A336506 -id:A336506 - cp's OEIS frontend

A336507 Lambda-practical numbers (A336506) that are not phi-practical (A260653).

45, 135, 225, 405, 675, 765, 855, 1035, 1125, 1215, 1275, 1305, 1395, 1665, 1845, 1935, 2025, 2115, 2295, 2565, 3105, 3375, 3645, 3825, 3915, 4185, 4275, 4995, 5175, 5535, 5625, 5805, 6075, 6345, 6375, 6525, 6885, 6975, 7155, 7695, 7965, 8235, 8325, 9045, 9225

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

Thompson (2012) proved that all phi-practical numbers are lambda-practical, that all the terms of this sequence are not squarefree numbers, and that this sequence is infinite: for example, 45 * Product_{i=10..k} prime(i) is a term for all k >= 10.

Amiram Eldar, Table of n, a(n) for n = 1..300
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.

Subsequence of A013929.

Complement of A260653 with respect to A336506.

phiPracticalQ[n_] := If[n<1, False, If[n==1, True, (lst = Sort @ EulerPhi @ Divisors[n]; ok=True; Do[If[lst[[m]]>Sum[lst[[l]], {l, 1, m-1}]+1, (ok=False; Break[])], {m, 1, Length[lst]}]; ok)]]; rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; lambdaPracticalQ[n_] := Module[{d = Divisors[n], lam, ns, r, x}, lam = CarmichaelLambda[d]; ns = EulerPhi[d]/lam; r = rep[lam, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] > 0]; Select[Range[1000], !phiPracticalQ[#] && lambdaPracticalQ[#] &] (* after Frank M Jackson at A260653 *)

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).

Original entry on oeis.org

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

Amiram Eldar, Jul 23 2020

nonn

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, ...

			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.

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.

Subsequence of A336506.

Cf. A002322, A005153, A260653.

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]

cp's OEIS Frontend

A336507 Lambda-practical numbers (A336506) that are not phi-practical (A260653).

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