A065339 -id:A065339 - cp's OEIS frontend

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

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

A072202 Same numbers of prime factors of forms 4k+1 and 4k+3, counted with multiplicity.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 30, 32, 35, 39, 51, 55, 60, 64, 70, 78, 87, 91, 95, 102, 110, 111, 115, 119, 120, 123, 128, 140, 143, 155, 156, 159, 174, 182, 183, 187, 190, 203, 204, 215, 219, 220, 222, 225, 230, 235, 238, 240, 246, 247, 256, 259, 267, 280, 286, 287, 291

Offset: 1

Reinhard Zumkeller, Jul 03 2002

nonn

Equivalently, numbers n such that A083025(n) = A065339(n), indices of zeros in A079635.
Closed under multiplication.
Closed with respect to permutation A267099. - Antti Karttunen, Feb 03 2016

			825 = 3*5*5*11 = [(4*0+3)*(4*2+3)]*[(4*1+1)*(4*1+1)], therefore 825 is a term.

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

Cf. A002144, A002145, A202237 (odd elements), A079635, A065339, A083025, A267099.
Primitive elements are {2} U A080774. - Franklin T. Adams-Watters, Dec 16 2011.
Subsequence of A078613 and of A268381.

a072202 n = a072202_list !! (n-1)
a072202_list = [x | x <- [1..], a083025 x == a065339 x]
-- Reinhard Zumkeller, Jan 10 2012

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

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

(define A072202 (ZERO-POS 1 1 A079635)) ;; [requires also my IntSeq-library] - Antti Karttunen, Feb 03 2016

A014709 The regular paper-folding (or dragon curve) sequence. Alphabet {1,2}.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane

nonn

Over the alphabet {a,b} this is aabaabbaaabbabbaaabaabbbaabbabbaaaba...

With offset 1, completely multiplicative modulo 3. - Peter Munn, Jun 20 2022

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 155, 182.
G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Gabriele Fici and Luca Q. Zamboni, On the least number of palindromes contained in an infinite word, Theoretical Computer Science, Volume 481, 2013, pp. 1-8. See page 1.
Index entries for sequences obtained by enumerating foldings

See A014577 for more references and more terms.
The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012
Cf. A065339.

(3 - JacobiSymbol[-1, Range[100]])/2 (* Paolo Xausa, May 26 2024 *)

a(n)=if(n%2==0, 1+bitand(1,n\2), a(n\2) );
for(n=0,122,print1(a(n),", "))

Set a=1, b=2, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).
a(4n) = 1, a(4n+2) = 2, a(2n+1) = a(n).
a(n) = (3-jacobi(-1,n+1))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012 [index adjusted by Peter Munn, Jun 22 2022]
a(n) = 1 + A065339(n+1) mod 2. - Peter Munn, Jun 20 2022

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

A249344 A(n,k) = exponent of the largest power of n-th prime which divides k, square array read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 28 2014

Square array A(n,k), where n = row, k = column, read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... (transpose of array A060175).
A(n,k) is the (p_n)-adic valuation of k, where p_n is the n-th prime, A000040(n).
Each row is effectively a ruler function, s, with s(1) = 0. - Peter Munn, Apr 30 2022

			The top-left corner of the array:
  0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...
  0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, ...
  0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, ...
  0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, ...
  ...
A(1,8) = 3, because 2^3 is the largest power of 2 (= p_1 = A000040(1)) that divides 8.
a(2,9) = 2, because 3^2 is the largest power of 3 (= p_2) that divides 9.
a(3,15) = 1, because 5^1 is the largest power of 5 (= p_3) that divides 15.

Transpose: A060175.
Row 1: A007814.
Row 2: A007949.
Row 3: A112765.
Row 4: A214411.
Cf. also A000040, A002260, A004736, A085604, A090622, A115627, A249421.
Completely additive sequences where more than one prime is mapped to 1, all other primes to 0: A065339, A083025, A087436, A169611.
Ruler functions, s, with s(1) = 0 that are not rows here: A122840, A122841, A235127, A244413.

A[n_, k_] := IntegerExponent[k, Prime[n]]; Table[A[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 01 2023 *)

a(n, k) = valuation(k, prime(n)); \\ Michel Marcus, Jun 24 2017

from sympy import prime
def a(n, k):
    p=prime(n)
    i=z=0
    while p**i<=k:
        if k%(p**i)==0: z=i
        i+=1
    return z
for n in range(1, 10): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 24 2017

(define (A249344 n) (A249344bi (A002260 n) (A004736 n)))
(define (A249344bi row col) (let ((p (A000040 row))) (let loop ((n col) (i 0)) (cond ((not (zero? (modulo n p))) i) (else (loop (/ n p) (+ i 1)))))))

Row n, as a sequence, is completely additive with A(n, prime(n)) = 1, A(n, prime(m)) = 0 for m <> n. - Peter Munn, Apr 30 2022

Sum_{k=1..m} A(n,k) ~ (1/(prime(n)-1)) * m. - Amiram Eldar, Oct 01 2023

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)).

