A005094 -id:A005094 - cp's OEIS frontend

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101

Offset: 1

Antti Karttunen, Feb 01 2016

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.
Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.
Sequences A072202 and A078613 are closed with respect to this permutation.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
Cf. A001481, A268377, A004431, A268378, A004613, A004614.
Cf. also A108548.

up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to),i=0,c=0); forprime(p=2,prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to,m,k) = { my(v=vector(up_to),i=0); forprime(p=2,, if(k==(p%m), i++; v[i] = p; if(i==up_to,return(v)))); };
v002144 = list_primes_of_the_form(2*up_to,4,1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to,4,3);
A002145(n) = v002145[n];
A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A267101(primepi(f[k,1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A267101(n).
a(2*n) = 2*a(n).
a(3*n) = 5*a(n).
a(5*n) = 3*a(n).
a(7*n) = 13*a(n).
a(11*n) = 17*a(n).
etc. See examples in A267101.
A000035(n) = A000035(a(n)). [Preserves the parity of n.]
A005094(a(n)) = -A005094(n).
A079635(a(n)) = -A079635(n).

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

A005091 Number of distinct primes = 3 mod 4 dividing n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Peter Koymans, On Dirichlet biquadratic fields, arXiv:2001.05350 [math.NT], 2020.

Cf. A001221, A005089, A005094.

Cf. A079261, A027748, A072437, A187811.

a005091 = sum . map a079261 . a027748_row
-- Reinhard Zumkeller, Jan 07 2013

[0] cat [#[p:p in PrimeDivisors(n)| p mod 4 eq 3]: n in [2..100]]; // Marius A. Burtea, Nov 19 2019

[0] cat [&+[Binomial(p,3) mod 2:p in PrimeDivisors(n)]:n in [2..100]]; // Marius A. Burtea, Nov 19 2019

with(numtheory): seq(add(binomial(p,3) mod 2, p in factorset(n)), n=1..100); # Ridouane Oudra, Nov 19 2019

f[n_]:=Length@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==3&]; Table[f[n],{n,102}] (* Ray Chandler, Dec 18 2011 *)

for(n=1,100,print1(sumdiv(n,d,isprime(d)*if((d-3)%4,0,1)),","))

from sympy import primefactors
def A005091(n): return sum(1 for p in primefactors(n) if p&3==3) # Chai Wah Wu, Jul 07 2024

Additive with a(p^e) = 1 if p = 3 (mod 4), 0 otherwise.
From Reinhard Zumkeller, Jan 07 2013: (Start)
a(n) = Sum_{k=1..A001221(n)} A079261(A027748(n,k)).
a(A072437(n)) = 0.
a(A187811(n)) > 0. (End)
a(n) = Sum_{p|n} (binomial(p,3) mod 2), where p is a prime. - Ridouane Oudra, Nov 19 2019

A005089 Number of distinct primes == 1 (mod 4) dividing n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Peter Koymans, On Dirichlet biquadratic fields, arXiv:2001.05350 [math.NT], 2020.

Cf. A001221, A005091, A005094, A083025 (with multiplicity).

Cf. A079260, A027748, A004144, A009003.

a005089 = sum . map a079260 . a027748_row
-- Reinhard Zumkeller, Jan 07 2013

[#[p:p in PrimeDivisors(n)|p mod 4 eq 1]: n in [1..100]]; // Marius A. Burtea, Jan 16 2020

A005089 := proc(n)
    local a,pe;
    a := 0 ;
    for pe in ifactors(n)[2] do
        if modp(op(1,pe),4) =1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end  proc:
seq(A005089(n),n=1..100) ; # R. J. Mathar, Jul 22 2021

f[n_]:=Length@Select[If[n==1,{},FactorInteger[n]],Mod[#[[1]],4]==1&]; Table[f[n],{n,102}] (* Ray Chandler, Dec 18 2011 *)
a[n_] := DivisorSum[n, Boole[PrimeQ[#] && Mod[#, 4] == 1]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

for(n=1,100,print1(sumdiv(n,d,isprime(d)*if((d-1)%4,0,1)),","))

