A135539 -id:A135539 - cp's OEIS frontend

A332085 Number of ordered pairs of divisors of n, (d1,d2), such that d1 is prime and d1 <= d2.

0, 1, 1, 2, 1, 5, 1, 3, 2, 5, 1, 9, 1, 5, 5, 4, 1, 9, 1, 8, 5, 5, 1, 13, 2, 5, 3, 8, 1, 18, 1, 5, 5, 5, 5, 15, 1, 5, 5, 12, 1, 17, 1, 8, 9, 5, 1, 17, 2, 9, 5, 8, 1, 13, 5, 12, 5, 5, 1, 29, 1, 5, 9, 6, 5, 17, 1, 8, 5, 18, 1, 21, 1, 5, 9, 8, 5, 17, 1, 16, 4, 5, 1, 28, 5, 5, 5, 11, 1, 30

Offset: 1

Wesley Ivan Hurt, Aug 22 2020

nonn

			a(7) = 1; There are two divisors of 7: {1,7}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (7,7). So a(7) = 1.
a(8) = 3; There are 4 divisors of 8: {1,2,4,8}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (2,2), (2,4) and (2,8). So a(8) = 3.
a(9) = 2; There are three divisors of 9: {1,3,9}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (3,3) and (3,9). So a(9) = 2.
a(10) = 5; There are four divisors of 10: {1,2,5,10}. If we list the ordered pairs of divisors of n, (d1,d2) where d1 is prime and d1 <= d2, we get (2,2), (2,5), (2,10), (5,5) and (5,10). So a(10) = 5.

Cf. A001221 (omega), A135539, A337228, A337320, A337322, A248577.

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ A135539
a(n) = my(v=row(n)); sumdiv(n, d, if (isprime(d), v[d])); \\ Michel Marcus, May 24 2025

a(n) = Sum_{d1|n, d2|n, d1 is prime, d1 <= d2} 1.
a(n) = A337320(n) + omega(n).
a(n) = Sum_{p|n, p prime} A135539(n,p). - Ridouane Oudra, May 24 2025
a(n) = A248577(n) - A337322(n). - Ridouane Oudra, May 30 2025

A086670 Sum of floor(d/2) where d is a divisor of n.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 3, 7, 5, 8, 5, 13, 6, 11, 10, 15, 8, 18, 9, 20, 14, 17, 11, 29, 14, 20, 18, 27, 14, 34, 15, 31, 22, 26, 22, 44, 18, 29, 26, 44, 20, 46, 21, 41, 36, 35, 23, 61, 27, 45, 34, 48, 26, 58, 34, 59, 38, 44, 29, 82, 30, 47, 49, 63, 40, 70, 33, 62, 46, 70, 35, 96, 36, 56

Offset: 1

Jon Perry, Jul 27 2003

nonn

Inverse Mobius transform of A004526. - R. J. Mathar, Jan 19 2009

			10 has divisors 1,2,5,10. floor(d/2) gives 0,1,2,5, therefore a(10)=8.

Cf. A000203, A001227, A004526, A079247.

Cf. A135539.

Table[Total[Floor[Divisors[n]/2]],{n,80}] (* Harvey P. Dale, Feb 13 2023 *)

for (n=1,100,s=0; fordiv(i=n,i,s+=floor(i/2)); print1(","s))

a(n) = my(f = factor(n)); (sigma(f) - (numdiv(f)/(valuation(n, 2)+1)))>>1 \\ David A. Corneth, Apr 15 2022 using Franklin T. Adams-Watters's formula

G.f.: Sum_{n>=1} floor(n/2)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
a(n) = (A000203(n) - A001227(n)) / 2. - Franklin T. Adams-Watters, Jan 05 2012
G.f.: Sum_{k>=1} x^(2*k) / ((1 + x^k) * (1 - x^k)^2). - Ilya Gutkovskiy, Aug 02 2021
a(n) = Sum_{i=1..floor(n/2)} A135539(n,2*i). - Ridouane Oudra, Apr 15 2022

A143356 A051731 * A006218.

Original entry on oeis.org

1, 4, 6, 12, 11, 23, 17, 32, 29, 41, 30, 66, 38, 61, 61, 82, 53, 104, 61, 115, 92, 107, 77, 170, 98, 132, 124, 170, 104, 216, 114, 201, 158, 183, 158, 287, 143, 210, 193, 293, 161, 318, 171, 291, 266, 266, 189, 418, 218, 335, 269, 357, 220, 426, 271, 429, 309, 354, 250

Offset: 1

Gary W. Adamson, Aug 10 2008

nonn

			a(4) = 12 = sum of row 4 terms of triangle A143355: (7, + 3 + 1 + 1).
a(4) = 12 = (1, 1, 0, 1) dot (1, 3, 5, 8) = (1 + 3 + 0 + 8), where (1, 1, 0, 1) = row 4 of A051731 and A006218 = (1, 3, 5, 8, 10, 14,...).

Cf. A006218, A051731, A143355.

Cf. A135539.

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ A135539
a(n) = my(v=row(n)); sum(i=1, n, numdiv(i)*v[i]); \\ Michel Marcus, Jul 26 2022

Inverse Mobius transform (A051731) of A006218. Row sums of triangle A143355.
a(n) = Sum_{i=1..n} tau(i)*A135539(n,i). - Ridouane Oudra, Jul 26 2022
a(n) = Sum_{d|n} A006218(d). - Ridouane Oudra, Jul 27 2022

Corrected typo in A-number in formula; added more terms - R. J. Mathar, Jan 19 2009

A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

Original entry on oeis.org

1, 6, 15, 36, 56, 111, 141, 240, 300, 446, 507, 791, 820, 1161, 1310, 1736, 1786, 2505, 2471, 3346, 3466, 4307, 4325, 5895, 5581, 7026, 7230, 8905, 8556, 11246, 10417, 13176, 13050, 15476, 15106, 19391, 17576, 21495, 21374, 25690, 23822, 30162, 27435, 33707, 32990, 37841, 35721

Offset: 1

Ilya Gutkovskiy, Oct 24 2018

Inverse Möbius transform of square pyramidal numbers (A000330).

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000203, A000330, A001157, A001158, A007437, A059358.

Cf. A135539.

a:=series(add(x^k*(1+x^k)/(1-x^k)^4,k=1..100),x=0,48): seq(coeff(a,x,n),n=1..47); # Paolo P. Lava, Apr 02 2019

nmax = 47; Rest[CoefficientList[Series[Sum[x^k (1 + x^k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[d (d + 1) (2 d + 1)/6, {d, Divisors[n]}], {n, 47}]
Table[(DivisorSigma[1, n] + 3 DivisorSigma[2, n] + 2 DivisorSigma[3, n])/6, {n, 47}]

a(n) = my(f = factor(n)); (2*sigma(f, 3) + 3*sigma(f, 2) + sigma(f, 1)) / 6; \\ Amiram Eldar, Jan 03 2025

G.f.: Sum_{k>=1} A000330(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d*(d + 1)*(2*d + 1)/6.
a(n) = (A000203(n) + 3*A001157(n) + 2*A001158(n))/6.
a(n) = Sum_{i=1..n} i^2*A135539(n,i). - Ridouane Oudra, Jul 22 2022
From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: zeta(s) * (2*zeta(s-3) + 3*zeta(s-2) + zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/12) * n^4. (End)

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A332085 Number of ordered pairs of divisors of n, (d1,d2), such that d1 is prime and d1 <= d2.

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A086670 Sum of floor(d/2) where d is a divisor of n.

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A143356 A051731 * A006218.

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A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

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