A348004 -id:A348004 - cp's OEIS frontend

A348200 Terms of A348004 having more unitary divisors than any smaller term.

1, 3, 12, 60, 660, 9240, 157080, 2984520, 68643960, 3226266120

Offset: 1

Amiram Eldar, Oct 06 2021

The corresponding numbers of unitary divisors are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (apparently, all the powers of 2).

a(11) > 7*10^10, if it exists.

			The sequence A348004 begins with 1, 3, 4, 5, 7, 8, 9, 11 and 12. The number of unitary divisors of these terms are 1, 2, 2, 2, 2, 2, 2, 2 and 4, respectively. The record values, 1, 2 and 4, occur at 1, 3 and 12, the first 3 terms of this sequence.

The unitary version of A348198.

Cf. A348002, A348004.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])] == Length[d]; dm = 0; s = {}; Do[If[q[n], d = 2^PrimeNu[n]; If[d > dm, dm = d; AppendTo[s, n]]], {n, 1, 10^6}]; s

A348173 a(n) is the sum of the distinct values obtained when the unitary totient function is applied to the unitary divisors of n.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 24, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 56, 57, 29, 59, 60, 61, 31, 63, 64, 65, 33, 67, 68, 69, 35, 71, 72, 73, 37, 75, 76, 77, 39, 79, 80, 81, 41, 83, 78

Offset: 1

Amiram Eldar, Oct 04 2021

nonn

First differs from A283971 at n = 84.

			The unitary divisors of 18 are {1, 2, 9, 18} and their uphi values are {1, 1, 8, 8}. The set of distinct values is {1, 8} whose sum is 9. Therefore, a(18) = 9.

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

The unitary version of A348158.

Cf. A047994, A077610, A283971, A348001, A348004.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Plus@@ DeleteDuplicates[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; Array[a, 100]

a(n) <= n, with equality if and only if n is in A348004.

A348003 Indices k of records of low value of the ratios A348001(k)/A034444(k) between the number of distinct values of the unitary totient function applied to the unitary divisors of k and the number of unitary divisors of k.

Original entry on oeis.org

1, 2, 546, 2730, 13650, 101010, 199290, 996450, 1919190, 7373730, 28020174, 32626230, 125353410, 140100870, 700504350, 2381714790, 11908573950, 15270994830

Offset: 1

Amiram Eldar, Sep 23 2021

The maximal possible value of the ratio A348001(k)/A034444(k) is 1 which occurs at the terms of A348004.
The rounded values of the corresponding record values are 1, 0.5, 0.438, 0.406, 0.375, 0.359, 0.344, 0.312, 0.281, 0.266, 0.258, 0.250, 0.242, 0.199, 0.195, 0.170, 0.168, 0.145, ...
a(19) > 2*10^10, if it exists.

The unitary version of A328859.

Cf. A034444, A047994, A348001, A348002, A348004.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; r[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])]/Length[d]; rm = 2; seq = {}; Do[r1 = r[n]; If[r1 < rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A348264 a(n) is the number of iterations that n requires to reach a fixed point under the map x -> A348173(x).

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

a(n) first differs from A011765(n+2) at n = 84.
The fixed points are terms of A348004, so a(n) = 0 if and only if n is a term of A348004.
Conjecture: essentially partial sums of A219977 (verified for n <= 5000).

			a(1) = 0 since 1 is in A348004.
a(2) = 1 since there is one iteration of the map x -> A348173(x) starting from 2: 2 -> 1.
a(84) = 2 since there are 2 iterations of the map x -> A348173(x) starting from 84: 84 -> 78 -> 39.

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

Cf. A047994, A011765, A348004, A348173.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := Plus @@ DeleteDuplicates[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; a[n_] := -2 + Length@ FixedPointList[s, n]; Array[a, 100]

A359418 Unitary phi-practical (A286906) whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

Original entry on oeis.org

1, 3, 15, 105, 165, 195, 255, 1155, 1785, 1995, 2145, 2415, 2805, 3045, 3135, 3255, 3315, 3705, 3795, 3885, 4305, 4485, 4515, 4785, 4845, 4935, 5115, 5565, 5655, 5865, 6045, 6105, 6195, 6405, 7035, 7095, 7215, 7395, 7455, 7665, 7755, 7905, 7995, 8295, 8385, 8715

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

A unitary phi-practical number k is a number k such that each number in the range 1..k is a subsum of a the multiset {uphi(d) : d | k, gcd(d, k/d) = 1}. 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 A286906 and A348004.