Additive with a(p^e) = 1 if p == 1 (mod 4), 0 otherwise.
From Reinhard Zumkeller, Jan 07 2013: (Start)
a(n) = Sum_{k=1..A001221(n)} A079260(A027748(n,k)).
a(A004144(n)) = 0.
a(A009003(n)) > 0. (End)

A079635 Sum of (2 - p mod 4) for all prime factors p of n (with repetition).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 30 2003

sign

a(n) = {number of primes of the form 4k+1 dividing n} minus {number of primes of the form 4k+3 dividing n}, both counted with multiplicity. - Antti Karttunen, Feb 03 2016, after the formula.

			a(55) = a(5*11) = (2 - 5 mod 4)+(2 - 11 mod 4) = (2-1)+(2-3) = (1)+(-1) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A065339, A083025.
Cf. A072202 (indices of zeros), A268379 (of strictly positive terms), A268380 (of negative terms), A268381 (of nonnegative terms).
Cf. A027746, A267099.
Cf. A005094 (difference when counting only distinct primes).

a079635 1 = 0
a079635 n = sum $ map ((2 - ) . (`mod` 4)) $ a027746_row n
-- Reinhard Zumkeller, Jan 10 2012

f:= proc(n) local t;
add(t[2]*(2-(t[1] mod 4)), t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Feb 05 2016

f[n_]:=Plus@@((2-Mod[#[[1]],4])*#[[2]]&/@If[n==1,{},FactorInteger[n]]); Table[f[n],{n,100}] (* Ray Chandler, Dec 20 2011 *)

(define (A079635 n) (- (A083025 n) (A065339 n))) ;; Antti Karttunen, Feb 03 2016

a(n) = A083025(n) - A065339(n).
Other identities. For all n >= 1:
a(A267099(n)) = -a(n). - Antti Karttunen, Feb 03 2016
Totally additive with a(2) = 0, a(p) = 1 if p == 1 (mod 4), and a(p) = -1 if p == 3 (mod 4). - Amiram Eldar, Jun 17 2024

Edited by Ray Chandler, Dec 20 2011

A078613 Same numbers of distinct prime factors of forms 4k+1 and 4k+3.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 30, 32, 35, 39, 45, 51, 55, 60, 64, 70, 75, 78, 87, 90, 91, 95, 102, 110, 111, 115, 117, 119, 120, 123, 128, 135, 140, 143, 150, 153, 155, 156, 159, 174, 175, 180, 182, 183, 187, 190, 203, 204, 215, 219, 220, 222, 225, 230, 234, 235, 238, 240

Offset: 1

Reinhard Zumkeller, Dec 10 2002

nonn

Equivalently, numbers n such that A005089(n)=A005091(n); A005094(a(n))=0.
A001221(a(n)) and a(n) are of opposite parity.
If m is in the sequence, then also 2*m.
Conjecture : a(n) is asymptotic to c*n where c is around 4 - Benoit Cloitre, Jan 06 2003

			n = 99 = [(4*0+3)^2]*[(4*1+1)], therefore 99 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072202.

Cf. A221264, A221265.

a078613 n = a078613_list !! (n-1)
a078613_list = filter ((== 0) . a005094) [1..]
-- Reinhard Zumkeller, Jan 07 2013

fQ[n_]:=Plus@@((Mod[#[[1]],4]-2)&/@If[n==1,{},FactorInteger[n]])==0; Select[Range[240],fQ] (* Ray Chandler, Dec 18 2011*)

Edited by Ray Chandler, Dec 18 2011

A221264 Numbers having fewer distinct prime factors of form 4k+1 than of 4k+3.

