cp's OEIS Frontend

A342594 Earliest occurrence of the next distinct width pattern (as listed in A342592) in the symmetric representation of sigma(n) not yet encountered as n increases.

%I A342594 #12 Jun 19 2025 23:51:18
%S A342594 1,3,6,9,15,18,21,30,45,60,63,72,75,78,81,90,105,120,135,147,150,162,
%T A342594 165,180,189,210,225,231,300,315,357,360,378,390,405,420,441,450,465,
%U A342594 495,504,525,540,567,630,648,666,675,690,693,729,735,770,810,825,840,855,858,882,900,903,945,975,990
%N A342594 Earliest occurrence of the next distinct width pattern (as listed in A342592) in the symmetric representation of sigma(n) not yet encountered as n increases.
%C A342594 The width pattern of the symmetric representation of sigma(a(n)) is the n-th row of the table of A342592.
%C A342594 Conjecture: If for some number n the symmetric representation of sigma(n) has the symmetric width pattern w in row n of A342592 then infinitely many numbers have that width pattern w.
%e A342594 a(1) = 1 is the smallest power of 2 whose symmetric representation of sigma has width pattern (1).
%e A342594 a(2) = 3 is the smallest odd prime whose symmetric representation of sigma has width pattern (1 0 1).
%e A342594 a(4) = 9 is the first number whose symmetric representation of sigma has width pattern (1 0 1 0 1). The infinitely many numbers 2^s * p^2, s >= 0 and p an odd prime larger than 2^(s+1), have the same width pattern.
%t A342594 (* function a341969[ ] is defined in A341969 *)
%t A342594 a342594[n_] := Module[{listW={}, listK={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; listK]
%t A342594 a342594[990] (* 64 entries; the 64th new pattern is encountered at n=990 *)
%Y A342594 Cf. A235791, A237048, A237270, A237591, A237593, A249223, A341969, A342592.
%K A342594 nonn
%O A342594 1,2
%A A342594 _Hartmut F. W. Hoft_, Mar 16 2021