A072436 -id:A072436 - cp's OEIS frontend

A097706 Part of n composed of prime factors of form 4k+3.

1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 11, 3, 1, 7, 3, 1, 1, 9, 19, 1, 21, 11, 23, 3, 1, 1, 27, 7, 1, 3, 31, 1, 33, 1, 7, 9, 1, 19, 3, 1, 1, 21, 43, 11, 9, 23, 47, 3, 49, 1, 3, 1, 1, 27, 11, 7, 57, 1, 59, 3, 1, 31, 63, 1, 1, 33, 67, 1, 69, 7, 71, 9, 1, 1, 3, 19, 77, 3, 79, 1, 81, 1, 83, 21

Offset: 1

Ralf Stephan, Aug 30 2004

Largest term of A004614 that divides n. - Peter Munn, Apr 15 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Equivalent sequence for distinct prime factors: A170819.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A170818 (4k+1), A072436 (not 4k+3), A248909 (6k+1), A343431 (6k+5).
Range of values: A004614.
Positions of 1's: A072437.
Cf. also A065338, A065339, A260728, A286363.

a:= n-> mul(`if`(irem(i[1], 4)=3, 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] == 3, p^e, 1], {pe, FactorInteger[n]}]; Array[a, 100] (* Jean-François Alcover, Jun 16 2015, updated May 29 2019 *)

a(n)=local(f); f=factor(n); prod(k=1, matsize(f)[1], if(f[k, 1]%4<>3, 1, f[k, 1]^f[k, 2]))

from sympy import factorint
from operator import mul
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 n/a072436(n) # Indranil Ghosh, May 08 2017

from math import prod
from sympy import factorint
def A097706(n): return prod(p**e for p, e in factorint(n).items() if p & 3 == 3) # Chai Wah Wu, Jun 28 2022

a(n) = n/A072436(n).
a(A004614(n)) = A004614(n).
a(A072437(n)) = 1.
a(n) = A000265(n)/A170818(n). - Peter Munn, Apr 15 2021

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

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)).

A072438 Remove prime factors of form 4*k+1.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 11, 12, 1, 14, 3, 16, 1, 18, 19, 4, 21, 22, 23, 24, 1, 2, 27, 28, 1, 6, 31, 32, 33, 2, 7, 36, 1, 38, 3, 8, 1, 42, 43, 44, 9, 46, 47, 48, 49, 2, 3, 4, 1, 54, 11, 56, 57, 2, 59, 12, 1, 62, 63, 64, 1, 66, 67, 4, 69, 14, 71, 72, 1, 2, 3, 76, 77, 6, 79, 16, 81, 2

Offset: 1

Reinhard Zumkeller, Jun 17 2002

a(n) <= n; a(a(n)) = a(n).

All factors p^m of a(n) are of the form p=2 or p=4*k+3.

			a(90) = a(2*3*3*5) = a(2*(4*0+3)^2*(4*1+1)^1) = 2*3^2*1 = 18.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A072436, A004144, A002144, A002145.

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

a[n_] := 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)); for (i=1, #f~, if ((f[i,1] % 4) == 1, f[i,1] = 1)); factorback(f); \\ Michel Marcus, Jun 09 2014

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

Multiplicative with a(p)=(if p==1 (mod 4) then 1 else p).

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A097706 Part of n composed of prime factors of form 4k+3.

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

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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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A072438 Remove prime factors of form 4*k+1.

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