A331598 -id:A331598 - cp's OEIS frontend

A331594 Number of prime factors (with multiplicity) of A331598(n), where A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 3, 0, 0, 0, 0, 0, 2, 1, 4, 0, 1, 2, 5, 1, 3, 0, 1, 0, 0, 2, 6, 1, 1, 0, 7, 3, 2, 0, 2, 0, 4, 0, 8, 0, 1, 3, 0, 4, 5, 0, 1, 0, 3, 5, 9, 0, 2, 0, 10, 1, 0, 1, 3, 0, 6, 6, 1, 0, 1, 0, 11, 1, 7, 2, 4, 0, 2, 1, 12, 0, 3, 2, 13, 7, 4, 0, 2, 1, 8, 8, 14, 3, 1, 0, 0, 2, 2, 0, 5, 0, 5, 0

Offset: 1

Antti Karttunen, Jan 25 2020

nonn

Apparently also the number of prime factors (with multiplicity) of A331599(n).

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

Cf. A001222, A331598, A331599.

Array[If[# == 1, 0, PrimeOmega[#1/GCD[#1, #2]] & @@ {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 25 2020, after JungHwan Min at A122111 *)

PARI
```
A331594(n) = bigomega(A331598(n));
```

a(n) = A001222(A331598(n)).

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

Original entry on oeis.org

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).

A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 2, 9, 1, 5, 1, 27, 1, 1, 1, 1, 1, 25, 3, 81, 1, 7, 4, 243, 2, 125, 1, 5, 1, 1, 9, 729, 2, 5, 1, 2187, 27, 49, 1, 25, 1, 625, 1, 6561, 1, 11, 8, 1, 81, 3125, 1, 3, 1, 343, 243, 19683, 1, 35, 1, 59049, 5, 1, 3, 125, 1, 15625, 729, 5, 1, 7, 1, 177147, 2, 78125, 4, 625, 1, 121, 2, 531441, 1, 245, 9, 1594323, 2187, 2401, 1, 21

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

It appears that these and the terms of A331598 have the same prime signatures, that is, A046523(a(n)) = A046523(A331598(n)) seems to hold for all n.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A241909, A241916, A331595, A331598.

Array[If[# == 1, 1, #2/GCD[#1, #2] & @@ {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[#]] &, 90] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)

A331599(n) = { my(u=A241909(n)); u/gcd(A122111(n), u); };

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

a(n) = A331598(A241916(n)).

cp's OEIS Frontend

A331594 Number of prime factors (with multiplicity) of A331598(n), where A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

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

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A331599 a(n) = A241909(n) / gcd(A122111(n),A241909(n)).

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