A322354 -id:A322354 - cp's OEIS frontend

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

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).

A322591 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 A007947(n) for any other number.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 5, 6, 3, 4, 3, 7, 8, 2, 3, 4, 3, 6, 9, 10, 3, 4, 11, 12, 5, 7, 3, 13, 3, 2, 14, 15, 16, 4, 3, 17, 18, 6, 3, 19, 3, 10, 8, 20, 3, 4, 21, 6, 22, 12, 3, 4, 23, 7, 24, 25, 3, 13, 3, 26, 9, 2, 27, 28, 3, 15, 29, 30, 3, 4, 3, 31, 8, 17, 32, 33, 3, 6, 5, 34, 3, 19, 35, 36, 37, 10, 3, 13, 38, 20, 39, 40, 41, 4, 3, 7, 14, 6, 3, 42, 3, 12, 43

Offset: 1

Antti Karttunen, Dec 18 2018

nonn

For all i, j:
  a(i) = a(j) => A066086(i) = A066086(j),
  a(i) = a(j) => A322354(i) = A322354(j).

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

Cf. A007947, A305801, A322587, A322588, A322589, A322590, 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; };
A007947(n) = factorback(factorint(n)[, 1]);
Aux322591(n) = if((n>2)&&isprime(n),0,A007947(n));
v322591 = rgs_transform(vector(up_to, n, Aux322591(n)));
A322591(n) = v322591[n];

A322356 Product of such primes p that both p and p-2 divide n, and p-2 is also prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 35

Offset: 1

Antti Karttunen, Dec 16 2018

nonn

Product of those distinct greater twin primes (A006512) that divide n for which the corresponding lesser twin prime (A001359) also divides n.

			For n = 105 = 3*5*7, a(105) = 5*7 = 35.

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

Cf. A001359, A006512, A322354, A322357, A322358.

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

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); };

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

A001221(a(n)) = A001222(a(n)) = A322358(n).

cp's OEIS Frontend

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

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A322591 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 A007947(n) for any other number.

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A322356 Product of such primes p that both p and p-2 divide n, and p-2 is also prime.

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