A097706 -id:A097706 - cp's OEIS frontend

A286363 Least number with the same prime signature as {the largest divisor of n with only prime factors of the form 4k+3} has: a(n) = A046523(A097706(n)).

1, 1, 2, 1, 1, 2, 2, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 1, 6, 2, 2, 2, 1, 1, 8, 2, 1, 2, 2, 1, 6, 1, 2, 4, 1, 2, 2, 1, 1, 6, 2, 2, 4, 2, 2, 2, 4, 1, 2, 1, 1, 8, 2, 2, 6, 1, 2, 2, 1, 2, 12, 1, 1, 6, 2, 1, 6, 2, 2, 4, 1, 1, 2, 2, 6, 2, 2, 1, 16, 1, 2, 6, 1, 2, 2, 2, 1, 4, 2, 2, 6, 2, 2, 2, 1, 4, 12, 1, 1, 2, 2, 1, 6, 1, 2, 8, 1, 2, 2, 2, 1, 6, 2, 1, 4, 2, 2, 2

Offset: 1

Antti Karttunen, May 08 2017

nonn

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

Cf. A046523, A097706, A286361, A286364, A286365.

Cf. also A065338, A065339, A260728, A267099.

from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a072436(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])
def a(n): return a046523(n/a072436(n)) # Indranil Ghosh, May 09 2017

(define (A286363 n) (A046523 (A097706 n)))

a(n) = A046523(A097706(n)).

a(n) = A286361(A267099(n)).

A324893 a(n) = sigma(A097706(n)), where A097706(n) is the part of n composed of prime factors of form 4k+3.

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 8, 1, 13, 1, 12, 4, 1, 8, 4, 1, 1, 13, 20, 1, 32, 12, 24, 4, 1, 1, 40, 8, 1, 4, 32, 1, 48, 1, 8, 13, 1, 20, 4, 1, 1, 32, 44, 12, 13, 24, 48, 4, 57, 1, 4, 1, 1, 40, 12, 8, 80, 1, 60, 4, 1, 32, 104, 1, 1, 48, 68, 1, 96, 8, 72, 13, 1, 1, 4, 20, 96, 4, 80, 1, 121, 1, 84, 32, 1, 44, 4, 12, 1, 13, 8, 24, 128, 48, 20, 4, 1, 57, 156, 1, 1, 4

Offset: 1

Antti Karttunen, Mar 27 2019

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A000593, A097706, A324891.

Array[DivisorSigma[1, Times @@ Power @@@ Select[FactorInteger[#], Mod[#[[1]], 4] == 3 &]] &, 102] (* Michael De Vlieger, Mar 30 2019 *)

A324893(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4<3, 1, ((f[i,1]^(1+f[i,2]))-1)/(f[i,1]-1))); };

A097706(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4<3, 1, f[i, 1])^f[i, 2]); };
A324893(n) = sigma(A097706(n));

Multiplicative with a(p^e) = (p^(e+1) - 1)/(p-1) if p == 3 (mod 4), otherwise a(p^e) = 1.
a(n) = A000203(A097706(n)).
a(n) = A000593(n) / A324891(n).

A034947 Jacobi (or Kronecker) symbol (-1/n).

Original entry on oeis.org

1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, 1, 1

Offset: 1

N. J. A. Sloane

Also the regular paper-folding sequence.
For a proof that a(n) equals the paper-folding sequence, see Allouche and Sondow, arXiv v4. - Jean-Paul Allouche and Jonathan Sondow, May 19 2015
It appears that, replacing +1 with 0 and -1 with 1, we obtain A038189. Alternatively, replacing -1 with 0 we obtain (allowing for offset) A014577. - Jeremy Gardiner, Nov 08 2004
Partial sums = A005811 starting (1, 2, 1, 2, 3, 2, 1, 2, 3, ...). - Gary W. Adamson, Jul 23 2008
The congruence in {-1,1} of the odd part of n modulo 4 (Cf. A099545). - Peter Munn, Jul 09 2022

			G.f. = x + x^2 - x^3 + x^4 + x^5 - x^6 - x^7 + x^8 + x^9 + x^10 - x^11 - x^12 + ...