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 14, 18, 19, 21, 22, 23, 24, 27, 28, 31, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 92, 93, 94, 96, 98, 99, 103, 105, 107, 108, 112, 114, 118, 121, 124, 126, 127, 129

Offset: 1

Reinhard Zumkeller, Jan 07 2013

A005089(a(n)) < A005091(a(n)); A005094(a(n)) < 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005089, A005091, A005094, A078613, A221265.

a221264 n = a221264_list !! (n-1)
a221264_list = filter ((< 0) . a005094) [1..]

is_A221264(n)={#(n=vecsort(factor(n>>valuation(n,2))[,1]%4))&&n[(1+#n)\2]==3} \\ M. F. Hasler, Dec 17 2014

A221265 Numbers having more distinct prime factors of form 4k+1 than of 4k+3.

Original entry on oeis.org

5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 97, 100, 101, 104, 106, 109, 113, 116, 122, 125, 130, 136, 137, 145, 146, 148, 149, 157, 160, 164, 169, 170, 173, 178, 181, 185, 193, 194, 195, 197, 200, 202

Offset: 1

Reinhard Zumkeller, Jan 07 2013

nonn

A005089(a(n)) > A005091(a(n)); A005094(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A078613, A221264.

Cf. A005089, A005091, A005094.

a221265 n = a221265_list !! (n-1)
a221265_list = filter ((> 0) . a005094) [1..]

mdpf1Q[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},Count[f,?(Mod[ #,4] == 1&)]>Count[f,?(Mod[#,4]==3&)]]; Select[Range[2,250],mdpf1Q] (* Harvey P. Dale, Mar 03 2016 *)

A368647 The number of distinct primes of the form 3k+2 dividing n minus the number of distinct primes of the form 3k+1 dividing n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, -1, 1, 0, 2, 1, 1, -1, 0, 1, 1, 1, 1, -1, 2, -1, 2, 1, 1, 1, 0, 0, 0, 1, 2, -1, 1, 1, 2, 0, 1, -1, 0, -1, 2, 1, 0, -1, 2, 1, 2, 1, 1, -1, 2, 1, 0, 1, 1, 2, 0, -1, 2, 1, 2, -1, 0, -1, 1, 0, 2, -1, 2, 1, 1, 1, 1, -1, 0, 1, 0, 0, 0, -1, 2, 0, 2

Offset: 1

Amiram Eldar, Jan 02 2024

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

Cf. A002476, A003627, A005094, A005088, A005090, A086241.

f[p_, e_] := Switch[Mod[p, 3], 0, 0, 1, -1, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(p = factor(n)[, 1]); sum(i = 1, #p, if(p[i]%3 == 0, 0, if(p[i]%3 == 1, -1, 1)));}

Additive with a(p^e) = 0 if p = 3, 1 if p == 2 (mod 3), and -1 if p == 1 (mod 3).
a(n) = A005090(n) - A005088(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A086241 = 0.641944... .

cp's OEIS Frontend

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(x*y) = a(x)*a(y) for x, y > 1.

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A005091 Number of distinct primes = 3 mod 4 dividing n.

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A005089 Number of distinct primes == 1 (mod 4) dividing n.

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A079635 Sum of (2 - p mod 4) for all prime factors p of n (with repetition).

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A078613 Same numbers of distinct prime factors of forms 4*k+1 and 4*k+3.

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Haskell

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A221264 Numbers having fewer distinct prime factors of form 4*k+1 than of 4*k+3.

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Haskell

PARI

A221265 Numbers having more distinct prime factors of form 4*k+1 than of 4*k+3.

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Author

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

A078613 Same numbers of distinct prime factors of forms 4k+1 and 4k+3.

A221264 Numbers having fewer distinct prime factors of form 4k+1 than of 4k+3.

A221265 Numbers having more distinct prime factors of form 4k+1 than of 4k+3.

A368647 The number of distinct primes of the form 3k+2 dividing n minus the number of distinct primes of the form 3k+1 dividing n.