A322318 -id:A322318 - cp's OEIS frontend

A322588 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A291750(n) for any other number.

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 23, 25, 3, 26, 27, 28, 3, 29, 3, 30, 31, 21, 3, 32, 33, 34, 21, 35, 3, 36, 21, 37, 38, 39, 3, 40, 3, 29, 41, 42, 43, 44, 3, 45, 29, 44, 3, 46, 3, 47, 48, 49, 29, 50, 3, 51, 52, 53, 3, 54, 55, 56, 57, 58, 3, 59, 60, 40, 61, 44, 57, 62, 3, 63, 64, 65, 3, 66, 3

Offset: 1

Antti Karttunen, Dec 18 2018

nonn

For all i, j: a(i) = a(j) => A322318(i) = A322318(j).

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

Cf. A291750, A291751, A322587, A322589, A322591, A322592.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
Aux322588(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n)));
v322588 = rgs_transform(vector(up_to, n, Aux322588(n)));
A322588(n) = v322588[n];

A322320 a(n) = gcd(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

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

Cf. A000010, A003557, A173557, A322321, A322351, A322352.

Cf. also A066086, A322318.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322320(n) = gcd(A173557(n), A003557(n));

a(n) = gcd(A003557(n), A173557(n)) = gcd(A322351(n), A322352(n)).

a(n) = A000010(n) / A322321(n).

A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 10, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

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

Cf. A003557, A034448, A322318, A323159.

Differs from A062760 for the first time at n=36, where a(36) = 2, while A062760(36) = 1.

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323163(n) = gcd(A003557(n), A034448(n));

a(n) = gcd(A003557(n), A034448(n)).

A322319 a(n) = lcm(A003557(n), A048250(n)).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 12, 14, 24, 24, 24, 18, 12, 20, 18, 32, 36, 24, 12, 30, 42, 36, 24, 30, 72, 32, 48, 48, 54, 48, 12, 38, 60, 56, 36, 42, 96, 44, 36, 24, 72, 48, 24, 56, 90, 72, 42, 54, 36, 72, 24, 80, 90, 60, 72, 62, 96, 96, 96, 84, 144, 68, 54, 96, 144, 72, 12, 74, 114, 120, 60, 96, 168, 80, 72

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A001615, A003557, A048250, A322318.

Cf. also A322321, A322359.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A322319(n) = lcm(A048250(n), A003557(n));

a(n) = lcm(A003557(n), A048250(n)).

a(n) = A001615(n) / A322318(n).

cp's OEIS Frontend

A322588 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A291750(n) for any other number.

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A322320 a(n) = gcd(A003557(n), A173557(n)).

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A323163 Greatest common divisor of product (1+(p^e)) and product p^(e-1), where p ranges over prime factors of n, with e corresponding exponent; a(n) = gcd(A034448(n), A003557(n)).

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A322319 a(n) = lcm(A003557(n), A048250(n)).

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