sumdiv(n, d, hammingweight(d)*(-1)^(d==n)) \\ Charles R Greathouse IV, Feb 07 2013
(Python)
from sympy import divisors
def A192895(n): return sum((d.bit_count() if d<n else -d.bit_count()) for d in divisors(n, generator=True)) # Chai Wah Wu, Jul 25 2023
CROSSREFS
Cf. A000120, A033879, A093653, A175522, A175524, A175526, A292257.
Cf. A257691 (positions where a(n) <= 0), A294905 (and its char.fun).
KEYWORD
sign
AUTHOR
Reinhard Zumkeller, Jul 12 2011
STATUS
approved
A294902 Number of proper divisors of n that are in A175526. +10
9
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 3, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 3, 0, 2, 2, 0, 0, 6, 0, 1, 0, 2, 0, 4, 0, 4, 0, 0, 0, 7, 0, 0, 2, 4, 0, 3, 0, 2, 0, 3, 0, 8, 0, 0, 1, 2, 0, 3, 0, 6, 2, 0, 0, 7, 0, 0, 0, 4, 0, 7, 0, 2, 0, 0, 0, 8, 0, 2, 2, 4, 0, 3, 0, 4, 3, 0, 0, 8, 0, 3, 0, 6, 0, 3, 0, 2, 2, 0, 0, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..25000
FORMULA
a(n) = Sum_{d|n, d<n} (1-A294905(d)).
a(n) = A294904(n) + A294905(n) - 1.
a(n) + A294901(n) = A032741(n).
MATHEMATICA
q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] > 2 * DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, 1 &, # < n && q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI)
A292257(n) = sumdiv(n, d, (d<n)*hammingweight(d));
A294905(n) = (A292257(n) <= hammingweight(n));
A294902(n) = sumdiv(n, d, (d<n)*(0==A294905(d)));
CROSSREFS
Cf. A000120, A032741, A175526, A292257, A294901, A294903, A294904, A294905.
Cf. also A294892.
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 10 2017
STATUS
approved
A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n. +10
9
1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 15, 10, 18, 2, 19, 2, 20, 21, 7, 22, 23, 2, 15, 21, 24, 2, 25, 2, 26, 27, 28, 2, 29, 30, 31, 10, 26, 2, 32, 33, 34, 21, 28, 2, 35, 2, 36, 37, 38, 33, 39, 2, 13, 40, 41, 2, 42, 2, 43, 44, 26, 45, 46, 2, 47, 48, 43, 2, 49, 50, 51, 40, 52, 2, 53, 45, 54, 55, 56, 33, 57, 2, 58, 59 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n
FORMULA
For all i, j:
a(i) = a(j) => A000005(i) = A000005(j).
a(i) = a(j) => A292257(i) = A292257(j).
a(i) = a(j) => A305426(i) = A305426(j).
a(i) = a(j) => A305435(i) = A305435(j).
PROG
(PARI)
\\ 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; };
A305792(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(A286622(d)-1))); (m); };
v305793 = rgs_transform(vector(up_to, n, A305792(n)));
A305793(n) = v305793[n];
CROSSREFS
Cf. A278222, A286622, A305792, A305795.
Cf. also A293215, A300833, A304103, A305813.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 11 2018
STATUS
approved
A300836 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n. +10
8
0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 4, 3, 5, 1, 7, 1, 7, 4, 4, 1, 11, 2, 3, 4, 8, 1, 10, 1, 7, 4, 5, 4, 14, 1, 5, 3, 11, 1, 10, 1, 8, 7, 4, 1, 15, 3, 8, 5, 7, 1, 12, 4, 12, 5, 4, 1, 21, 1, 5, 7, 10, 3, 13, 1, 8, 4, 11, 1, 19, 1, 4, 8, 10, 5, 10, 1, 16, 7, 5, 1, 20, 5, 5, 4, 12, 1, 20, 4, 10, 5, 4, 5, 21, 1, 9, 10, 16, 1, 13, 1, 11, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
a(n) = Sum_{d|n, d<n} A007895(d).
a(n) = A300837(n) - A007895(n).
a(n) = A001222(A300834(n)).
For all n >=1, a(n) >= A293435(n).
