A331736 The largest odd divisor of A225546(n).

1, 1, 1, 3, 1, 1, 1, 3, 9, 1, 1, 3, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 3, 81, 1, 9, 3, 1, 1, 1, 5, 1, 1, 1, 27, 1, 1, 1, 3, 1, 1, 1, 3, 9, 1, 1, 5, 6561, 81, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 1, 9, 15, 1, 1, 1, 3, 1, 1, 1, 27, 1, 1, 81, 3, 1, 1, 1, 5, 25, 1, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 5, 1, 6561, 9, 243, 1, 1, 1, 3, 1

Offset: 1

Author

Antti Karttunen, Feb 02 2020

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1680

Crossrefs

Cf. A000265, A005117, A008833, A052928, A225546.

Programs

Mathematica

Array[#/2^IntegerExponent[#, 2] &@ If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] (* Michael De Vlieger, Feb 12 2020 *)

PARI

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A331736(n) = { my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] = A019565((f[i, 2]>>1)<<1); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); };

PARI

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331736(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=2,e); for(i=2,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=2,u,prime(i)^A048675(prods[i])));

Formula

Multiplicative, with a(prime(i)^j) = A000265(A019565(j))^A000079(i-1).

Equally, with a(prime(i)^j) = A019565(A052928(j))^A000079(i-1).

a(n) = A000265(A225546(n)).

a(n) = A225546(A008833(n)).