A378194 -id:A378194 - cp's OEIS frontend

A065339 Number of primes congruent to 3 modulo 4 dividing n (with multiplicity).

0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 1, 1, 1, 0, 0, 3, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 2, 1, 1, 2, 1, 1, 1, 2, 0, 1, 0, 0, 3, 1, 1, 2, 0, 1, 1, 0, 1, 3, 0, 0, 2, 1, 0, 2, 1, 1, 2, 0, 0, 1, 1, 2, 1, 1, 0, 4, 0, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 1, 0, 2, 3, 0, 0, 1, 1, 0, 2

Offset: 1

Reinhard Zumkeller, Oct 29 2001

nonn

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001222, A007814, A065338, A005091, A007949, A083025 (analogous for 4k+1 primes), A097706, A378194.

Cf. A020639, A027746, A028234, A032742, A067029, A260728, A260730.

a065339 1 = 0
a065339 n = length [x | x <- a027746_row n, mod x 4 == 3]
-- Reinhard Zumkeller, Jan 10 2012

A065339 := proc(n)
    a := 0 ;
    for f in ifactors(n)[2] do
        if op(1,f) mod 4 = 3 then
            a := a+op(2,f) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Dec 16 2011

f[n_]:=Plus@@Last/@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==3&]; Table[f[n],{n,100}] (* Ray Chandler, Dec 18 2011 *)

A065339(n)=sum(i=1,#n=factor(n)~,if(n[1,i]%4==3,n[2,i]))  \\ M. F. Hasler, Apr 16 2012

;; using memoization-macro definec
(definec (A065339 n) (cond ((< n 3) 0) ((even? n) (A065339 (/ n 2))) (else (+ (/ (- (modulo (A020639 n) 4) 1) 2) (A065339 (A032742 n))))))
;; Antti Karttunen, Aug 14 2015

;; using memoization-macro definec
(definec (A065339 n) (cond ((< n 3) 0) ((even? n) (A065339 (/ n 2))) ((= 1 (modulo (A020639 n) 4)) (A065339 (A032742 n))) (else (+ (A067029 n) (A065339 (A028234 n))))))
;; Antti Karttunen, Aug 14 2015

a(n) = A001222(n) - A007814(n) - A083025(n).
(2^A007814(n)) * (3^a(n)) = A065338(n).
From Antti Karttunen, Aug 14 2015: (Start)
a(1) = a(2) = 0; thereafter, if n is even, a(n) = a(n/2), otherwise a(n) = ((A020639(n) mod 4)-1)/2 + a(n/A020639(n)). [Where A020639(n) gives the smallest prime factor of n.]
Other identities and observations. For all n >= 1:
a(n) = A007949(A065338(n)).
a(n) = A001222(A097706(n)).
a(n) >= A260728(n). [See A260730 for the positions of differences.] (End)
Totally additive with a(2) = 0, a(p) = 1 if p == 3 (mod 4), and a(p) = 0 if p == 1 (mod 4). - Amiram Eldar, Jun 17 2024

A378193 Rectangular array read by descending antidiagonals: row n shows the integers m such that the number of Pythagorean primes (including multiplicities) that divide m is n-1.

Original entry on oeis.org

1, 2, 5, 3, 10, 25, 4, 13, 50, 125, 6, 15, 65, 250, 625, 7, 17, 75, 325, 1250, 3125, 8, 20, 85, 375, 1625, 6250, 15625, 9, 26, 100, 425, 1875, 8125, 31250, 78125, 11, 29, 130, 500, 2125, 9375, 40625, 156250, 390625, 12, 30, 145, 650, 2500, 10625, 46875, 203125, 781250, 1953125

Offset: 1

Clark Kimberling, Jan 14 2025

Every positive integer occurs exactly once.

			Corner:
     1     2     3     4      6       7
     5    10    13    15     17      20
    25    50    65    75     85     100
   125   250   325   375    425     500
   625  1250  1625  1875   2125    2500
  3125  6250  8125  9375  10625   12500

Cf. A083025, A002144, A002145, A378194.

A378193 := proc(n,k)
    option remember;
    local a;
    if k = 0 then
       0;
    else
        for a from procname(n,k-1)+1 do
            if A083025(a) = n-1 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(seq( A378193(n,d-n),n=1..d-1),d=2..10) ;

u = Map[Map[#[[1]] &, #] &, GatherBy[
    SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 1)/4] && PrimeQ[#] &,
             Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
             FactorInteger[#]]]], True]} &,
      Range[13000]], #[[2]] &], #[[2]] &]];
r[m_] := Take[u[[m]], 6];
w[m_, n_] := r[m][[n]];
Grid[Table[w[m, n], {m, 1, 6}, {n, 1, 6}]] (* array *)
Table[w[n - k + 1, k], {n, 6}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Nov 19 2024 *)

cp's OEIS Frontend

A065339 Number of primes congruent to 3 modulo 4 dividing n (with multiplicity).

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