A260653 -id:A260653 - cp's OEIS frontend

A330871 Numbers k such that k and k+1 are both phi-practical numbers (A260653).

1, 2, 3, 15, 255, 735, 2624, 3135, 4095, 4784, 5264, 5984, 7215, 7424, 7904, 9344, 10064, 10335, 10815, 11024, 11984, 12375, 12495, 13695, 16184, 16575, 22575, 22784, 22815, 26144, 26264, 27104, 30015, 30855, 30975, 32384, 33824, 34335, 34544, 38024, 38415, 39104

Offset: 1

Amiram Eldar, Apr 29 2020

nonn

			1 is a term since both 1 and 2 are phi-practical numbers.

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

Cf. A260653, A287681.

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)]]; Select[Range[40000], phiPracticalQ[#] && phiPracticalQ[#+1] &] (* after Frank M Jackson at A260653 *)

A359417 Phi-practical numbers (A260653) whose divisors have distinct values of the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 15, 105, 165, 195, 255, 495, 525, 735, 975, 1155, 1485, 1785, 1815, 1995, 2145, 2415, 2535, 2625, 2805, 3045, 3135, 3255, 3315, 3675, 3705, 3795, 3885, 4305, 4455, 4485, 4515, 4785, 4845, 4875, 4935, 5115, 5145, 5445, 5565, 5655, 5865, 6045, 6105, 6195, 6405

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

A phi-practical number k is a number k such that each number in the range 1..k is a subsum of a the multiset {phi(d) : d | k}. This sequence is restricted to cases in which all the values in this multiset are distinct.

Are all the terms above 3 divisible by 5?

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

Intersection of A260653 and A326835.

Cf. A000010.

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)]]; (* Frank M Jackson's code at A260653 *)
Select[Range[40000], UnsameQ @@ EulerPhi[Divisors[#]] && phiPracticalQ[#]  &]

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

Original entry on oeis.org

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 *)

A359420 Numbers that are both practical (A005153) and phi-practical (A260653).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 208, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 272, 280, 288

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

First differs from A325795 at n = 45, and from A325781 at n = 36.

Numbers k such that each number in the range 1..sigma(k) is a sum of distinct divisors of k, and each number in the range 1..k is a subsum of the multiset {phi(d) : d | k}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, Uniform Distribution Theory, Vol. 18, No. 2 (2023), pp. 19-30; arXiv preprint, arXiv:2303.16819 [math.NT], 2023. Mentions this sequence.

Intersection of A005153 and A260653.
Cf. A000010 (phi), A000203 (sigma).
Cf. A325781, A325795.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {};
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)]; (* Frank M Jackson's code at A260653 *)
Select[Range[300], pracQ[#] && phiPracticalQ[#] &]

A286906 Unitary phi-practical numbers: numbers k such that each m < k is a sum of a subset of {uphi(d) : d | k, gcd(d,k/d)=1}, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 2, 3, 6, 12, 15, 30, 42, 60, 84, 105, 120, 132, 156, 165, 195, 210, 240, 255, 330, 390, 420, 462, 510, 546, 570, 660, 690, 714, 780, 798, 840, 870, 924, 930, 966, 1020, 1050, 1092, 1140, 1155, 1218, 1302, 1320, 1365, 1380, 1428, 1554, 1560, 1596, 1680

Offset: 1

Amiram Eldar, May 15 2017

nonn

The unitary version of A260653.

			The unitary divisors of 12 are 1, 3, 4 and 12, and the set of their uphi values is {1, 2, 3, 6}. Each number below 12 is the sum of a subset, e.g., 11 = 2 + 3 + 6, 10 = 1 + 3 + 6, etc.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicholas Schwab and Lola Thompson, A generalization of the practical numbers, arXiv:1701.08504 [math.NT], 2017.

Cf. A005153, A047994, A260653.

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)]]; Select[Range[10000], uPhiPracticalQ]

A336503 2-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_2[x].

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 45, 48, 54, 56, 60, 63, 64, 70, 72, 80, 84, 90, 96, 100, 105, 108, 112, 120, 124, 126, 128, 132, 135, 136, 140, 144, 147, 150, 154, 156, 160, 162, 165, 168, 176, 180, 182, 186, 189, 192

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 2-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007733(d) * n_d, where A007733(d) is the multiplicative order of 2 modulo the odd part of d, and 0 <= n_d <= phi(d)/A007733(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 6, 34, 243, 1790, 14703, 120276, 1030279, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
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.
Eric Weisstein's World of Mathematics, Finite Field.
Wikipedia, Finite field.

