A066086 -id:A066086 - cp's OEIS frontend

A322359 Least common multiple of product (p-1) and product (p+1), where p ranges over distinct prime divisors of n.

1, 3, 4, 3, 12, 12, 24, 3, 4, 36, 60, 12, 84, 24, 24, 3, 144, 12, 180, 36, 96, 180, 264, 12, 12, 84, 4, 24, 420, 72, 480, 3, 240, 432, 48, 12, 684, 180, 168, 36, 840, 96, 924, 180, 24, 792, 1104, 12, 24, 36, 288, 84, 1404, 12, 360, 24, 720, 1260, 1740, 72, 1860, 480, 96, 3, 336, 720, 2244, 432, 1056, 144, 2520, 12

Offset: 1

Antti Karttunen, Dec 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to lcm's

Cf. A048250, A066086, A173557, A322360.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[Times@@((#-1)& @@@ f), Times@@((#+1)& @@@ f)]]]; Array[a, 100] (* Amiram Eldar, Dec 05 2018 *)

A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322359(n) = lcm(A048250(n), A173557(n));

a(n) = lcm(A048250(n), A173557(n)).

a(n) = A322360(n)/A066086(n).

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).

A342459 a(n) = gcd(A048250(n), A342001(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 1, 3, 8, 1, 1, 1, 1, 6, 2, 1, 1, 1, 2, 3, 1, 8, 1, 1, 1, 1, 2, 1, 12, 2, 1, 3, 8, 1, 1, 1, 1, 12, 1, 1, 1, 2, 2, 9, 4, 14, 1, 3, 8, 1, 2, 1, 1, 2, 1, 3, 1, 3, 6, 1, 1, 18, 2, 1, 1, 1, 1, 3, 1, 20, 6, 1, 1, 2, 4, 1, 1, 2, 2, 3, 8, 1, 1, 1, 4, 24, 2, 1, 24, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001615, A003415, A003557, A048250, A342001, A342458, A342919.

Cf. also A066086, A342416.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A342459(n) = gcd(A048250(n), A342001(n));

a(n) = A342458(n) / A003557(n) = gcd(A048250(n), A342001(n)).

a(n) = A342001(n) / A342919(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

A322359 Least common multiple of product (p-1) and product (p+1), where p ranges over distinct prime divisors of 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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A342459 a(n) = gcd(A048250(n), A342001(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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