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A322371 a(n) is the least practical number that is divisible by prime(n).

%I A322371 #30 Feb 03 2019 21:02:18
%S A322371 2,6,20,28,66,78,204,228,276,348,496,666,820,860,1128,1272,1416,1464,
%T A322371 2010,2130,2190,2844,2988,3204,4074,4848,4944,5136,5232,5424,7620,
%U A322371 7860,8220,8340,8940,9060,9420,9780,10020,12456,12888,13032,13752,13896,16548,16716,17724
%N A322371 a(n) is the least practical number that is divisible by prime(n).
%H A322371 Charlie Neder, <a href="/A322371/b322371.txt">Table of n, a(n) for n = 1..3000</a>
%H A322371 B. M. Stewart, <a href="http://www.jstor.org/stable/2372651">Sums of distinct divisors</a>, Amer. J. Math., 76 (1954), 779-785 [<a href="http://www.ams.org/mathscinet-getitem?mr=64800">MR64800</a>]
%F A322371 From _Charlie Neder_, Jan 30 2019: (Start)
%F A322371 Let p = prime(n). Then a(n) = p*k, where k is the least practical number such that sigma(k)+1 >= p.
%F A322371 Proof: By Stewart's theorem (see link), since a(n) is practical, each prime factor of a(n) is at most 1 plus the sum of divisors of the product of the smaller primes in a(n). In particular, dividing a(n) by its largest prime factor will leave a practical number, since the criterion applies inductively on the product of smaller primes, so if the largest prime factor of a(n) is greater than p or the exponent of p is greater than 1, then a(n) can be reduced to a smaller multiple of p and is thus not minimal.
%F A322371 The choice of k is then the least practical number allowed by the theorem. (End)
%e A322371 For n = 4 we have prime(n) = prime(4) = 7. The least k such that k*prime(4) is practical is k = 4. Therefore, a(4) = 4*prime(4) = 28.
%t A322371 PracticalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
%t A322371 a[n_] := Module[{p = Prime[n]}, For[i = 1, True, i++, If[PracticalQ[p i], Return[p i]]]];
%t A322371 Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Jan 19 2019, after _T. D. Noe_ in A005153 *)
%o A322371 (PARI) a(n) = my(p = prime(n)); for(i = 1, oo, if(is_A005153(p * i), return(p * i))) \\ _David A. Corneth_, Dec 31 2018
%Y A322371 Cf. A005153, A322372.
%K A322371 nonn
%O A322371 1,1
%A A322371 _David A. Corneth_, Dec 05 2018
%E A322371 New name by _Michel Marcus_, Jan 18 2019