A297108 If n is prime(k)^e, e >= 1, then a(n) = 2^(k-1), otherwise 0; Möbius transform of A048675.

0, 1, 2, 1, 4, 0, 8, 1, 2, 0, 16, 0, 32, 0, 0, 1, 64, 0, 128, 0, 0, 0, 256, 0, 4, 0, 2, 0, 512, 0, 1024, 1, 0, 0, 0, 0, 2048, 0, 0, 0, 4096, 0, 8192, 0, 0, 0, 16384, 0, 8, 0, 0, 0, 32768, 0, 0, 0, 0, 0, 65536, 0, 131072, 0, 0, 1, 0, 0, 262144, 0, 0, 0, 524288, 0, 1048576, 0, 0, 0, 0, 0, 2097152, 0, 2, 0, 4194304

Offset

1,3

Comments

This is also Xor-Moebius transform of A248663, in other words, the unique sequence satisfying SumXOR_{d divides n} a(d) = A248663(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of this transform.

Links

Antti Karttunen, Table of n, a(n) for n = 1..1024

Formula

If A001221(n) = 1 [when n is in A000961], then a(n) = 2^(A297109(n)-1) = 2^(A055396(n)-1), otherwise 0.

a(n) = Sum_{d|n} A048675(d)*A008683(n/d).

Prog

(PARI)
A297108(n) = if(1==omega(n), 2^(primepi(factor(n)[1, 1])-1), 0);
\\ A more complicated way which demonstrates the Moebius transform:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ This function after Michel Marcus
A297108(n) = sumdiv(n, d, moebius(n/d)*A048675(d));
\\ And yet another way demonstrating the comment:
A248663(n) = A048675(core(n));
A297108(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A248663(d)))); (v); } \\ after code in A295901.

Crossrefs

Cf. A008683, A048675, A248663, A295901, A297106, A297109.

Sequence in context: A348508 A077954 A077979 * A307626 A122161 A067164

Adjacent sequences: A297105 A297106 A297107 * A297109 A297110 A297111

Keyword

nonn,base

Author

Antti Karttunen, Dec 25 2017

Status

approved