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 155, 182.
H. Cohen, Course in Computational Number Theory, p. 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J.-P. Allouche, G.-N. Han, and Jeffrey Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.
J.-P. Allouche and Jonathan Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, Electron. J. Combin., 22 #1 (2015) P1.59; see p. 8.
J.-P. Allouche and Jonathan Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, arXiv:1408.5770 [math.NT] v4, 2015; see p. 9.
Jean-Paul Allouche and Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.
L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (1)
Joerg Arndt, Matters Computational (The Fxtbook), section 38.8.4 Differences of the sum of Gray code digits, coefficients of polynomials L.
Danielle Cox and K. McLellan, A problem on generation sets containing Fibonacci numbers, Fib. Quart., 55 (No. 2, 2017), 105-113.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted with addendum in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614. a(n) = d(n) at equation 3.1.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Jeffrey Shallit, Cloitre's Self-Generating Sequence, arXiv:2501.00784 [math.CO], 2025. See page 4.
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Index entries for sequences obtained by enumerating foldings.

Cf. A000265, A005811, A000364, A008836, A065339, A097706, A099545, A209615.
Moebius transform of A035184.
Cf. A091072 (indices of 1), A091067 (indices of -1), A371594 (indices of run starts).
The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012

[KroneckerSymbol(-1,n): n in [1..100]]; // Vincenzo Librandi, Aug 16 2016

with(numtheory): A034947 := n->jacobi(-1,n);

Table[KroneckerSymbol[ -1, n], {n, 0, 100}] (* Corrected by Jean-François Alcover, Dec 04 2013 *)

PARI
```
{a(n) = kronecker(-1, n)};
```

for(n=1, 81, f=factor(n); print1((-1)^sum(s=1, omega(n), f[s, 2]*(Mod(f[s, 1], 4)==3)), ", ")); \\ Arkadiusz Wesolowski, Nov 05 2013

a(n)=direuler(p=1,n,if(p==2,1/(1-kronecker(-4, p)*X)/(1-X),1/(1-kronecker(-4, p)*X))) /* Ralf Stephan, Mar 27 2015 */

a(n) = if(n%2==0, a(n/2), (n+2)%4-2) \\ Peter Munn, Jul 09 2022

def A034947(n):
    s = bin(n)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return 1-2*int(s[m-i-2]) if m-i-2 >= 0 else 1 # Chai Wah Wu, Apr 08 2021

def A034947(n): return -1 if n>>(-n&n).bit_length()&1 else 1 # Chai Wah Wu, Feb 26 2025

Multiplicative with a(2^e) = 1, a(p^e) = (-1)^(e*(p-1)/2) if p>2.
a(2*n) = a(n), a(4*n+1) = 1, a(4*n+3) = -1, a(-n) = -a(n). a(n) = 2*A014577(n-1)-1.
a(prime(n)) = A070750(n) for n > 1. - T. D. Noe, Nov 08 2004
This sequence can be constructed by starting with w = "empty string", and repeatedly applying the map w -> w 1 reverse(-w) [See Allouche and Shallit p. 182]. - N. J. A. Sloane, Jul 27 2012
a(n) = (-1)^A065339(n) = lambda(A097706(n)), where A065339(n) is number of primes of the form 4*m + 3 dividing n (counted with multiplicity) and lambda is Liouville's function, A008836. - Arkadiusz Wesolowski, Nov 05 2013 and Peter Munn, Jun 22 2022
Sum_{n>=1} a(n)/n = Pi/2, due to F. von Haeseler; more generally, Sum_{n>=1} a(n)/n^(2*d+1) = Pi^(2*d+1)*A000364(d)/(2^(2*d+2)-2)(2*d)! for d >= 0; see Allouche and Sondow, 2015. - Jean-Paul Allouche and Jonathan Sondow, Mar 20 2015
Dirichlet g.f.: beta(s)/(1-2^(-s)) = L(chi_2(4),s)/(1-2^(-s)). - Ralf Stephan, Mar 27 2015
a(n) = A209615(n) * (-1)^(v2(n)), where v2(n) = A007814(n) is the 2-adic valuation of n. - Jianing Song, Apr 24 2021
a(n) = 2 - A099545(n) == A000265(n) (mod 4). - Peter Munn, Jun 22 2022 and Jul 09 2022

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

Original entry on oeis.org

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

A072437 Numbers with no prime factors of form 4*k+3.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 16, 17, 20, 25, 26, 29, 32, 34, 37, 40, 41, 50, 52, 53, 58, 61, 64, 65, 68, 73, 74, 80, 82, 85, 89, 97, 100, 101, 104, 106, 109, 113, 116, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 157, 160, 164, 169, 170, 173, 178, 181, 185, 193, 194

Offset: 1

Reinhard Zumkeller, Jun 17 2002

nonn

m is a term iff A072436(m) = m.

