A336508 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).

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, 84, 90, 96, 105, 108, 112, 120, 126, 128, 132, 135, 140, 144, 150, 160, 162, 165, 168, 176, 180, 192, 195, 198, 200, 210, 216, 220, 224, 225, 234, 240, 252, 256, 260

Offset: 1

Author

Amiram Eldar, Jul 23 2020

Keywords

• nonn

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, ...

Examples

			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.
		

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.

Crossrefs

Subsequence of A336506.

Cf. A002322, A005153, A260653.

Programs

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]