OFFSET
1,2
COMMENTS
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).
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, ...
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, ...
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicholas Schwab and Lola Thompson, A generalization of the practical numbers, International Journal of Number Theory, Vol. 14, No. 5 (2018), pp. 1487-1503.
EXAMPLE
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.
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 23 2020
STATUS
approved