These numbers have density zero (Pollack).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 119. alternate link

Cf. A004144, A002144, A002145, A004613 (odd terms).
A097706(a(n)) = 1.
Cf. A187811 (complement).

import Data.List (elemIndices)
a072437 n = a072437_list !! (n-1)
a072437_list = map (+ 1) $ elemIndices 0 a005091_list
-- Reinhard Zumkeller, Jan 07 2013

npfQ[n_]:=Count[Transpose[FactorInteger[n]][[1]],?(Mod[#,4]==3&)]==0; Select[Range[200],npfQ] (* _Harvey P. Dale, Nov 12 2013 *)

is(n)=n==1||vecmax(factor(n)[,1]%4)<3 \\ Charles R Greathouse IV, Apr 16 2012

n>0 such that A001842(n)=0. - Benoit Cloitre, Apr 24 2003
A005091(a(n)) = 0. - Reinhard Zumkeller, Jan 07 2013
A065339(a(n)) = 0 . - R. J. Mathar, Jan 28 2025

A170818 a(n) is the product of primes (with multiplicity) of form 4*k+1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 13, 1, 5, 1, 17, 1, 1, 5, 1, 1, 1, 1, 25, 13, 1, 1, 29, 5, 1, 1, 1, 17, 5, 1, 37, 1, 13, 5, 41, 1, 1, 1, 5, 1, 1, 1, 1, 25, 17, 13, 53, 1, 5, 1, 1, 29, 1, 5, 61, 1, 1, 1, 65, 1, 1, 17, 1, 5, 1, 1, 73, 37, 25, 1, 1, 13, 1, 5, 1, 41, 1, 1, 85, 1, 29, 1

Offset: 1

N. J. A. Sloane, Dec 22 2009

Completely multiplicative with a(p) = p if p = 4k+1 and a(p) = 1 otherwise. - Tom Edgar, Mar 05 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340.
Index to divisibility sequences

Cf. A170817-A170819, A097706, A083025, A072438, A286361.

a:= n-> mul(`if`(irem(i[1], 4)=1, i[1]^i[2], 1), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 1, p^e, 1], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, May 29 2019 *)

a(n)=my(f=factor(n)); prod(i=1,#f~,if(f[i,1]%4>1,1,f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import factorint, prod
def a072438(n):
    f = factorint(n)
    return 1 if n == 1 else prod(i**f[i] for i in f if i % 4 != 1)
def a(n): return n//a072438(n) # Indranil Ghosh, May 08 2017

a(n) = n/A072438(n). - Michel Marcus, Mar 05 2015

A343430 Part of n composed of prime factors of the form 3k-1.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 29, 10, 1, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 53, 2, 55, 8, 1, 58, 59, 20, 1, 2, 1, 64, 5, 22, 1, 68, 23, 10, 71, 8, 1, 2, 25, 4, 11, 2, 1, 80, 1, 82, 83, 4, 85

Offset: 1

Peter Munn, Jun 08 2021

Largest term of A004612 that divides n.
Modulo 6, the prime numbers are partitioned into 4 nonempty sets: {2}, {3}, primes of the form 6k-1 (A007528) and primes of the form 6k+1 (A002476). The modulo 3 partition is nearly the same, but unites the only even prime, 2, with primes of the form 6k-1 in the set of primes we use here.
A positive integer m is a Loeschian number (a term of A003136) if and only if a(A007913(m)) = 1, that is the squarefree part of m has no prime factors of the form 3k-1.

			n = 60 has prime factorization 60 = 2 * 2 * 3 * 5. Factors 2 = 3*1 - 1 and 5 = 3*2 - 1 have form 3k-1, whereas 3 does not (having form 3k). Multiplying the factors of form 3k-1, we get 2 * 2 * 5 = 20. So a(60) = 20.