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)).

Original entry on oeis.org

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)).

A373591 Number of primes congruent to 1 modulo 3 dividing n (with multiplicity).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001222, A007949, A248909, A373592.

Cf. also A065339, A083025.

f[p_, e_] := If[Mod[p, 3] == 1, e, 0]; f[3, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 17 2024 *)

A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]); \\ After code in A083025

a(n) = A001222(A248909(n)).
a(n) = A001222(n) - (A007949(n)+A373592(n)).
Totally additive with a(3) = 0, a(p) = 1 if p == 1 (mod 3), and a(p) = 0 if p == 2 (mod 3). - Amiram Eldar, Jun 17 2024

A286473 Compound filter (for counting primes of form 4k+1, 4k+2 and 4k+3): a(n) = 4*A032742(n) + (A020639(n) mod 4), a(1) = 1.

Original entry on oeis.org

1, 6, 7, 10, 5, 14, 7, 18, 15, 22, 7, 26, 5, 30, 23, 34, 5, 38, 7, 42, 31, 46, 7, 50, 21, 54, 39, 58, 5, 62, 7, 66, 47, 70, 29, 74, 5, 78, 55, 82, 5, 86, 7, 90, 63, 94, 7, 98, 31, 102, 71, 106, 5, 110, 45, 114, 79, 118, 7, 122, 5, 126, 87, 130, 53, 134, 7, 138, 95, 142, 7, 146, 5, 150, 103, 154, 47, 158, 7, 162, 111, 166, 7, 170, 69, 174, 119, 178, 5, 182, 55

Offset: 1

Antti Karttunen, May 11 2017

nonn

For all i, j: a(i) = a(j) => A079635(i) = A079635(j). This follows because A079635(n) can be computed by recursively invoking a(n), without needing any other information.

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

Cf. A020639, A032742, A285729, A286472, A286474, A286475, A286476.

Cf. A001511, A007814, A065339, A079635, A083025 (some of the matched sequences).

With[{k = 4}, Table[Function[{p, d}, k d + Mod[p, k] - k Boole[n == 1]] @@ {#, n/#} &@ FactorInteger[n][[1, 1]], {n, 91}]] (* Michael De Vlieger, May 12 2017 *)

from sympy import divisors, primefactors
def a(n): return 1 if n==1 else 4*divisors(n)[-2] + (min(primefactors(n))%4) # Indranil Ghosh, May 12 2017

(define (A286473 n) (if (= 1 n) n (+ (* 4 (A032742 n)) (modulo (A020639 n) 4))))

a(1) = 1, for n > 1, a(n) = 4*A032742(n) + (A020639(n) mod 4).

A373592 Number of primes congruent to 2 modulo 3 dividing n (with multiplicity).

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 2, 0, 1, 1, 4, 1, 1, 0, 3, 0, 2, 1, 3, 2, 1, 0, 2, 1, 2, 0, 5, 1, 2, 1, 2, 0, 1, 0, 4, 1, 1, 0, 3, 1, 2, 1, 4, 0, 3, 1, 2, 1, 1, 2, 3, 0, 2, 1, 3, 0, 1, 0, 6, 1, 2, 0, 3, 1, 2, 1, 3, 0, 1, 2, 2, 1, 1, 0, 5, 0, 2, 1, 2, 2, 1, 1, 4, 1, 2, 0, 3, 0, 2, 1, 5, 0, 1, 1, 4, 1, 2, 0, 3, 1

Offset: 1

Antti Karttunen, Jun 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001222, A343430, A373591.
Cf. also A065339, A083025.
Differs from A257991 for the first time at n=29, where a(29) = 1, while A257991(29) = 0.

f[p_, e_] := If[Mod[p, 3] == 2, e, 0]; f[3, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 17 2024 *)

A373592(n) = sum(i=1, #n=factor(n)~, (2==n[1, i]%3)*n[2, i]); \\ After code in A083025

a(n) = A001222(A343430(n)).
a(n) = A001222(n) - (A007949(n)+A373591(n)).
Totally additive with a(3) = 0, a(p) = 1 if p == 2 (mod 3), and a(p) = 0 if p == 1 (mod 3). - Amiram Eldar, Jun 17 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A072202 Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Scheme

A014709 The regular paper-folding (or dragon curve) sequence. Alphabet {1,2}.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Scheme

Formula

Extensions

A249344 A(n,k) = exponent of the largest power of n-th prime which divides k, square array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

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)).

A072202 Same numbers of prime factors of forms 4k+1 and 4k+3, counted with multiplicity.