A032741 -id:A032741 - cp's OEIS frontend

A338648 Number of divisors of n which are greater than 4.

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 3, 2, 2, 3, 5, 1, 2, 2, 5, 1, 5, 1, 3, 4, 2, 1, 6, 2, 4, 2, 3, 1, 5, 3, 5, 2, 2, 1, 8, 1, 2, 4, 4, 3, 5, 1, 3, 2, 6, 1, 8, 1, 2, 4, 3, 3, 5, 1, 7, 3, 2, 1, 8, 3, 2, 2, 5, 1, 9, 3, 3, 2, 2, 3, 8, 1, 4, 4, 6, 1, 5, 1, 5, 6, 2, 1, 8, 1, 6

Offset: 1

Ilya Gutkovskiy, Apr 22 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers.

Column k=5 of A135539.

Cf. A000005, A001620, A023645, A032741, A083040, A321014, A338649, A338650, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 4 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(5 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 1] &
nmax = 110; CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 5, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>4); \\ Michel Marcus, Apr 22 2021; corrected Jun 13 2022

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=5, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(5*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=5} (1 - x^k)^(1/k) ).
a(n) = A000005(n) - A083040(n).
G.f.: Sum_{k>=5} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 37/12), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(4) prepended by David A. Corneth, Jun 13 2022

A338649 Number of divisors of n which are greater than 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 4, 4, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 3, 5, 1, 6, 3, 2, 1, 8, 2, 2, 2, 5, 1, 8, 3, 3, 2, 2, 2, 8, 1, 4, 4, 5, 1, 5, 1, 5, 5, 2, 1, 8, 1, 5

Offset: 1

Ilya Gutkovskiy, Apr 22 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers.

Column k=6 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338650, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 5 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(6 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 1] &
nmax = 110; CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 6, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>5); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=6, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(6*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=6} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=6} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 197/60), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(5) prepended by David A. Corneth, Jun 13 2022

A354960 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of the number of proper divisors of a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 5, 7, 10, 15, 18, 20, 25, 14, 21, 24, 28, 30, 35, 27, 33, 36, 16, 32, 40, 42, 49, 22, 39, 45, 50, 55, 48, 54, 56, 63, 60, 11, 13, 17, 19, 23, 26, 51, 57, 66, 70, 77, 69, 72, 44, 65, 75, 80, 81, 52, 85, 78, 84, 88, 91, 87, 90, 99, 95, 93, 96, 110, 98, 100, 64, 102, 105, 112

Offset: 1

Scott R. Shannon, Jul 23 2022

The terms are concentrated along numerous lines, some of which curve upward while others curve downward. See the first linked image. Surprisingly these lines are not shared by terms which are a multiple of a given proper divisor count, but predominantly by terms sharing a certain prime factor. See the second linked image.
The sequence is conjectured to be a permutation of the positive integers although it may take an extremely large number of terms for the primes to appear; e.g., 263 has not occurred after 500000 terms. Also although the vast majority of primes will appear in their natural order, some may not; e.g., a(455) = 840, which has 31 proper divisors, so a(456) = 31, and then a(457) = 29.
In the first 500000 terms the only fixed points beyond the first two are 3, 4, 1159, 1207. It is possible that no more exist, although this is unknown.

			a(3) = 3 as a(2) = 2 which has one proper divisor, and 2 is the smallest unused multiple of 1.
a(5) = 6 as a(4) = 4 which has two proper divisors, and 6 is the smallest unused multiple of 2.
a(9) = 5 as a(8) = 12 which has five proper divisors, and 5 is the smallest unused multiple of 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n - 1..2^16, showing records in red, local minima in blue, fixed points in gold, primes in green, and prime powers in cyan.
Michael De Vlieger, Log-log scatterplot of a(n) n = 1..2^12, with a color function indicating tau(a(n-1))-1.
Scott R. Shannon, Image of the first 50000 terms. The green line is y = n.
Scott R. Shannon, Image of the first 50000 terms with color. Terms containing prime factors 17, 13, 11, 7, 5, 3, 2 are colored purple, blue, green, yellow, orange, red, and white respectively. All other terms are colored gray.
Scott R. Shannon, Image of the first 500000 terms.

Cf. A128556 (all divisors), A339793, A000120, A032741, A005179, A027751, A335372.

nn = 76; c[] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 2; Do[k = DivisorSigma[0, a[n - 1]] - 1; m = 1 + Floor[u/k]; While[c[m k], m++]; a[n] = m k; c[m k] = True; If[a[n] == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jul 24 2022 *)

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Original entry on oeis.org

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240

Offset: 1

Seiichi Manyama, Jun 11 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000292, A032741, A065608, A069153, A363605, A363606.
Cf. A059358, A363607, A363611.
Cf. A000203, A001158, A013662, A092348.

a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.
G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) - zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A084114 Number of divisions when calculating A084110(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 1, 2, 1, 2, 0, 1, 1, 2, 0, 4, 0, 1, 1, 1, 1, 2, 0, 3, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 1, 1, 1, 1, 4, 0, 1, 1, 2, 0, 2, 0, 2, 2

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A000005(n) - 1 - A084113(n) = A032741(n) - A084113(n) = (A032741(n)-A084115(n))/2;

a(n) = 0 iff n is prime or a square of prime (A000430).

