A243381 -id:A243381 - cp's OEIS frontend

A087691 Squares of primes of the form 4*k+3.

9, 49, 121, 361, 529, 961, 1849, 2209, 3481, 4489, 5041, 6241, 6889, 10609, 11449, 16129, 17161, 19321, 22801, 26569, 27889, 32041, 36481, 39601, 44521, 49729, 51529, 57121, 63001, 69169, 73441, 80089, 94249, 96721, 109561, 120409, 128881

Offset: 1

Cino Hilliard, Sep 27 2003

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A002145, A243379, A243381.

Select[4*Range[0,100]+3,PrimeQ]^2 (* Harvey P. Dale, Sep 10 2012 *)

p4np3(n)= forprime(x=3,n,if(x%4==3,y=x*x; print1(y, ", ")));

a(n) = A002145(n)^2.
a(n) ~ 4n^2 * (log n)^2. - Charles R Greathouse IV, Sep 20 2016
From Amiram Eldar, Dec 02 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A243381.
Product_{n>=1} (1 - 1/a(n)) = A243379. (End)

More terms from Ray Chandler, Oct 26 2003

A327122 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 + x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 3, 7, 7, 9, 7, 15, 10, 21, 11, 21, 15, 21, 21, 31, 19, 30, 19, 49, 21, 33, 23, 45, 38, 45, 30, 49, 31, 63, 31, 63, 33, 57, 49, 70, 39, 57, 45, 105, 43, 63, 43, 77, 70, 69, 47, 93, 50, 114, 57, 105, 55, 90, 77, 105, 57, 93, 59, 147, 63, 93, 70, 127, 105

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

Inverse Moebius transform of A050469.

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

Cf. A000203, A050469, A101455, A167181 (fixed points), A262209, A288417, A327123.

Cf. A072691, A175647, A243381.

nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
A050469[n_] := DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 4]] &] - DivisorSum[n, # &, MemberQ[{3}, Mod[n/#, 4]] &]; a[n_] := DivisorSum[n, A050469[#] &]; Table[a[n], {n, 1, 65}]
f[p_, e_] := If[Mod[p, 4] == 1, (p^(e+2)-(e+2)*p+e+1)/(p-1)^2, (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1))]; f[2, e_] := 2^(e+1)-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Aug 28 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p == 2, 2^(e+1)-1, if(p%4 == 1, (p^(e+2)-(e+2)*p+e+1)/(p-1)^2, (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1))))); } \\ Amiram Eldar, Aug 28 2023

a(n) = Sum_{d|n} A050469(d).
From Amiram Eldar, Aug 28 2023: (Start)
Multiplicative with a(2^e) = 2^(e+1)-1, and if p is an odd prime a(p^e) = (p^(e+2)-(e+2)*p+e+1)/(p-1)^2 if p == 1 (mod 4) and (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1)) otherwise.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 * (A175647/A243381) = 0.753351504961... . (End)

cp's OEIS Frontend

A087691 Squares of primes of the form 4*k+3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions