A260653 -id:A260653 - cp's OEIS frontend

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

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]

A336509 Even squarefree numbers k such that d_{i+1}/d_i < 2 for all 1 < i < tau(k) - 1, where 1 = d_1 < d_2 < ... < d_tau(k) = k are the divisors of k, and tau(k) is their number (A000005).

Original entry on oeis.org

6, 30, 210, 330, 390, 510, 570, 690, 870, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 8610, 8970, 9030, 9570, 9690, 9870, 10230, 11130, 11310, 11730, 12090, 12210, 12390, 12810, 13110, 13530, 14070, 14190, 14430, 14790, 14910

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

Thompson (2012) called these numbers "strictly 2-dense numbers" and proved that they are all phi-practical numbers (A260653).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Lola Thompson, Polynomials with divisors of every degree, Journal of Number Theory, Vol. 132, No. 5 (2012), pp. 1038-1053.
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 A174973 and A260653.

Cf. A000005, A005117, A039956.

sdQ[n_] := SquareFreeQ[n] && Length[(d = Rest @ Most @ Divisors[n])] >0 && Max[Rest[d]/Most[d]] < 2; Select[Range[2, 15000, 2], sdQ]

A359419 Nonsquarefree numbers that are both phi-practical and unitary phi-practical.

Original entry on oeis.org

12, 60, 84, 120, 132, 156, 240, 420, 660, 780, 840, 924, 1020, 1050, 1092, 1140, 1320, 1380, 1428, 1560, 1596, 1680, 1716, 1740, 1860, 1932, 2040, 2100, 2220, 2244, 2280, 2436, 2460, 2508, 2580, 2604, 2640, 2652, 2760, 2820, 2940, 2964, 3036, 3108, 3120, 3180

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

The squarefree numbers (A005117) are excluded from this sequence since every squarefree phi-practical number is also a unitary phi-practical number.

The least odd term in this sequence is a(104) = 8085.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A013929, A260653 and A286906.

Cf. A005117.

phiPracticalQ[n_] := 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)];
uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]];
uDivisors[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
uPhiPracticalQ[n_] := If[n < 1, False, If[n == 1, True, (lst = Sort@Map[uphi, uDivisors[n]]; ok = True; Do[If[lst[[m]] > Sum[lst[[l]], {l, 1, m - 1}] + 1, (ok = False; Break[])], {m, 1, Length[lst]}]; ok)]];  (* Frank M Jackson's code at A260653 *)
Select[Range[3200], ! SquareFreeQ[#] && phiPracticalQ[#] && uPhiPracticalQ[#] &]

cp's OEIS Frontend

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).

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A336509 Even squarefree numbers k such that d_{i+1}/d_i < 2 for all 1 < i < tau(k) - 1, where 1 = d_1 < d_2 < ... < d_tau(k) = k are the divisors of k, and tau(k) is their number (A000005).

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Mathematica

A359419 Nonsquarefree numbers that are both phi-practical and unitary phi-practical.

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Mathematica