A323410 -id:A323410 - cp's OEIS frontend

A362181 Number of numbers k such that A323410(k) = n.

0, 0, 1, 0, 2, 1, 2, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 5, 4, 5, 4, 5, 3, 4, 4, 5, 3, 5, 3, 5, 5, 5, 4, 6, 4, 6, 4, 6, 2, 7, 4, 6, 4, 6, 3, 7, 3, 5, 4, 6, 3, 8, 2, 6, 6, 7, 4, 8, 4, 6, 6, 7, 3, 9, 4, 7, 4, 5, 5, 9, 6, 9, 4, 7, 3

Offset: 2

Amiram Eldar, Apr 10 2023

nonn

The offset is 2 since A323410(p) = 1 for all prime powers p (A246655).

a(0) = 1, since there is only one solution, x = 1, to A323410(x) = 0.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Row lengths of A362180.
The unitary version of A063740.
Cf. A246655, A323410, A362182 (positions of 0's), A362183 (indices of records), A362184, A362185 (positions of 1's), A362186.
Similar sequences: A014197, A361967.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 100}, ucot = Table[ucototient[n], {n, 1, max^2}]; Table[Length[Position[ucot, n]], {n, 2, max}] // Flatten]

a(A362182(n)) = 0.
a(A362185(n)) = 1.
a(A362186(n)) = n.

A362180 Irregular table read by rows in which the n-th row consists of all the numbers m such that A323410(m) = n.

Original entry on oeis.org

6, 10, 12, 15, 14, 20, 21, 18, 24, 28, 35, 22, 36, 40, 33, 45, 26, 44, 56, 39, 55, 63, 52, 72, 65, 77, 34, 48, 88, 51, 91, 99, 38, 68, 80, 104, 57, 85, 117, 30, 76, 112, 95, 119, 143, 46, 136, 144, 69, 133, 153, 50, 92, 152, 176, 75, 115, 171, 187, 54, 100, 208

Offset: 2

Amiram Eldar, Apr 10 2023

The offset is 2 since A323410(p) = 1 for all prime powers p (A246655).

The 0th row consists of one term, 1, since 1 is the only solution to A323410(x) = 0.

			The table begins:
  n   n-th row
  --  -----------
   2
   3
   4  6;
   5
   6  10, 12;
   7  15;
   8  14, 20;
   9  21;
  10  18, 24, 28;
  11  35;
  12  22, 36, 40;

Amiram Eldar, Table of n, a(n) for n = 2..12684 (rows 2..1000)

Cf. A246655, A323410, A362181 (row lengths).

Similar sequences: A032447, A361966, A362213.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0;
With[{max = 28}, ucot = Table[ucototient[n], {n, 1, max^2}]; row[n_] := Position[ucot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

A362183 Unitary highly cototient numbers: numbers k that have more solutions x to the equation A323410(x) = k than any smaller k.

Original entry on oeis.org

0, 6, 10, 20, 31, 47, 53, 65, 77, 89, 113, 119, 149, 167, 179, 209, 293, 299, 329, 359, 389, 419, 479, 509, 599, 629, 779, 839, 989, 1049, 1139, 1259, 1469, 1559, 1649, 1679, 1889, 2099, 2309, 2729, 3149, 3359, 3569, 3989, 4289, 4409, 4619, 5249, 5459, 6089, 6509

Offset: 1

Amiram Eldar, Apr 10 2023

nonn

Indices of records of A362181.

The corresponding numbers of solutions are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 21, ... (A362184).

Amiram Eldar, Table of n, a(n) for n = 1..96 (terms below 10^5)

Cf. A323410, A362181, A362184.
The unitary version of A100827.
Similar sequences: A097942, A361968.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; s = {0}; solmax=1; Do[sol = solnum[[k]]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 2, max}]; s]

A362185 Numbers k with a single solution x to the equation A323410(x) = k.

Original entry on oeis.org

0, 4, 7, 9, 11, 216, 218, 220, 546, 652, 666, 700, 834, 850, 906, 924, 996, 1242, 1386, 1476, 1506, 1516, 1596, 1646, 1662, 1758, 1770, 1858, 1890, 1900, 1946, 2046, 2170, 2262, 2352, 2422, 2578, 2626, 2668, 2682, 2814, 2842, 2980, 2992, 3010, 3048, 3100, 3154

Offset: 1

Amiram Eldar, Apr 10 2023

nonn

Numbers k such that A362181(k) = 1.

