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A253824 Numbers m = concat(s,t) such that m = sigma(s) * sigma(t), where sigma(x) is the sum of the divisors of x.

%I A253824 #39 May 26 2025 00:24:46
%S A253824 540,2352,28224,82890,737856,1524096,1531152,3429216,17062920,
%T A253824 22264200,23268600,49447728,104941200,162496048,197499456,267450144,
%U A253824 502334784,619672032,2347826040,2942021520,4045874976,4302305280,9876226752,22712348160,24705882348,33114541824,34144545792,45916416000
%N A253824 Numbers m = concat(s,t) such that m = sigma(s) * sigma(t), where sigma(x) is the sum of the divisors of x.
%H A253824 Max Alekseyev, <a href="/A253824/b253824.txt">Table of n, a(n) for n = 1..40</a>
%e A253824 540 = concat(5,40) -> sigma(5) = 6, sigma(40) = 90 and 6*90 = 540.
%e A253824 2352 = concat(23,52) -> sigma(23) = 24, sigma(52) = 98 and 24*98 = 2352.
%e A253824 28224 = concat(28,224) -> sigma(28) = 56, sigma(224) = 504 and 56*504 = 28222.
%e A253824 82890 = concat(8,2890) -> sigma(8) = 15, sigma(2890) = 5526 and 15*5526 = 82890.
%p A253824 with(numtheory): P:=proc(q) local s, t, k, n;
%p A253824 for n from 1 to q do for k from 1 to ilog10(n) do s:=n mod 10^k; t:=trunc(n/10^k); if s*t>0 then if sigma(s)*sigma(t)=n
%p A253824 then print(n); break; fi; fi; od; od; end: P(10^6);
%t A253824 fQ[n_] := Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, MemberQ[ Table[ DivisorSigma[1, FromDigits@ Take[ idn, {1, i}]] DivisorSigma[1, FromDigits@ Take[ idn, {i + 1, lng + 1}]], {i, lng}], n]]; k = 1; lst = {}; While[k < 1310000001, If[fQ@ k, AppendTo[ lst, k]; Print@ k]; k++] (* _Robert G. Wilson v_, Jan 19 2015 *)
%o A253824 (PARI) isok(n) = {len = #Str(n); for (k=1, len-1, na = n\10^k; nb = n % 10^k; if (nb && (n == sigma(na)*sigma(nb)), return (1)););} \\ _Michel Marcus_, Jan 15 2015
%Y A253824 Cf. A000203, A159000, A253825, A260144.
%K A253824 nonn,base
%O A253824 1,1
%A A253824 _Paolo P. Lava_, Jan 15 2015
%E A253824 a(8) from _Michel Marcus_, Jan 15 2015
%E A253824 a(9)-a(17) from _Robert G. Wilson v_, Jan 18 2015
%E A253824 Missing a(14) and a(19)-a(23) from _Giovanni Resta_, Jul 17 2015
%E A253824 Terms a(24) onward from _Max Alekseyev_, May 25 2025