A336508 - OEIS

%I #13 Feb 17 2024 11:28:01

%S 1,2,3,4,6,8,12,15,16,18,20,24,30,32,36,40,42,45,48,54,56,60,64,72,80,

%T 84,90,96,105,108,112,120,126,128,132,135,140,144,150,160,162,165,168,

%U 176,180,192,195,198,200,210,216,220,224,225,234,240,252,256,260

%N Numbers m such that every number 1 <= k <= s is the sum of a subset of the set {lambda(d) : d | m}, where s is the total sum of the set and lambda is the Carmichael lambda function (A002322).

%C Schwab and Thompson (2018) named these numbers "lambda*-practical". The asterisk in the terminology was chosen to emphasize that this notion differs from the definition of lambda-practical numbers (A336506).

%C This sequence is in fact a subsequence of the lambda-practical numbers. Lambda-practical numbers that are not in this sequence are 100, 156, 208, 255, 272, 294, 380, 392, 408, 456, 500, ...

%C The number of terms of this sequence that do not exceed 10^k for k = 1, 2, ... are 6, 28, 164, 1015, 7128, 52326, 409714, ...

%H Amiram Eldar, <a href="/A336508/b336508.txt">Table of n, a(n) for n = 1..10000</a>

%H Nicholas Schwab and Lola Thompson, <a href="https://doi.org/10.1142/S1793042118500902">A generalization of the practical numbers</a>, International Journal of Number Theory, Vol. 14, No. 5 (2018), pp. 1487-1503.

%e 6 is a term since the values of the Carmichael lambda function at its divisors, {1, 2, 3, 6}, are {1, 1, 2, 2}, and every number 1 <= k <= 6 is a sum of elements of this set: 1 = 1, 2 = 2, 3 = 1 + 2, 4 = 2 + 2, 5 = 1 + 2 + 2 and 6 = 1 + 1 + 2 + 2.

%t lamPracQ[n_] := Module[{d = Divisors[n], sm}, lam = CarmichaelLambda[d]; sm = Plus @@ lam; Min @ Rest @ CoefficientList[Series[Product[1 + x^lam[[i]], {i, Length[lam]}], {x, 0, sm}], x] > 0]; Select[Range[300], lamPracQ]

%Y Subsequence of A336506.

%Y Cf. A002322, A005153, A260653.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 23 2020