A296513 a(n) is the smallest subpart of the symmetric representation of sigma(n).

1, 3, 2, 7, 3, 1, 4, 15, 3, 9, 6, 5, 7, 12, 1, 31, 9, 2, 10, 3, 5, 18, 12, 13, 5, 21, 6, 1, 15, 3, 16, 63, 7, 27, 3, 10, 19, 30, 8, 11, 21, 4, 22, 42, 1, 36, 24, 29, 7, 15, 10, 49, 27, 3, 8, 9, 11, 45, 30, 6, 31, 48, 5, 127, 9, 1, 34, 63, 13, 13, 36, 7, 37, 57, 3

Offset: 1

Author

Omar E. Pol, Feb 10 2018

Keywords

• nonn

Comments

If n is an odd prime (A065091) then a(n) = (n + 1)/2.

If n is a power of 2 (A000079) then a(n) = 2*n - 1.

If n is a perfect number (A000396) then a(n) = 1 assuming there are no odd perfect numbers.

a(n) is also the smallest number in the n-th row of the triangles A279391 and A280851.

a(n) is also the smallest nonzero term in the n-th row of triangle A296508.

The symmetric representation of sigma(n) has A001227(n) subparts.

For the definition of the "subpart" see A279387.

For a diagram with the subparts for the first 16 positive integers see A296508.

It appears that a(n) = 1 if and only if n is a hexagonal number (A000384). - Omar E. Pol, Sep 08 2021

The above conjecture is true. See A280851 for a proof. - Omar E. Pol, Mar 10 2022

Examples

			For n = 15 the subparts of the symmetric representation of sigma(15) are [8, 7, 1, 8], the smallest subpart is 1, so a(15) = 1.
		

Crossrefs

Shares infinitely many terms with A241558, A241559, A241838, A296512 (and possibly more).

Cf. A000079, A000203 (sum of subparts), A000225, A000384, A000396, A001227 (number of subparts), A065091, A099378, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A279387, A279391, A280850, A280851, A296508.

Programs

Mathematica

(* a280851[] and support function are defined in A280851 *)
a296513[n_]:=Min[a280851[n]]
Map[a296513,Range[75]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

Extensions

More terms from Omar E. Pol, Aug 28 2021