A334990 a(1) = 1 and for any n > 1 with prime factorization Product_{k = 1..m} prime(k)^e_k (where prime(k) denotes the k-th prime number and e_m > 0), a(n) = Product_{k = 1..m-1} prime(k)^(e_k XOR e_{k+1}) (where XOR denotes the bitwise XOR operator).

1, 1, 2, 1, 3, 1, 5, 1, 4, 6, 7, 8, 11, 10, 2, 1, 13, 8, 17, 12, 30, 14, 19, 4, 9, 22, 8, 20, 23, 1, 29, 1, 42, 26, 3, 1, 31, 34, 66, 24, 37, 15, 41, 28, 108, 38, 43, 32, 25, 18, 78, 44, 47, 4, 105, 40, 102, 46, 53, 8, 59, 58, 180, 1, 165, 21, 61, 52, 114, 6

Offset: 1

Author

Rémy Sigrist, May 18 2020

Keywords

• nonn
• base

Comments

This sequence has similarities with A038554; here we consider prime exponents, there binary digits.

Links

• Index entries for sequences related to XOR-triangles

Crossrefs

Cf. A006530, A038554, A100778.

Programs

PARI

a(n) = { my (v=1, p=2, e=valuation(n,p)); n/=p^e; forprime (q=p+1, oo, if (n==1, return (v), my (f=valuation(n,q)); n/=q^f; v*=p^bitxor(e,f); [p,e]=[q,f])) }

Formula

a(n) = 1 iff n belongs to A100778.

a(n^2) = a(n)^2.

a(n^k) = a(n)^k for any k >= 0 and any squarefree number n.

a(prime(n+1)) = prime(n).

A006530(a(n)) < A006530(n) for any n > 1.