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

Author

N. J. A. Sloane, Dec 22 2009

Keywords

• nonn
• mult

Comments

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

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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