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A248909 Completely multiplicative with a(p) = p if p = 6k+1 and a(p) = 1 otherwise. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Index to divisibility sequences

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

PROG

(Sage)

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]]

(PARI) 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

(Python)

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

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

CROSSREFS

Sequences used in a definition of this sequence: A002476, A004611, A007528, A020639, A028234, A032742.

Cf. A065330, A170824.

Sequence in context: A325470 A240831 A170824 * A140213 A331927 A285483

Adjacent sequences:  A248906 A248907 A248908 * A248910 A248911 A248912

KEYWORD

nonn,mult,easy

AUTHOR

Tom Edgar, Mar 06 2015

STATUS

approved

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Last modified June 15 22:06 EDT 2021. Contains 345053 sequences. (Running on oeis4.)