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

Cf. A080257.

Cf. A027750, A084110, A084113.

a084114 = g 0 1 . tail . a027750_row where
   g c _ []     = c
   g c x (d:ds) = if r > 0 then g c (x * d) ds else g (c + 1) x' ds
                  where (x', r) = divMod x d
-- Reinhard Zumkeller, Jul 31 2014

A084115 A084113(n) minus A084114(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 2, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 3, 4, 1, 3, 1, 3, 3

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A084113(n) - A084114(n) = 2*A084113(n) - A032741(n) = A032741(n) - 2*A084114(n);

a(A084116(n)) = 1.

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

Cf. A066423, A084116.

Cf. A084110, A084113, A084114.

a084115 n = a084113 n - a084114 n -- Reinhard Zumkeller, Jul 31 2014

Definition fixed by Reinhard Zumkeller, Jul 31 2014

A294138 Number of compositions (ordered partitions) of n into proper divisors of n.

Original entry on oeis.org

1, 0, 1, 1, 5, 1, 24, 1, 55, 19, 128, 1, 1627, 1, 741, 449, 5271, 1, 45315, 1, 83343, 3320, 29966, 1, 5105721, 571, 200389, 26425, 5469758, 1, 154004510, 1, 47350055, 226019, 9262156, 51885, 15140335649, 1, 63346597, 2044894, 14700095925, 1, 185493291000, 1, 35539518745, 478164162

Offset: 0

Ilya Gutkovskiy, Oct 23 2017

nonn

			a(4) = 5 because 4 has 3 divisors {1, 2, 4} among which 2 are proper divisors {1, 2} therefore we have [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Amiram Eldar, Table of n, a(n) for n = 0..3359
Index entries for sequences related to compositions

Cf. A027750, A032741, A065205, A100346, A210442, A294137.

Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[d[[k]] != n] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 45}]

a(n) = [x^n] 1/(1 - Sum_{d|n, d < n} x^d).

a(n) = A100346(n) - 1.

A294901 Number of proper divisors of n that are in A257691.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 10 2017

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A000120, A032741, A257691, A292257, A294902, A294903, A294904, A294905.

Cf. also A294891.

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 *)

A292257(n) = sumdiv(n,d,(dA294905(n) = (A292257(n) <= hammingweight(n));
A294901(n) = sumdiv(n,d,(dA294905(d));

a(n) = Sum_{d|n, dA294905(d).
a(n) = A294903(n) - A294905(n).
a(n) + A294902(n) = A032741(n).

A304102 a(n) = Product_{d|n, dA304101(d)-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 12, 4, 8, 2, 120, 2, 12, 8, 24, 2, 200, 2, 120, 12, 12, 2, 1680, 4, 8, 20, 180, 2, 2000, 2, 120, 12, 44, 12, 12600, 2, 44, 8, 1680, 2, 1200, 2, 180, 200, 20, 2, 42000, 6, 440, 44, 120, 2, 7800, 12, 3960, 44, 12, 2, 3234000, 2, 44, 120, 840, 8, 10200, 2, 264, 20, 3000, 2, 630000, 2, 20, 440, 1452, 18, 2000, 2, 109200, 260, 44, 2, 1386000

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304103 (restricted growth sequence transform of this sequence), A304104.

\\ Needs also code from A304101:
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };

a(n) = Product_{d|n, dA000040(A304101(d)-1).
a(n) = 2*A304104(n) / A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A293435(n).
A007949(a(n)) = A304095(n).

A336942 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 5, 95, 8823, 4952323, 20285515801, 714092378624317

Offset: 0

Gus Wiseman, Aug 14 2020

			The a(0) = 1 through a(2) = 5 chains:
  {1}  {2,1}  {12,1}
              {12,2,1}
              {12,3,1}
              {12,4,1}
              {12,4,2,1}

A076954 can be used instead of A006939 (cf. A307895, A325337).
A336423 and A336571 are not restricted to A006939.
A336941 is the version not restricted by A130091.
A337075 is the version for factorials.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
Cf. A000005, A001055, A002033, A032741, A067824, A124010, A167865, A336419, A336420, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[chern[n]],{n,0,3}]

a(n) = A336423(A006939(n)) = A336571(A006939(n)).

cp's OEIS Frontend

A338648 Number of divisors of n which are greater than 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A338649 Number of divisors of n which are greater than 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A354960 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier that is a multiple of the number of proper divisors of a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A084114 Number of divisions when calculating A084110(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A084115 A084113(n) minus A084114(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Extensions

A294138 Number of compositions (ordered partitions) of n into proper divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A294901 Number of proper divisors of n that are in A257691.

Views

Author

Keywords

Links