A032741 -id:A032741 - cp's OEIS frontend

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

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

A051777 Triangle read by rows, where row (n) = n mod n, n mod (n-1), n mod (n-2), ...n mod 1.

Original entry on oeis.org

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

Offset: 1

Asher Auel, Dec 09 1999

Also, rectangular array read by antidiagonals, a(n, k) = k mod n (k >= 0, n >= 1). Cf. A048158, A051127. - David Wasserman, Oct 01 2008

Central terms: a(2*n - 1, n) = n - 1. - Reinhard Zumkeller, Jan 25 2011

			row (5) = 5 mod 5, 5 mod 4, 5 mod 3, 5 mod 2, 5 mod 1 = 0, 1, 2, 1, 0.
0 ;
0  0 ;
0  1  0 ;
0  1  0  0 ;
0  1  2  1  0;
0  1  2  0  0  0 ;
0  1  2  3  1  1  0 ;
0  1  2  3  0  2  0  0;
0  1  2  3  4  1  0  1  0 ;
0  1  2  3  4  0  2  1  0  0 ;
0  1  2  3  4  5  1  3  2  1  0 ;
0  1  2  3  4  5  0  2  0  0  0  0 ;
0  1  2  3  4  5  6  1  3  1  1  1  0 ;

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Cf. A051778. Row sums give A004125. Number of 0's in row n gives A000005 (tau(n)). Number of 1's in row n+1 gives A032741(n).

a051777 n k = a051777_row n !! (k-1)
a051777_row n = map (mod n) [n, n-1 .. 1]
a051777_tabl = map a051777_row [1..]
-- Reinhard Zumkeller, Jan 25 2011

Flatten[Table[Mod[n,Range[n,1,-1]],{n,20}]] (* Harvey P. Dale, Nov 30 2011 *)

A276634 Sum of cubes of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 9, 1, 36, 1, 73, 28, 134, 1, 316, 1, 352, 153, 585, 1, 981, 1, 1198, 371, 1340, 1, 2556, 126, 2206, 757, 3160, 1, 4752, 1, 4681, 1359, 4922, 469, 8605, 1, 6868, 2225, 9710, 1, 12600, 1, 12052, 4257, 12176, 1, 20476, 344, 16759, 4941, 19846, 1, 26496, 1457, 25624, 6887, 24398, 1

Offset: 1

Ilya Gutkovskiy, Sep 08 2016

More generally, the Dirichlet generating function for the sum of k-th powers of proper divisors of n is zeta(s-k)*(zeta(s) - 1).

			a(10) = 1^3 + 2^3 + 5^3 = 134, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^3 = 1, because 11 has 1 proper divisor {1}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Proper divisors.

Cf. A000035, A000578, A001065, A001158, A032741, A053867, A067558.

[DivisorSigma(3, n) - n^3: n in [1..70]]; // Vincenzo Librandi, Sep 09 2016

Table[DivisorSigma[3, n] - n^3, {n, 70}]

a(n) = sigma(n, 3) - n^3; \\ Michel Marcus, Sep 08 2016

a(n) = 1 if n is prime.
a(p^k) = (p^(3*k) - 1)/(p^3 - 1) for p prime.
Dirichlet g.f.: zeta(s-3)*(zeta(s) - 1).
a(n) = A001158(n) - A000578(n).
A000035(a(n)) = A053867(n).
Sum_{n=1..k} a(n) ~ k^2*(Pi^4*k^2/90 - (k + 1)^2)/4.
G.f.: -x*(1 + 4*x + x^2)/(1 - x)^4 + Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 17 2017

A284288 Numbers n such that the average of the strong divisors of n is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 28, 29, 31, 37, 41, 43, 47, 49, 53, 54, 56, 59, 61, 64, 67, 68, 71, 73, 79, 81, 83, 89, 91, 97, 98, 99, 100, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 138, 139, 148, 149, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 188, 191, 192, 193, 197, 199

Offset: 1

Ilya Gutkovskiy, Mar 24 2017

nonn

We say d is a strong divisor of n iff d is a divisor of n and d > 1.
Numbers n such that A032741(n) divides A039653(n).
All primes and squares of primes are in this sequence.
Positions of ones in A296082 and A296084. - Antti Karttunen, Dec 05 2017