Cf. A000265, A002326, A007733, A007814.

Cf. A000010, A260653, A336504, A336505.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 2] &]

A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 26, 27, 30, 32, 36, 39, 40, 42, 44, 45, 48, 52, 54, 56, 60, 63, 64, 66, 72, 78, 80, 81, 84, 88, 90, 96, 99, 100, 104, 105, 108, 112, 117, 120, 126, 128, 130, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 3-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007734(d) * n_d, where A007734(d) is the multiplicative order of 3 modulo the largest divisor of d not divisible by 3, and 0 <= n_d <= phi(d)/A007734(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 41, 258, 1881, 15069, 127350, 1080749, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
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.
Eric Weisstein's World of Mathematics, Finite Field.
Wikipedia, Finite field.

Cf. A038502, A007734, A007949.

Cf. A000010, A260653, A336503, A336505.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 3] &]

A336505 5-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_5[x].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 50, 52, 54, 55, 56, 60, 62, 64, 65, 66, 70, 72, 75, 78, 80, 84, 88, 90, 93, 96, 100, 104, 105, 108, 110, 112, 117, 120, 124, 125, 126, 128, 130, 132, 135, 140, 144, 150

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 5-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007736(d) * n_d, where A007736(d) is the multiplicative order of 5 modulo the largest divisor of d not divisible by 5, and 0 <= n_d <= phi(d)/A007736(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 46, 286, 2179, 16847, 141446, 1223577, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
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.
Eric Weisstein's World of Mathematics, Finite Field.
Wikipedia, Finite field.

Cf. A132739, A007736, A112765.

Cf. A000010, A260653, A336503, A336504.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 5] &]

A336506 Lambda-practical numbers: numbers that are p-practical for every rational prime p.

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, 100, 105, 108, 112, 120, 126, 128, 132, 135, 140, 144, 150, 156, 160, 162, 165, 168, 176, 180, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 234, 240

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p. See A336503, A336504 and A336505 for examples.
A number m is lambda-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} lambda(d) * n_d, where lambda(d) = A002322(d) is the Carmichael lambda function, and 0 <= n_d <= phi(d)/lambda(d).
A squarefree number is lambda-practical if and only if it is phi-practical (A260653). All phi-practical numbers are lambda-practical, but there are infinitely many lambda-practical numbers that are not phi-practical: 45, 135, 225, 405, 675, ... (A336507).
If N(x) is the number of terms not exceeding, x then there are two positive constants c_1 and c_2 such that c_1 * x/log(x) <= N(x) <= c_2 * x/log(x) for all x >= 2.

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

Disjoint union of A260653 and A336507.
Cf. A336503, A336504, A336505.
Cf. A000010, A002322, A005117.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; lpQ[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[250], lpQ]

A361922 Infinitary phi-practical numbers: numbers m such that each k <= m is a subsum of a the multiset {iphi(d) : d infinitary divisor of m}, where iphi is an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 15, 24, 30, 40, 42, 56, 60, 72, 84, 105, 108, 120, 132, 135, 156, 165, 168, 195, 210, 216, 240, 255, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 420, 440, 456, 462, 480, 504, 510, 520, 540, 546, 552, 570, 600, 616, 640, 660, 672, 680, 690

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

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

Cf. A077609, A091732.

Similar sequences: A260653, A286906, A334901.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
iPhiPracticalQ[n_] := Module[{s = Sort@ Map[iphi, idivs[n]], ans = True}, Do[If[s[[j]] > Sum[s[[i]], {i, 1, j - 1}] + 1, ans = False; Break[]], {j, 1, Length[s]}]; ans]; Select[Range[700], iPhiPracticalQ]

cp's OEIS Frontend

A330871 Numbers k such that k and k+1 are both phi-practical numbers (A260653).

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A359417 Phi-practical numbers (A260653) whose divisors have distinct values of the Euler totient function (A000010).

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

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A359420 Numbers that are both practical (A005153) and phi-practical (A260653).

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A286906 Unitary phi-practical numbers: numbers k such that each m < k is a sum of a subset of {uphi(d) : d | k, gcd(d,k/d)=1}, where uphi is the unitary totient function (A047994).

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A336503 2-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_2[x].

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A336504 3-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_3[x].

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A336505 5-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_5[x].

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A336506 Lambda-practical numbers: numbers that are p-practical for every rational prime p.

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