EXAMPLE
For n=12, its proper divisors are 1, 2, 3, 4 and 6. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101 and 1001. Total number of 1's present is 7, thus a(12) = 7.
PROG
(PARI)
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); }
A300836(n) = sumdiv(n, d, (d<n)*A007895(d));
CROSSREFS
Cf. A000045, A007895, A014417, A072649, A300834, A300837.
Cf. also A292257, A293435.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 18 2018
STATUS
approved
A294901 Number of proper divisors of n that are in A257691. +10
7
0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 3, 1, 3, 3, 3, 1, 3, 2, 3, 2, 3, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 3, 3, 1, 4, 1, 3, 3, 3, 1, 3, 2, 4, 3, 3, 1, 3, 3, 3, 3, 3, 1, 4, 1, 3, 3, 2, 3, 4, 1, 3, 3, 4, 1, 3, 1, 3, 4, 3, 3, 4, 1, 3, 2, 3, 1, 4, 3, 3, 3, 3, 1, 4, 3, 3, 3, 3, 3, 3, 1, 3, 3, 4, 1, 4, 1, 3, 4, 3, 1, 3, 1, 4, 3, 3, 1, 4, 3, 3, 3, 3, 3, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..25000
FORMULA
a(n) = Sum_{d|n, d<n} A294905(d).
a(n) = A294903(n) - A294905(n).
a(n) + A294902(n) = A032741(n).
MATHEMATICA
q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] <= 2*DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, 1 &, # < n && q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI)
A292257(n) = sumdiv(n, d, (d<n)*hammingweight(d));
A294905(n) = (A292257(n) <= hammingweight(n));
A294901(n) = sumdiv(n, d, (d<n)*A294905(d));
CROSSREFS
Cf. A000120, A032741, A257691, A292257, A294902, A294903, A294904, A294905.
Cf. also A294891.
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 10 2017
STATUS
approved
A294904 Number of divisors of n that are in A175526. +10
7
0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 1, 0, 5, 0, 1, 2, 3, 0, 4, 0, 4, 1, 1, 1, 6, 0, 1, 1, 5, 0, 4, 0, 3, 3, 1, 0, 7, 1, 2, 1, 3, 0, 5, 1, 5, 1, 1, 0, 8, 0, 1, 3, 5, 1, 4, 0, 3, 1, 4, 0, 9, 0, 1, 2, 3, 1, 4, 0, 7, 3, 1, 0, 8, 1, 1, 1, 5, 0, 8, 1, 3, 1, 1, 0, 9, 0, 3, 3, 5, 0, 4, 0, 5, 4, 1, 0, 9, 0, 4, 0, 7, 0, 4, 1, 3, 3, 1, 0, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
Number of terms of A175526 that divide n.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..25000
FORMULA
a(n) = Sum_{d|n} (1-A294905(d)).
a(n) = 1 + (A294902(n)-A294905(n)).
a(n) + A294903(n) = A000005(n).
MATHEMATICA
q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] > 2 * DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI)
A292257(n) = sumdiv(n, d, (d<n)*hammingweight(d));
A294905(n) = (A292257(n) <= hammingweight(n));
A294904(n) = sumdiv(n, d, (0==A294905(d)));
CROSSREFS
Cf. A000120, A175526, A292257, A294901, A294902, A294903, A294905.
Cf. also A294894.
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 10 2017
STATUS
approved
A294903 Number of divisors of n that are in A257691. +10
5
1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 4, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 4, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 3, 2, 3, 3, 4, 2, 4, 2, 3, 4, 3, 2, 3, 2, 4, 4, 3, 2, 4, 3, 3, 3, 3, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Number of terms of A257691 that divide n.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..25000
FORMULA
a(n) = Sum_{d|n} A294905(d).
a(n) = A294901(n) + A294905(n).
a(n) + A294904(n) = A000005(n).
MATHEMATICA
q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] <= 2*DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
PROG
(PARI)
A292257(n) = sumdiv(n, d, (d<n)*hammingweight(d));
A294905(n) = (A292257(n) <= hammingweight(n));
A294903(n) = sumdiv(n, d, A294905(d));
CROSSREFS
Cf. A000120, A257691, A292257, A294901, A294902, A294904, A294905.
Cf. also A294893.
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 10 2017
STATUS
approved
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