			28 is in the sequence because 28 has 6 divisors {1, 2, 4, 7, 14, 28} therefore 5 strong divisors {2, 4, 7, 14, 28}, 2 + 4 + 7 + 14 + 28 = 55 and 5 divides 55.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A000203, A000430, A003601, A023884, A023886, A032741, A039653, A296082, A296084 (characteristic function).

filter:= proc(n) local d,t;
  d:= numtheory:-divisors(n) minus {1};
  convert(d,`+`) mod nops(d) = 0
end proc:
select(filter, [$2..1000]); # Robert Israel, Mar 27 2017

Select[Range[2, 200], Mod[DivisorSigma[1, #1] - 1, DivisorSigma[0, #1] - 1] == 0 &]

for(n=2, 200, if((sigma(n) - 1)%(numdiv(n) - 1)==0, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

from sympy.ntheory import divisor_sigma, divisor_count
print([n for n in range(2, 201) if (divisor_sigma(n) - 1)%(divisor_count(n) - 1) == 0]) # Indranil Ghosh, Mar 24 2017

A293524 a(n) = Product_{d|n, d>1} prime(A052409(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 30, 6, 8, 2, 48, 2, 8, 8, 210, 2, 48, 2, 48, 8, 8, 2, 480, 6, 8, 30, 48, 2, 128, 2, 2310, 8, 8, 8, 864, 2, 8, 8, 480, 2, 128, 2, 48, 48, 8, 2, 6720, 6, 48, 8, 48, 2, 480, 8, 480, 8, 8, 2, 3072, 2, 8, 48, 30030, 8, 128, 2, 48, 8, 128, 2, 17280, 2, 8, 48, 48, 8, 128, 2, 6720, 210, 8, 2, 3072, 8, 8, 8, 480

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A052409, A294873, A294874, A294875.

Cf. also A293514, A293442.

A293524(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(!e, m += m, m *= prime(e)))); m; };

a(n) = Product_{d|n, d>1} A000040(A052409(d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A183096(n).
A064989(a(n)) = A294875(n).

A294891 Number of proper divisors d of n such that Stern polynomial B(d,x) is irreducible.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n=50, with proper divisors [1, 2, 5, 10, 25], 2, 5, and 25 are larger than one and included in A186891, thus a(50) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..22001

Cf. A186891, A283991, A294892, A294893, A294894.
Cf. also A294881, A294901.
Differs from A087624 for the first time at n=50.

ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A283991(n) = polisirreducible(ps(n));
A294891(n) = sumdiv(n,d,(dA283991(d));

a(n) = Sum_{d|n, dA283991(d).
a(n) + A294892(n) = A032741(n).
a(n) = A294893(n) - A283991(n).

A294892 Number of proper divisors d of n such that either d=1 or Stern polynomial B(d,x) is reducible.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 10 2017

nonn

			For n=48, its proper divisors are [1, 2, 3, 4, 6, 8, 12, 16, 24]. After 1, the divisors 4, 6, 8, 12, 16 and 24 are not found in A186891, thus a(48) = 1+6 = 7.
For n=50, its proper divisors are [1, 2, 5, 10, 25]. After 1, only 10 is not found in A186891, thus a(50) = 1+1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..22001

Cf. A186891, A283991, A294891, A294893, A294894.

Cf. also A294882, A294902.

ps(n) = if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)));
A283991(n) = polisirreducible(ps(n));
A294892(n) = sumdiv(n,d,(dA283991(d)));

a(n) = Sum_{d|n, dA283991(d)).
a(n) + A294891(n) = A032741(n).
a(n) = A294894(n) + A283991(n) - 1.

cp's OEIS Frontend

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

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

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A051777 Triangle read by rows, where row (n) = n mod n, n mod (n-1), n mod (n-2), ...n mod 1.

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A276634 Sum of cubes of proper divisors of n.

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A284288 Numbers n such that the average of the strong divisors of n is an integer.

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