A147811 -id:A147811 - cp's OEIS frontend

A147807 Partial sums of A147810(n) = tau(n^2 + 1)/2.

1, 2, 4, 5, 7, 8, 11, 13, 15, 16, 18, 20, 24, 25, 27, 28, 32, 35, 37, 38, 42, 44, 48, 49, 51, 52, 56, 58, 60, 62, 66, 69, 73, 75, 77, 78, 82, 85, 87, 88, 91, 93, 99, 101, 103, 105, 113, 115, 117, 119, 121, 123, 127, 128, 132, 133, 141, 143, 145, 147, 149, 151, 155, 157

Offset: 1

M. F. Hasler, Dec 13 2008

Also, number of inequivalent (i.e., q < r) integer solutions to 1/pqr = 1/p - 1/q - 1/r with p <= n; cf. A147811.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A093582, A147806, A147809, A147810, A147811.

Accumulate[DivisorSigma[0, Range[64]^2 + 1]/2] (* Amiram Eldar, Oct 25 2019 *)

s=0;A147807=vector(99,n,s+=numdiv(n^2+1))/2

a(n) = Sum_{p = 1..n} tau(1 + p^2)/2 = n + A147806(n) > n.

a(n) ~ c * n * log(n), where c = 3/(2*Pi) = 0.477464... (A093582). - Amiram Eldar, Dec 01 2023

A147806 Partial sums of A147809(n) = tau(n^2 + 1)/2 - 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 5, 6, 6, 7, 8, 11, 11, 12, 12, 15, 17, 18, 18, 21, 22, 25, 25, 26, 26, 29, 30, 31, 32, 35, 37, 40, 41, 42, 42, 45, 47, 48, 48, 50, 51, 56, 57, 58, 59, 66, 67, 68, 69, 70, 71, 74, 74, 77, 77, 84, 85, 86, 87, 88, 89, 92, 93, 94, 94, 97, 100, 101, 103, 104, 107

Offset: 1

M. F. Hasler, Dec 13 2008

It seems that a(10^n) = (6, 168, 2754, 38561, 495569, ...) ~ 1.1*(n-0.5)*10^n; otherwise said, a(n) ~ 1.1*(log_10(n)-0.5)*n, asymptotically.

The exact value of the coefficient above is 3*log(10)/(2*Pi) = 1.09940339... . - Amiram Eldar, Dec 01 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A093582, A147807, A147809, A147810, A147811.

Accumulate[DivisorSigma[0, Range[72]^2 + 1]/2 - 1] (* Amiram Eldar, Oct 25 2019 *)

s=0;a147806=vector(99,n,s+=numdiv(n^2+1))/2
A147806(n)=sum(p=1,n,numdiv(n^2+1))/2-n

a(n) = A147807(n) - n.

a(n) ~ c * n * log(n), where c = 3/(2*Pi) = 0.477464... (A093582). - Amiram Eldar, Dec 01 2023

cp's OEIS Frontend

A147807 Partial sums of A147810(n) = tau(n^2 + 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula