A170817 -id:A170817 - cp's OEIS frontend

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

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

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

A170824 a(n) = product of distinct primes of form 6k+1 that divide n.

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, 7, 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, 1, 1, 7, 1, 43, 1, 1, 1, 1, 91, 1, 31, 1

Offset: 1

N. J. A. Sloane, Dec 25 2009, following a suggestion from Jonathan Vos Post

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A002476, A170817.
Cf. A140213. [R. J. Mathar, Jan 21 2010]
Differs from A248909 for the first time at n=49, where a(49) = 7, while A248909(49) = 49.

A170824 := proc(n) a := 1 ; for p in numtheory[factorset](n) do if p mod 6 = 1 then a := a*p ; end if ; end do ; a ; end proc:
A140213 := proc(n) a := 1 ; for p in numtheory[divisors](n) do if p mod 6 = 1 then a := a*p ; end if ; end do ; a ; end proc:
seq(A170824(n),n=1..120) ; # R. J. Mathar, Jan 21 2010

test[p_] := IntegerQ[(p - 1)/6]; a[n_]:= Module[{aux = FactorInteger[n]}, Product[If[test[aux[[i, 1]]],aux[[i, 1]],1],{i, Length[aux]}]]; Table[a[n], {i, 1, 200}] (* Jose Grau, Feb 16 2010 *)
Table[Times@@Select[Transpose[FactorInteger[n]][[1]],IntegerQ[(#-1)/6]&],{n,100}] (* Harvey P. Dale, Jul 29 2013 *)

a(n) = my(f=factor(n)); prod(k=1, #f~, if (((p=f[k,1])%6) == 1, p, 1)); \\ Michel Marcus, Jul 10 2017

(define (A170824 n) (if (= 1 n) n (* (if (= 1 (modulo (A020639 n) 6)) (A020639 n) 1) (A170824 (A028234 n))))) ;; Antti Karttunen, Jul 09 2017

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

More terms from R. J. Mathar, Jan 21 2010

cp's OEIS Frontend

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

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

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A170824 a(n) = product of distinct primes of form 6k+1 that divide n.

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