A352063 -id:A352063 - cp's OEIS frontend

A378646 Dirichlet convolution of A000027 and A378643.

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 4, 1, 3, 3, 8, 1, 4, 1, 6, 3, 3, 1, 8, 2, 3, 4, 8, 1, 1, 1, 16, 3, 3, 3, 10, 1, 3, 3, 10, 1, 1, 1, 8, 8, 3, 1, 16, 2, 8, 3, 8, 1, 12, 3, 12, 3, 3, 1, 10, 1, 3, 8, 32, 3, 1, 1, 8, 3, 9, 1, 16, 1, 3, 8, 8, 3, 1, 1, 20, 8, 3, 1, 8, 3, 3, 3, 16, 1, 14, 3, 8, 3, 3, 3, 32, 1, 8, 8, 14, 1, 1, 1, 18, 13

Offset: 1

Antti Karttunen, Dec 03 2024

sign

First negative value is a(140) = -2.

Odd bisection is not A352063: a(472*2 + 1) = 102 != A352063(472) = A074206(945) = 132. Note that 945 is the first odd abundant number, A005231.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000027, A005101, A005231, A074206, A103977, A378643, A378645 (Dirichlet inverse).

A378646(n) = sumdiv(n,d,d*A378643(n/d));

a(n) = Sum_{d|n} d*A378643(n/d).

a(n) = A074206(n) if n is a non-abundant number (A263837), and apparently never when n is abundant, A005101.

A378222 Number of ordered factorizations of the odd part of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 3, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 1, 8, 1, 1, 1, 2, 2, 3, 1, 1, 4, 3, 1, 3, 1, 1, 3, 1, 1, 8, 1, 3, 3, 1, 1, 3, 3, 1, 2, 1, 1, 8, 1, 3, 3, 1, 1, 8, 1, 1, 3, 3, 1, 3, 1, 1, 8, 3, 1, 3, 1, 3, 1, 1, 2, 8, 2, 1, 3, 1, 1, 13

Offset: 1

Antti Karttunen, Nov 24 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000265, A002033, A074206.

Bisections: A352063, and the sequence itself.

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; a[n_] := f[n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, May 01 2025 *)

A000265(n) = (n>>valuation(n,2));
A074206(n) = if(n>1, sumdiv(n, i, if(iA074206(i))), n); \\ From A074206
A378222(n) = A074206(A000265(n));

a(n) = A074206(A000265(n)).
For all n >= 1, a(2n) = a(n).
For all n >= 0, a(2n+1) = A352063(n) = A002033(2*n) = A074206(2*n+1).

cp's OEIS Frontend

A378646 Dirichlet convolution of A000027 and A378643.

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