A322362 -id:A322362 - cp's OEIS frontend

A322361 a(n) = gcd(n, A003961(n)), where A003961 is completely multiplicative with a(prime(k)) = prime(k+1).

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 7, 9, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 9, 1, 1, 5, 1, 11, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 35

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003961, A064989, A318668, A322362.

a[n_] := If[n == 1, 1, GCD[n, Times@@(NextPrime[First[#]]^Last[#] &/@FactorInteger[n])]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A322361(n) = gcd(n, A003961(n));

from math import gcd, prod
from sympy import nextprime, factorint
def A322361(n): return gcd(n,prod(nextprime(p)**e for p, e in factorint(n).items())) # Chai Wah Wu, Dec 26 2022

a(n) = gcd(n, A003961(n)).

a(n) = A003961(gcd(n, A064989(n))).

A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 6, 2, 6, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 2, 8, 2, 2, 2, 2, 2, 8, 6, 2, 2, 8, 6, 4, 2, 2, 8, 2, 6, 4, 1, 12, 4, 2, 2, 4, 24, 2, 2, 2, 6, 8, 6, 12, 24, 2, 2, 2, 2, 2, 12, 4, 6, 8, 2, 2, 8, 8, 2, 4, 2, 24, 2, 2, 6, 4

Offset: 1

Labos Elemer, Dec 04 2001

nonn

Frequently equal, but not identical, to A009223 (i.e. GCD of sigma and phi of n).

Antti Karttunen, Table of n, a(n) for n = 1..23374
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A048250, A173557, A023900, A000203, A007947, A000010, A009223, A066087, A322359, A322360, A322362.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Table[g2[w], {w, 1, 128}]
a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

a(n)=my(f=factor(n)[,1]);gcd(prod(i=1,#f,f[i]+1),prod(i=1,#f,f[i]-1)) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = gcd(A048250(n), A023900(n)) = gcd(A000203(A007947(n)), A000010(A007947(n))).

a(n) = A322360(n) / A322359(n). - Antti Karttunen, Dec 04 2018

Name edited, part of the old name transferred to the formula section by Antti Karttunen, Dec 04 2018

A126865 a(n) = gcd(Product_{p|n} (p+1)^b(p,n), Product_{p|n} (p-1)^b(p,n)), where the products are over the distinct primes, p, that divide n and p^b(p,n) is the highest power of p dividing n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 6, 8, 1, 2, 4, 2, 2, 4, 2, 2, 2, 4, 6, 8, 6, 2, 8, 2, 1, 4, 2, 24, 4, 2, 6, 8, 2, 2, 12, 2, 2, 16, 2, 2, 2, 4, 4, 8, 6, 2, 8, 8, 6, 4, 2, 2, 8, 2, 6, 8, 1, 12, 4, 2, 2, 4, 24, 2, 4, 2, 6, 16, 18, 12, 24, 2, 2, 16, 2, 2, 12, 4, 6, 8, 2, 2, 16, 8, 2, 4, 2, 24, 2, 2, 12, 8

Offset: 1

Leroy Quet, Mar 15 2007

nonn

First occurrence of k or 0 if not possible (including all odd primes k): 2, 1, 0, 9, 0, 14, 0, 15, 0, 0, 0, 42, 0, 0, 0, 45, 0, 76, 0, 589, 0, 0, 0, 35, 0, 0, 0, 4381, 0, 0, ..., . - Robert G. Wilson v, Sep 08 2007

			400 = 2^4 * 5^2. So a(400) = gcd((2+1)^4 * (5+1)^2, (2-1)^4 * (5-1)^2) = gcd(2916, 16) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..12441
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A003958, A003959, A066086, A322362.

f[n_] := Block[{fi = FactorInteger@n}, GCD[Times @@ ((First /@ fi - 1)^Last /@ fi), Times @@ ((First /@ fi + 1)^Last /@ fi)]]; Array[f, 99] (* Robert G. Wilson v, Sep 08 2007 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A126865(n) = gcd(A003958(n),A003959(n)); \\ Antti Karttunen, Dec 17 2018

From Antti Karttunen, Dec 17 2018: (Start)
a(n) = gcd(A003958(n), A003959(n)).
a(A007947(n)) = A066086(n).
(End)

More terms from Robert G. Wilson v, Sep 08 2007

A322354 Greatest common divisor of product p and product (p+2), where p ranges over distinct prime divisors of n; a(n) = gcd(A007947(n), A166590(A007947(n))).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 7, 2, 1, 2, 3, 2, 1, 6, 1, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 10, 1, 2, 3, 2, 5, 2, 1, 2, 1, 14, 1, 2, 1, 2, 5, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 10, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 105

Offset: 1

Antti Karttunen, Dec 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..15015
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A007947, A166590, A322356, A322357, A322362.