Equivalent sequences for prime factors of other forms: A006519 (2 only), A000265 (2k+1), A038500 (3 only), A038502 (3k+/-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A343431 (6k-1).
Range of terms: A004612 (closure under multiplication of A003627).
Cf. A002476, A007528, squarefree part (A007913) of terms of A003136.
First 28 terms are the same as A247503.

f[p_, e_] := If[Mod[p, 3] == 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 11 2021 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 3, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343430(n): return prod(p**e for p, e in factorint(n).items() if p%3==2) # Chai Wah Wu, Dec 23 2022

Completely multiplicative with a(p) = p if p is of the form 3k-1, otherwise a(p) = 1.
For k >= 1, a(n) = a(k*n) / gcd(k, a(k*n)).
a(n) = A006519(n) * A343431(n).
a(n) = (n / A038500(n)) / A248909(n) = A038502(n) / A248909(n).
A006519(a(n)) = a(A006519(n)) = A006519(n).
A343431(a(n)) = a(A343431(n)) = A343431(n).
A038500(a(n)) = a(A038500(n)) = 1.
A248909(a(n)) = a(A248909(n)) = 1.

A248909 Completely multiplicative with a(p) = p if p = 6k+1 and a(p) = 1 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 19, 1, 7, 1, 1, 1, 1, 13, 1, 7, 1, 1, 31, 1, 1, 1, 7, 1, 37, 19, 13, 1, 1, 7, 43, 1, 1, 1, 1, 1, 49, 1, 1, 13, 1, 1, 1, 7, 19, 1, 1, 1, 61, 31, 7, 1, 13, 1, 67, 1, 1, 7, 1, 1, 73, 37, 1, 19, 7, 13, 79, 1, 1

Offset: 1

Tom Edgar, Mar 06 2015

To compute a(n) replace primes not of the form 6k+1 in the prime factorization of n by 1.
The first place this sequence differs from A170824 is at n = 49.
For p prime, a(p) = p if p is a term in A002476 and a(p) = 1 if p = 2, p = 3 or p is a term in A007528.
a(n) is the largest term of A004611 that divides n. - Peter Munn, Mar 06 2021

			a(49) = 49 because 49 = 7^2 and 7 = 6*1 + 1.
a(15) = 1 because 15 = 3*5 and neither of these primes is of the form 6k+1.
a(62) = 31 because 62 = 31*2, 31 = 6*5 + 1, and 2 is not of the form 6k+1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Sequences used in a definition of this sequence: A002476, A004611, A007528, A020639, A028234, A032742.
Equivalent sequence for distinct prime factors: A170824.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A343431 (6k-1), A065330 (6k+/-1), A065331 (<= 3).

A248909 := proc(n)
    local a,pf;
    a := 1 ;
    for pf in ifactors(n)[2] do
        if modp(op(1,pf),6) = 1 then
            a := a*op(1,pf)^op(2,pf) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 14 2015

f[p_, e_] := If[Mod[p, 6] == 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i,1] - 1) % 6, f[i, 1] = 1);); factorback(f);} \\ Michel Marcus, Mar 11 2015

from sympy import factorint
def A248909(n):
    y = 1
    for p,e in factorint(n).items():
        y *= (1 if (p-1) % 6 else p)**e
    return y # Chai Wah Wu, Mar 15 2015

n=100; sixnplus1Primes=[x for x in primes_first_n(100) if (x-1)%6==0]
[prod([(x^(x in sixnplus1Primes))^y for x,y in factor(n)]) for n in [1..n]]

(define (A248909 n) (if (= 1 n) n (* (if (= 1 (modulo (A020639 n) 6)) (A020639 n) 1) (A248909 (A032742 n))))) ;; Antti Karttunen, Jul 09 2017

a(1) = 1; for n > 1, if A020639(n) = 1 (mod 6), a(n) = A020639(n) * a(A032742(n)), otherwise a(n) = a(A028234(n)). - Antti Karttunen, Jul 09 2017

a(n) = a(A065330(n)). - Peter Munn, Mar 06 2021

A260728 Bitwise-OR of the exponents of all 4k+3 primes in the prime factorization of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

A001481 (numbers that are the sum of 2 squares) gives the positions of even terms in this sequence, while its complement A022544 (numbers that are not the sum of 2 squares) gives the positions of odd terms.

