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

%I A296513 #56 Jun 18 2025 06:10:48
%S A296513 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 A296513 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 A296513 11,45,30,6,31,48,5,127,9,1,34,63,13,13,36,7,37,57,3
%N A296513 a(n) is the smallest subpart of the symmetric representation of sigma(n).
%C A296513 If n is an odd prime (A065091) then a(n) = (n + 1)/2.
%C A296513 If n is a power of 2 (A000079) then a(n) = 2*n - 1.
%C A296513 If n is a perfect number (A000396) then a(n) = 1 assuming there are no odd perfect numbers.
%C A296513 a(n) is also the smallest number in the n-th row of the triangles A279391 and A280851.
%C A296513 a(n) is also the smallest nonzero term in the n-th row of triangle A296508.
%C A296513 The symmetric representation of sigma(n) has A001227(n) subparts.
%C A296513 For the definition of the "subpart" see A279387.
%C A296513 For a diagram with the subparts for the first 16 positive integers see A296508.
%C A296513 It appears that a(n) = 1 if and only if n is a hexagonal number (A000384). - _Omar E. Pol_, Sep 08 2021
%C A296513 The above conjecture is true. See A280851 for a proof. - _Omar E. Pol_, Mar 10 2022
%e A296513 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 A296513 (* a280851[] and support function are defined in A280851 *)
%t A296513 a296513[n_]:=Min[a280851[n]]
%t A296513 Map[a296513,Range[75]] (* _Hartmut F. W. Hoft_, Sep 05 2021 *)
%Y A296513 Shares infinitely many terms with A241558, A241559, A241838, A296512 (and possibly more).
%Y A296513 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 A296513 nonn
%O A296513 1,2
%A A296513 _Omar E. Pol_, Feb 10 2018
%E A296513 More terms from _Omar E. Pol_, Aug 28 2021