A394790 Product of the sum of divisors of n with the exponent of the highest power of 2 dividing 2n: a(n) = A000203(n)*A001511(n).

1, 6, 4, 21, 6, 24, 8, 60, 13, 36, 12, 84, 14, 48, 24, 155, 18, 78, 20, 126, 32, 72, 24, 240, 31, 84, 40, 168, 30, 144, 32, 378, 48, 108, 48, 273, 38, 120, 56, 360, 42, 192, 44, 252, 78, 144, 48, 620, 57, 186, 72, 294, 54, 240, 72, 480, 80, 180, 60, 504, 62, 192, 104, 889, 84, 288, 68, 378, 96, 288

Offset

1,2

Comments

a(n) is the total area of a set formed by A001511(n) copies of the diagram called symmetric representation of sigma(n) with subparts. The set has the property that the total number of subparts equals A000005(n).

a(n) is also the total volume of a object whose base is the symmetric representation of sigma(n) with subparts and its height equals the ruler function A001511(n). After dissecting the object into horizontal slices of height 1, the total number of subparts equals A000005(n), the number of divisors of n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

For n >= 0, a((4*n+2))/3 = A005880(n) = 2*A000203(2*n+1). - Antti Karttunen, Apr 06 2026

From Amiram Eldar, Apr 06 2026: (Start)

Multiplicative with a(2^e) = (e+1)*(2^(e+1)-1), and a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p.

Dirichlet g.f.: ((4^s-2)/(4^s-3*2^s+2)) * zeta(s) * zeta(s-1).

Sum_{k=1..n} a(k) ~ 7*Pi^2*n^2/36. (End)

Mathematica

a[n_] := IntegerExponent[2*n, 2] * DivisorSigma[1, n]; Array[a, 70] (* Amiram Eldar, Apr 06 2026 *)

Prog

(PARI) A394790(n) = (sigma(n)*(1+valuation(n, 2))); \\ Antti Karttunen, Apr 06 2026

Crossrefs

Cf. A008438 is an odd bisection.

Some other sequences relating A000203 and A001511 are A286357 and A286460.

Cf. A000005, A001227, A005880, A237271, A237593, A279387 (subparts).

Sequence in context: A212891 A107983 A009278 * A213573 A321417 A185734

Adjacent sequences: A394787 A394788 A394789 * A394791 A394792 A394793

Keyword

nonn,mult,easy

Author

Omar E. Pol, Apr 01 2026

Status

approved