%I #10 Apr 20 2023 13:20:17
%S 0,1,2,3,5,7,9,10,13,15,16,20,22,23,28,34,46,53,60,62,78,81,113,115,
%T 122,132,154,184,185,222,248,254,343,346,350,354,497,569,701,711,860,
%U 941,1088,1221,1222,1235,1263,1306,1572,1721,1737,1948,2191,2315,2418,2877
%N Number of times that the number A362402(n) occurs as a sum of divisors that have a square factor (A162296).
%H Amiram Eldar, <a href="/A362403/b362403.txt">Table of n, a(n) for n = 1..60</a>
%t s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {0}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[2]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
%o (PARI) s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) - prod(i = 1, #f~, f[i, 1] + 1);}
%o lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(0, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(v[k], ", "))); }
%Y Cf. A162296, A362401, A362402.
%Y Similar sequences: A131934, A101373, A206027, A238896.
%K nonn
%O 1,3
%A _Amiram Eldar_, Apr 18 2023