A338648 -id:A338648 - cp's OEIS frontend

A135539 Triangle read by rows: T(n,k) = number of divisors of n that are >= k.

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 4, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Oct 30 2007

Row sums give A000203.

Left border is A000005.

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  2, 1, 1, 1, 1;
  4, 3, 2, 1, 1, 1;
  2, 1, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1;
  3, 2, 2, 1, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Column k=1..10 give A000005, A032741, A023645, A321014, A338648, A338649, A338650, A338651, A338652, A338653.
Cf. A051731, A000203.
Cf. A001008, A001620, A002805.

with(numtheory);
f1:=proc(n) local d,s1,t1,t2,i;
d:=tau(n);
s1:=sort(divisors(n));
t1:=Array(1..n,0);
for i from 1 to d do t1[n-s1[i]+1]:=1; od:
t2:=PSUM(convert(t1,list));
[seq(t2[n+1-i],i=1..n)];
end proc;
for n from 1 to 15 do lprint(f1(n)); od: # N. J. A. Sloane, Nov 09 2018

T[n_, k_] := DivisorSum[n, Boole[# >= k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 15 2023 *)

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ Michel Marcus, Jul 23 2022

Triangle read by rows, partial sums of A051731 starting from the right. A051731 as a lower triangular matrix times an all 1's lower triangular matrix.
From Seiichi Manyama, Jan 07 2023: (Start)
G.f. of column k: Sum_{j>=1} x^(k*j)/(1 - x^j).
G.f. of column k: Sum_{j>=k} x^j/(1 - x^j). (End)
Sum_{j=1..n} T(j, k) ~ n * (log(n) + 2*gamma - 1 - H(k-1)), where gamma is Euler's constant (A001620), and H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 08 2024

Clearer definition from N. J. A. Sloane, Nov 09 2018

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

A338650 Number of divisors of n which are greater than 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 4, 1, 3, 2, 5, 1, 7, 1, 2, 3, 3, 3, 4, 1, 6, 3, 2, 1, 7, 2, 2, 2, 5, 1, 7, 3, 3, 2, 2, 2, 7, 1, 4, 4, 5, 1, 4, 1, 5, 5, 2, 1, 7, 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=7 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 6 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(7 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, 7, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>6); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=7, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(7*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=7} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=7} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 69/20), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(6) prepended by David A. Corneth, Jun 13 2022

A338651 Number of divisors of n which are greater than 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 2, 4, 1, 6, 3, 2, 1, 6, 2, 2, 2, 5, 1, 7, 2, 3, 2, 2, 2, 7, 1, 3, 4, 5, 1, 4, 1, 5, 4, 2, 1, 7, 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=8 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 7 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(8 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, 8, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>7); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0], Vec(sum(k=8, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(8*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=8} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=8} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 503/140), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(7) prepended by David A. Corneth, Jun 13 2022

A338652 Number of divisors of n which are greater than 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 6, 1, 2, 3, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4, 1, 7, 2, 3, 2, 2, 2, 6, 1, 3, 4, 5, 1, 4, 1, 4, 4, 2, 1, 7, 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=9 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338651, A338653.

Table[DivisorSum[n, 1 &, # > 8 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(9 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, 9, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>8); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=9, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(9*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=9} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=9} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 1041/280), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(8) prepended by David A. Corneth, Jun 13 2022

A338653 Number of divisors of n which are greater than 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 6, 1, 2, 2, 3, 2, 4, 1, 3, 2, 4, 1, 5, 1, 2, 3, 3, 2, 4, 1, 5, 2, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 4, 1, 4, 4, 2, 1, 6, 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=10 of A135539.

Cf. A000005, A001620, A023645, A032741, A321014, A338648, A338649, A338650, A338651, A338652.

Table[DivisorSum[n, 1 &, # > 9 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(10 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, 10, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &
Table[Count[Divisors[n],?(#>9&)],{n,120}] (* _Harvey P. Dale, Jan 09 2025 *)

a(n) = sumdiv(n, d, d>9); \\ Michel Marcus, Apr 22 2021

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=10, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(10*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=10} (1 - x^k)^(1/k) ).
G.f.: Sum_{k>=10} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 9649/2520), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(9) prepended by David A. Corneth, Jun 13 2022

A359956 a(n) is the smallest number with exactly n divisors that are greater than or equal to 5.

Original entry on oeis.org

5, 10, 18, 24, 30, 48, 80, 60, 90, 192, 144, 120, 210, 180, 450, 240, 560, 1600, 5120, 360, 630, 3465, 900, 960, 2240, 720, 2800, 840, 1890, 10010, 5184, 1260, 3150, 17325, 36864, 1680, 5670, 2880, 11200, 15360, 3600, 19600, 99225, 2520, 6930, 6480, 70000, 61440

Offset: 1

Ilya Gutkovskiy, Jan 19 2023

nonn

Index entries for sequences related to divisors of numbers

Cf. A005179, A338648, A359955, A359957.

DeleteDuplicates[SortBy[Table[{n,Count[Divisors[n],?(#>4&)]},{n,100000}],Last],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* _Harvey P. Dale, Aug 18 2024 *)

a(n) = my(k=1); while (sumdiv(k, d, (d>=5)) != n, k++); k; \\ Michel Marcus, Jan 20 2023

cp's OEIS Frontend

A135539 Triangle read by rows: T(n,k) = number of divisors of n that are >= k.

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

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A338650 Number of divisors of n which are greater than 6.

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A338651 Number of divisors of n which are greater than 7.

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Mathematica

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A338652 Number of divisors of n which are greater than 8.

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Mathematica

PARI

PARI

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A338653 Number of divisors of n which are greater than 9.

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Mathematica

PARI

PARI

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A359956 a(n) is the smallest number with exactly n divisors that are greater than or equal to 5.

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