A268379 -id:A268379 - cp's OEIS frontend

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

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

A268381 Numbers having at least the same number of prime factors of the form 4k+1 than of the form 4k+3, when counted with multiplicity.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 15, 16, 17, 20, 25, 26, 29, 30, 32, 34, 35, 37, 39, 40, 41, 50, 51, 52, 53, 55, 58, 60, 61, 64, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 91, 95, 97, 100, 101, 102, 104, 106, 109, 110, 111, 113, 115, 116, 119, 120, 122, 123, 125, 128, 130, 136, 137, 140, 143, 145, 146, 148, 149, 150

Offset: 1

Antti Karttunen, Feb 05 2016

nonn

Numbers n for which A083025(n) >= A065339(n) or equally, for which A079635(n) >= 0.

Closed under multiplication.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A268380.
Disjoint union of A072202 and A268379.
Cf. A079635, A065339, A083025.

Position[Array[Map[Length, {Select[#, Mod[#, 4] == 1 &], Select[#, Mod[#, 4] == 3 &]}] &@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ #, 1] &, {150}], {a_, b_} /; a >= b] // Flatten (* Michael De Vlieger, Feb 05 2016 *)

A268380 Numbers having fewer prime factors of the form 4k+1 than of the form 4k+3, when counted with multiplicity.

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, 45, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84, 86, 88, 90, 92, 93, 94, 96, 98, 99, 103, 105, 107, 108, 112, 114, 117, 118, 121, 124, 126, 127, 129, 131, 132, 133, 134, 135, 138, 139, 141

Offset: 1

Antti Karttunen, Feb 05 2016

nonn

Numbers n for which A083025(n) < A065339(n) or equally, for which A079635(n) < 0.

Closed under multiplication.

			45 = 3*3*5 is included as there are more prime factors of the form 4k+3 (here two 3's) than prime factors of the form 4k+1 (here just one 5).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A268381.
Cf. A079635, A065339, A083025.
Cf. also A072202, A268379.
Differs from A221264 for the first time at n=23, where a(23) = 45, a value missing from A221264.

Position[Array[Map[Length, {Select[#, Mod[#, 4] == 1 &], Select[#, Mod[#, 4] == 3 &]}] &@ Flatten@ Apply[Table[#1, {#2}] &, FactorInteger@ #, 1] &, {141}], {a_, b_} /; a < b] // Flatten (* Michael De Vlieger, Feb 05 2016 *)

isok(n) = {my(f = factor(n)); sum(k=1, #f~, ((f[k,1] % 4)==1)*f[k,2]) < sum(k=1, #f~, ((f[k,1] % 4)==3)*f[k,2]);} \\ Michel Marcus, Feb 05 2016

cp's OEIS Frontend

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

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A268381 Numbers having at least the same number of prime factors of the form 4k+1 than of the form 4k+3, when counted with multiplicity.

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A268380 Numbers having fewer prime factors of the form 4k+1 than of the form 4k+3, when counted with multiplicity.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Haskell

Maple

Mathematica

Scheme

Formula

Extensions

A268381 Numbers having at least the same number of prime factors of the form 4*k+1 than of the form 4*k+3, when counted with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A268380 Numbers having fewer prime factors of the form 4*k+1 than of the form 4*k+3, when counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A268381 Numbers having at least the same number of prime factors of the form 4k+1 than of the form 4k+3, when counted with multiplicity.

A268380 Numbers having fewer prime factors of the form 4k+1 than of the form 4k+3, when counted with multiplicity.