Cf. A047994, A359417.

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[9000], UnsameQ @@ uphi /@ Divisors[#] && uPhiPracticalQ[#] &]

A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

First differs from A003159 at n = 57.
Numbers k such that A361923(k) = A037445(k).
Since Sum_{d infinitary divisor of k} iphi(d) = k, these are numbers k such that the multiset {iphi(d) | d infinitary divisor of k} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345) and all the powers of 4 (A000302).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 66, 651, 6497, 64894, 648641, 6485605, 64851632, 648506213, 6485025363, ... . Apparently, this sequence has an asymptotic density 0.6485...

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

Cf. A000302, A003159, A037445, A061345, A077609, A091732, A361923.

Similar sequences: A326835, A348004.

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};
q[n_] := Length @ Union[iphi /@ (d = idivs[n])] == Length[d]; Select[Range[100], q]

iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
is(k) = {my(d = idivs(k)); #Set(apply(x->iphi(x), d)) == #d;}

A348174 Indices k of records of low value in the ratios A348173(k)/k.

Original entry on oeis.org

1, 2, 546, 2730, 13650, 51870, 101010, 199290, 505050, 881790, 996450, 1919190, 32626230, 140100870, 654443790, 865554690

Offset: 1

Amiram Eldar, Oct 04 2021

The maximal possible value of the ratio A348173(k)/k is 1 which occurs at the terms of A348004.
The rounded values of the corresponding records are 1, 0.5, 0.478, 0.469, 0.466, 0.465, 0.4642, 0.4638, 0.4621, 0.4620, 0.460, 0.453, 0.450, 0.447, 0.446, 0.445, ...
a(17) > 1.4*10^9.

The unitary version of A348159.

Cf. A348003, A348004, A348173.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; r[n_] := Plus @@ DeleteDuplicates[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]/n; rm = 2; seq = {}; Do[r1 = r[n]; If[r1 < rm, rm = r1; AppendTo[seq, n]], {n, 1, 2*10^5}]; seq

A348265 a(n) is the least number k such that A348264(k) = n, or -1 if no such number exists.

Original entry on oeis.org

1, 2, 84, 1596, 410652, 9114300, 10628100, 3012406320

Offset: 0

Amiram Eldar, Oct 09 2021

			n The n iterations of a(n) under the map x -> A348264(x)
- --------------------------------------------------------------------------------
0 1
1 2 -> 1
2 84 -> 78 -> 39
3 1596 -> 1428 -> 1326 -> 663
4 410652 -> 410172 -> 366996 -> 340782 -> 170391
5 9114300 -> 8871252 -> 8680848 -> 8661648 -> 8487018 -> 4243509
6 10628100 -> 10344684 -> 10309980 -> 10287900 -> 10013556 -> 9298302 -> 4649151
7 3012406320 -> 2999958360 -> 2927545572 -> 2917724340 -> 2911475700 -> 2833836348 -> 2631419466 -> 1315709733

The unitary version of A348214.

Cf. A348004, A348264.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; sum[n_] := Plus @@ DeleteDuplicates[uphi /@ Select[Divisors[n], CoprimeQ[#, n/#] &]]; s[n_] := -2 + Length@FixedPointList[sum, n]; seq[m_, lim_] := Module[{t = Table[0, {m}], c = 0, n = 1}, While[c < m && n < lim, i = s[n] + 1; If[i <= m && t[[i]] == 0, c++; t[[i]] = n]; n++]; TakeWhile[t, # > 0 &]]; seq[5, 10^6]

cp's OEIS Frontend

A348200 Terms of A348004 having more unitary divisors than any smaller term.

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A348173 a(n) is the sum of the distinct values obtained when the unitary totient function is applied to the unitary divisors of n.

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A348003 Indices k of records of low value of the ratios A348001(k)/A034444(k) between the number of distinct values of the unitary totient function applied to the unitary divisors of k and the number of unitary divisors of k.

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A348264 a(n) is the number of iterations that n requires to reach a fixed point under the map x -> A348173(x).

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A359418 Unitary phi-practical (A286906) whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

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A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

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A348174 Indices k of records of low value in the ratios A348173(k)/k.

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A348265 a(n) is the least number k such that A348264(k) = n, or -1 if no such number exists.

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