A296513 - OEIS

%I #52 Mar 11 2022 12:54:58

%S 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,

%T 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,

%U 11,45,30,6,31,48,5,127,9,1,34,63,13,13,36,7,37,57,3

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

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

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

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

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

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

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

%C For the definition of the "subpart" see A279387.

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

%C It appears that a(n) = 1 if and only if n is a hexagonal number (A000384). - _Omar E. Pol_, Sep 08 2021

%C The above conjecture is true. See A280851 for a proof. - _Omar E. Pol_, Mar 10 2022

%e 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.

%t (* a280851[] and support function are defined in A280851 *)

%t a296513[n_]:=Min[a280851[n]]

%t Map[a296513,Range[75] (* _Hartmut F. W. Hoft_, Sep 05 2021 *)

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

%Y 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.

%K nonn

%O 1,2

%A _Omar E. Pol_, Feb 10 2018

%E More terms from _Omar E. Pol_, Aug 28 2021