A322371 a(n) is the least practical number that is divisible by prime(n).
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
Keywords
Examples
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.
Links
- Charlie Neder, Table of n, a(n) for n = 1..3000
- B. M. Stewart, Sums of distinct divisors, Amer. J. Math., 76 (1954), 779-785 [MR64800]
Programs
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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 *) -
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
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)
Extensions
New name by Michel Marcus, Jan 18 2019
Comments