A331597 -id:A331597 - cp's OEIS frontend

A331595 a(n) = gcd(A122111(n), A241909(n)).

1, 2, 4, 3, 8, 3, 16, 5, 3, 3, 32, 5, 64, 3, 18, 7, 128, 15, 256, 5, 18, 3, 512, 7, 3, 3, 5, 5, 1024, 15, 2048, 11, 18, 3, 18, 7, 4096, 3, 18, 7, 8192, 15, 16384, 5, 50, 3, 32768, 11, 3, 45, 18, 5, 65536, 7, 108, 7, 18, 3, 131072, 7, 262144, 3, 50, 13, 108, 15, 524288, 5, 18, 45, 1048576, 11, 2097152, 3, 15, 5, 18, 15, 4194304, 11, 7, 3

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A241916, A331596 (number of distinct prime factors), A331597, A331598, A331599, A331600.

Cf. also A280489, A280491.

Array[If[# == 1, 1, GCD @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 82] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331595(n) = gcd(A122111(n), A241909(n));

a(n) = gcd(A122111(n), A241909(n)).

a(A241916(n)) = a(n).

A331596 Number of distinct prime factors of gcd(A122111(n), A241909(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A001221, A331596, A331597.

Array[PrimeNu@ If[# == 1, 1, GCD @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 105] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111. *)

A331596(n) = omega(gcd(A122111(n), A241909(n)));

a(n) = A001221(A331596(n)) = A001221(gcd(A122111(n), A241909(n))).

a(n) = A001222(A331597(n)).

A331730 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A331595(n) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 5, 4, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 5, 6, 4, 3, 7, 4, 4, 5, 5, 3, 8, 3, 9, 6, 4, 6, 7, 3, 4, 6, 7, 3, 8, 3, 5, 10, 4, 3, 9, 4, 11, 6, 5, 3, 7, 12, 7, 6, 4, 3, 7, 3, 4, 10, 13, 12, 8, 3, 5, 6, 11, 3, 9, 3, 4, 8, 5, 6, 8, 3, 9, 7, 4, 3, 7, 12, 4, 6, 7, 3, 7, 12, 5, 6, 4, 12, 13, 3, 14, 10, 7, 3, 8, 3, 7, 15

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A331597(i) = A331597(j) => A331596(i) = A331596(j),
  a(i) = a(j) => A331731(i) = A331731(j) => A331600(i) = A331600(j).

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

Cf. A122111, A241909, A305801, A331594, A331595, A331596, A331597, A331600, A331731.

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; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331595(n) = gcd(A122111(n), A241909(n));
Aux331730(n) = if((n%2)&&isprime(n),0,A331595(n));
v331730 = rgs_transform(vector(up_to, n, Aux331730(n)));
A331730(n) = v331730[n];

cp's OEIS Frontend

A331595 a(n) = gcd(A122111(n), A241909(n)).

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A331596 Number of distinct prime factors of gcd(A122111(n), A241909(n)).

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A331730 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A331595(n) for all other n, except for odd primes p, f(p) = 0.

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