login
A322371
a(n) is the least practical number that is divisible by prime(n).
2
2, 6, 20, 28, 66, 78, 204, 228, 276, 348, 496, 666, 820, 860, 1128, 1272, 1416, 1464, 2010, 2130, 2190, 2844, 2988, 3204, 4074, 4848, 4944, 5136, 5232, 5424, 7620, 7860, 8220, 8340, 8940, 9060, 9420, 9780, 10020, 12456, 12888, 13032, 13752, 13896, 16548, 16716, 17724
OFFSET
1,1
LINKS
B. M. Stewart, Sums of distinct divisors, Amer. J. Math., 76 (1954), 779-785 [MR64800]
FORMULA
From Charlie Neder, Jan 30 2019: (Start)
Let p = prime(n). Then a(n) = p*k, where k is the least practical number such that sigma(k)+1 >= p.
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.
The choice of k is then the least practical number allowed by the theorem. (End)
EXAMPLE
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.
MATHEMATICA
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]]];
a[n_] := Module[{p = Prime[n]}, For[i = 1, True, i++, If[PracticalQ[p i], Return[p i]]]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 19 2019, after T. D. Noe in A005153 *)
PROG
(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
CROSSREFS
Sequence in context: A104749 A062281 A267124 * A131441 A035142 A064701
KEYWORD
nonn
AUTHOR
David A. Corneth, Dec 05 2018
EXTENSIONS
New name by Michel Marcus, Jan 18 2019
STATUS
approved