A253824 Numbers m = concat(s,t) such that m = sigma(s) * sigma(t), where sigma(x) is the sum of the divisors of x.
540, 2352, 28224, 82890, 737856, 1524096, 1531152, 3429216, 17062920, 22264200, 23268600, 49447728, 104941200, 162496048, 197499456, 267450144, 502334784, 619672032, 2347826040, 2942021520, 4045874976, 4302305280, 9876226752, 22712348160, 24705882348, 33114541824, 34144545792, 45916416000
Offset: 1
Examples
540 = concat(5,40) -> sigma(5) = 6, sigma(40) = 90 and 6*90 = 540. 2352 = concat(23,52) -> sigma(23) = 24, sigma(52) = 98 and 24*98 = 2352. 28224 = concat(28,224) -> sigma(28) = 56, sigma(224) = 504 and 56*504 = 28222. 82890 = concat(8,2890) -> sigma(8) = 15, sigma(2890) = 5526 and 15*5526 = 82890.
Links
- Max Alekseyev, Table of n, a(n) for n = 1..40
Programs
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Maple
with(numtheory): P:=proc(q) local s, t, k, n; 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 then print(n); break; fi; fi; od; od; end: P(10^6);
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Mathematica
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 *) -
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
Extensions
a(8) from Michel Marcus, Jan 15 2015
a(9)-a(17) from Robert G. Wilson v, Jan 18 2015
Missing a(14) and a(19)-a(23) from Giovanni Resta, Jul 17 2015
Terms a(24) onward from Max Alekseyev, May 25 2025