A339969 -id:A339969 - cp's OEIS frontend

A364255 a(n) = gcd(n, A163511(n)).

1, 1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 3, 2, 1, 24, 5, 2, 1, 4, 1, 2, 1, 32, 3, 2, 5, 36, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 10, 1, 4, 1, 2, 11, 8, 3, 2, 1, 4, 1, 2, 1, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 5, 4, 7, 2, 1, 16, 27, 2, 1, 12, 5, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 96, 1, 2, 1, 20, 1, 2, 1, 8, 105

Offset: 0

Antti Karttunen, Jul 16 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A163511, A364257 (Dirichlet inverse), A364258, A364491, A364492, A364493.

Cf. also A364254, A364256, A339969, A364500, A364949.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364255(n) = gcd(n, A163511(n)); \\ Antti Karttunen, Sep 01 2023

from math import gcd
from sympy import nextprime
def A364255(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return gcd(c*p,n) # Chai Wah Wu, Jul 25 2023

From Antti Karttunen, Sep 01 2023: (Start)
a(n) = gcd(n, A364258(n)) = gcd(A163511(n), A364258(n)).
a(n) = n / A364491(n) = A163511(n)/ A364492(n).
(End)

A364500 a(n) = gcd(n, A005940(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 1, 10, 1, 12, 1, 2, 3, 16, 1, 2, 1, 20, 7, 2, 1, 24, 1, 2, 3, 4, 1, 6, 1, 32, 1, 2, 1, 4, 1, 2, 3, 40, 1, 14, 1, 4, 5, 2, 1, 48, 1, 2, 3, 4, 1, 6, 5, 8, 1, 2, 1, 12, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 15, 4, 11, 6, 1, 80, 1, 2, 1, 28, 5, 2, 3, 8, 1, 10, 7, 4, 1, 2, 5, 96, 1, 2, 33, 4

Offset: 1

Antti Karttunen, Jul 28 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A005940, A364499, A364501, A364502.

Cf. also A324198, A339969, A364255.

nn = 100; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364500(n) = gcd(n, A005940(n));

A364500(n) = { my(orgn=n,p=2,rl=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), rl++; if(1==(n%4), z *= p^min(rl,valuation(orgn,p)); rl=0)); n>>=1); (z); };

a(n) = gcd(n, A364499(n)) = gcd(A005940(n), A364499(n)).

a(n) = n / A364501(n) = A005940(n) / A364502(n).

A340364 a(n) = gcd(A005940(n), A324106(n)), where A324106(n) is multiplicative with a(p^e) = A005940(p^e).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 3, 16, 11, 14, 21, 20, 1, 30, 45, 24, 49, 50, 75, 36, 125, 6, 81, 32, 1, 22, 3, 28, 55, 42, 3, 40, 77, 2, 105, 60, 35, 90, 135, 48, 121, 98, 3, 100, 245, 150, 75, 72, 7, 250, 375, 12, 625, 162, 9, 64, 1, 2, 39, 44, 5, 6, 99, 56, 91, 110, 3, 84, 5, 6, 189, 80, 143, 154, 231

Offset: 1

Antti Karttunen, Jan 06 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005940, A324106, A340362, A340365, A340366.

Cf. also A339969.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324106(n) = { my(f=factor(n)); prod(i=1, #f~, A005940(f[i,1]^f[i,2])); };
A340364(n) = gcd(A005940(n), A324106(n));

a(n) = gcd(A005940(n), A324106(n)) = gcd(A005940(n), A340362(n)).

cp's OEIS Frontend

A364255 a(n) = gcd(n, A163511(n)).

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A364500 a(n) = gcd(n, A005940(n)).

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A340364 a(n) = gcd(A005940(n), A324106(n)), where A324106(n) is multiplicative with a(p^e) = A005940(p^e).

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