If instead of bitwise-oring (A003986) we added in ordinary way the exponents of 4k+3 primes together, we would get the sequence A065339. For the positions where these two sequences differ see A260730.

			For n = 21 = 3^1 * 7^1 we compute A003986(1,1) = 1, thus a(21) = 1.
For n = 63 = 3^2 * 7^1 we compute A003986(2,1) = A003986(1,2) = 3, thus a(63) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000035, A001481, A022544, A003986, A020639, A028234, A032742, A067029, A097706, A229062, A260730.
Cf. also A267113, A267116, A267099.
Differs from A065339 for the first time at n=21, where a(21) = 1, while A065339(21)=2.

Table[BitOr @@ (Map[Last, FactorInteger@ n /. {p_, } /; MemberQ[{0, 1, 2}, Mod[p, 4]] -> Nothing]), {n, 0, 120}] (* _Michael De Vlieger, Feb 07 2016 *)

from functools import reduce
from operator import or_
from sympy import factorint
def A260728(n): return reduce(or_,(e for p, e in factorint(n).items() if p & 3 == 3),0) # Chai Wah Wu, Jun 28 2022

(define (A260728 n) (cond ((< n 3) 0) ((even? n) (A260728 (/ n 2))) ((= 1 (modulo (A020639 n) 4)) (A260728 (A032742 n))) (else (A003986bi (A067029 n) (A260728 (A028234 n)))))) ;; A003986bi implements bitwise-or (see A003986).

If n < 3, a(n) = 0; thereafter, for any even n: a(n) = a(n/2), for any n with its smallest prime factor (A020639) of the form 4k+1: a(n) = a(A032742(n)), otherwise [when A020639(n) is of the form 4k+3] a(n) = A003986(A067029(n),a(A028234(n))).
Other identities. For all n >= 0:
A229062(n) = 1 - A000035(a(n)). [Reduced modulo 2 and complemented, the sequence gives the characteristic function of A001481.]
a(n) = a(A097706(n)). [The result depends only on the prime factors of the form 4k+3.]
a(n) = A267116(A097706(n)).
a(n) = A267113(A267099(n)).

A170819 a(n) = product of distinct primes of the form 4k-1 that divide n.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 3, 1, 11, 3, 1, 7, 3, 1, 1, 3, 19, 1, 21, 11, 23, 3, 1, 1, 3, 7, 1, 3, 31, 1, 33, 1, 7, 3, 1, 19, 3, 1, 1, 21, 43, 11, 3, 23, 47, 3, 7, 1, 3, 1, 1, 3, 11, 7, 57, 1, 59, 3, 1, 31, 21, 1, 1, 33, 67, 1, 69, 7, 71, 3, 1, 1, 3, 19, 77, 3, 79, 1, 3, 1, 83, 21, 1

Offset: 1

N. J. A. Sloane, Dec 23 2009

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A011765, A065339, A097706, A170817, A170818, A363340.

A170819 := proc(n) a := 1 ; for p in numtheory[factorset](n) do if p mod 4 = 3 then a := a*p ; end if; end do: a ; end proc:
seq(A170819(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Array[Times @@ Select[FactorInteger[#][[All, 1]], Mod[#, 4] == 3 &] &, 85] (* Michael De Vlieger, Feb 19 2019 *)

for(n=1,99, t=select(x->x%4==3, factor(n)[,1]); print1(prod(i=1,#t,t[i])","))

Multiplicative with a(p^e) = p^A011765(p+1), e > 0. - R. J. Mathar, Jun 07 2011

a(n) = A007947(A097706(n)) = A097706(A007947(n)). - Peter Munn, Jul 06 2023

Extended with PARI program by M. F. Hasler, Dec 23 2009

cp's OEIS Frontend

A286363 Least number with the same prime signature as {the largest divisor of n with only prime factors of the form 4k+3} has: a(n) = A046523(A097706(n)).

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A324893 a(n) = sigma(A097706(n)), where A097706(n) is the part of n composed of prime factors of form 4k+3.

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A034947 Jacobi (or Kronecker) symbol (-1/n).

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

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A072437 Numbers with no prime factors of form 4*k+3.

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A170818 a(n) is the product of primes (with multiplicity) of form 4*k+1 that divide n.

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