Cf. also A066086.

f[n_] := If[n == 1, 1, Times @@ Power @@@ ({#[[1]] + 2, #[[2]]} & /@ FactorInteger[n])]; rad[n_] := Times @@ (First@# & /@ FactorInteger@n); a[n_] := GCD[rad[n], f[rad[n]]]; Array[a, 120] (* Amiram Eldar, Dec 16 2018 *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));
A007947(n) = factorback(factorint(n)[, 1]);
A322354(n) = A322362(A007947(n));
\\ Alternatively as:
A322354(n) = gcd(A007947(n), A166590(A007947(n)));

a(n) = A322362(A007947(n)) = gcd(A007947(n), A166590(A007947(n))).

a(n) = A322356(n) * A322357(n).

A322357 a(n) = A322354(n) / A322356(n).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Dec 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

f[n_] := If[n == 1, 1, Times @@ Power @@@ ({#[[1]] + 2, #[[2]]} & /@ FactorInteger [n])]; rad[n_] := Times @@ (First@# & /@ FactorInteger@n); fun[p_, n_] := If[ PrimeQ[p + 2] && Divisible[n, p + 2], p + 2, 1]; a[n_] := GCD[rad[n], f[rad[n]]]/ Times @@ (fun[#, n] & /@ FactorInteger[n][[;; , 1]]); Array[a, 120] (* Amiram Eldar, Dec 16 2018 *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));
A007947(n) = factorback(factorint(n)[, 1]);
A322354(n) = A322362(A007947(n));
A322356(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(isprime(f[i,1]+2)&&!(n%(f[i,1]+2)), m *= (f[i,1]+2))); (m); };
A322357(n) = (A322354(n)/A322356(n));

a(n) = A322354(n) / A322356(n).

A323406 Greatest common divisor of Product (p_i^e_i)-1 and Product (p_i^e_i)+1, when n = Product (p_i^e_i): a(n) = gcd(A047994(n), A034448(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 6, 4, 2, 2, 2, 2, 6, 2, 2, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 30, 4, 2, 2, 2, 2, 6, 8, 2, 2, 2, 8, 6, 4, 2, 2, 24, 2, 6, 16, 1, 12, 4, 2, 6, 4, 24, 2, 2, 2, 6, 8, 2, 12, 24, 2, 6, 2, 2, 2, 4, 4, 6, 8, 2, 2, 4, 8, 6, 4, 2, 24, 2, 2, 6, 40, 2, 2, 8, 2, 42, 48

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A047994.

Cf. also A009223, A066086, A126865, A322362, A323166, A323409.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A034448(n) = { my(f=factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323406(n) = gcd(A034448(n), A047994(n));

a(n) = gcd(A034448(n), A047994(n)), where A034448 is unitary sigma, and A047994 is unitary phi.

cp's OEIS Frontend

A322361 a(n) = gcd(n, A003961(n)), where A003961 is completely multiplicative with a(prime(k)) = prime(k+1).

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A066086 Greatest common divisor of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n; a(n) = gcd(A048250(n), A173557(n)).

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A126865 a(n) = gcd(Product_{p|n} (p+1)^b(p,n), Product_{p|n} (p-1)^b(p,n)), where the products are over the distinct primes, p, that divide n and p^b(p,n) is the highest power of p dividing n.

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A322354 Greatest common divisor of product p and product (p+2), where p ranges over distinct prime divisors of n; a(n) = gcd(A007947(n), A166590(A007947(n))).

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A322357 a(n) = A322354(n) / A322356(n).

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A323406 Greatest common divisor of Product (p_i^e_i)-1 and Product (p_i^e_i)+1, when n = Product (p_i^e_i): a(n) = gcd(A047994(n), A034448(n)).

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