			0 is a term since there is only one solution, x = 1, to A323410(x) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..2335 (terms below 10^5)

Cf. A323410, A362181.
The unitary version of A131825.
Similar sequence: A361969.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; Join[{0}, Position[solnum, 1] // Flatten]]

A362182 Unitary noncototient numbers: numbers k such that A323410(x) = k has no solution.

Original entry on oeis.org

2, 3, 5, 330, 1206, 1210, 1656, 1718, 1806, 1866, 1926, 2376, 2982, 3162, 3186, 3342, 4012, 4062, 4194, 4326, 4502, 4662, 4810, 5322, 5466, 6172, 6402, 6462, 6534, 6546, 6672, 6756, 7266, 7430, 7866, 8030, 8140, 8286, 8386, 8562, 8586, 8860, 9114, 9370, 9516, 9906

Offset: 1

Amiram Eldar, Apr 10 2023

nonn

Numbers k such that A362181(k) = 0.

Are 3 and 5 the only odd terms? There are no other odd terms below 10^5.

Amiram Eldar, Table of n, a(n) for n = 1..534 (terms below 10^5)

Cf. A323410, A362181.
The unitary version of A005278.
Similar sequences: A007617, A347771.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 2000}, Complement[Range[max], Table[ucototient[n], {n, 1, max^2}]]]

A362186 a(n) is the least number k such that the equation A323410(x) = k has exactly n solutions, or -1 if no such k exists.

Original entry on oeis.org

2, 0, 6, 10, 20, 31, 47, 53, 65, 77, 89, 113, 125, 119, 149, 173, 167, 179, 233, 279, 239, 209, 439, 293, 365, 299, 329, 359, 455, 521, 467, 389, 461, 419, 479, 773, 539, 509, 599, 845, 671, 791, 749, 719, 659, 629, 809, 1055, 881, 779, 899, 965, 929, 1121, 839, 1403

Offset: 0

Amiram Eldar, Apr 10 2023

nonn

Is there any n for which a(n) = -1?

Amiram Eldar, Table of n, a(n) for n = 0..1292

The unitary version of A063507.
Cf. A323410, A362181.
Similar sequences: A007374, A361970.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 300}, solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, solnum[[i]]++], {k, 2, max^2}]; Join[{2, 0}, TakeWhile[FirstPosition[ solnum, #] & /@ Range[2, max] // Flatten, NumberQ]]]

A362181(a(n)) = n.

A362211 a(n) is the unique solution to A323410(x) = A362185(n).

Original entry on oeis.org

1, 6, 15, 21, 35, 11392, 1688, 10048, 53632, 101632, 5272, 2632, 6616, 50368, 1386, 102016, 1716, 1722, 161152, 4356, 11992, 92992, 4716, 101312, 589312, 2634, 644608, 3538, 3778, 898048, 30896, 16312, 3610, 3510, 4702, 1432576, 4626, 606976, 8908, 3738, 343936

Offset: 1

Amiram Eldar, Apr 11 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2335

Cf. A323410, A362185.

Similar sequences: A131826, A362212.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 3000}, sol = solnum = Table[0, {n, 1, max}]; Do[If[(i = ucototient[k]) <= max, sol[[i]] = k; solnum[[i]]++], {k, 2, max^2}]; Join[{1}, sol[[Position[solnum, 1] // Flatten]]]]

A323410(a(n)) = A362185(n).

A323409 Greatest common divisor of Product (p_i^e_i)-1 and n, when n = Product (p_i^e_i); a(n) = gcd(n, A047994(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 12, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 4, 1, 2, 5, 14, 3, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 3, 2, 1, 6, 1, 20, 1, 2, 1, 12, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Records 1, 2, 6, 12, 14, 20, 24, 84, 120, 168, 240, 468, 720, 1008, 1240, 1488, 1632, 7440, 9360, 14880, 32640, ... occur at n = 1, 6, 12, 36, 56, 80, 144, 168, 240, 504, 720, 1404, 3600, 4032, 4960, 8928, 13056, 14880, 28080, 44640, 65280, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A047994, A323410.

Cf. also A009195, A323166, A323406.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A323409(n) = gcd(n, A047994(n));

a(n) = gcd(n, A047994(n)), where A047994 is unitary phi.

A323413 Infinitary analog of cototient function A051953: a(n) = n - A091732(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 5, 1, 6, 1, 6, 1, 8, 7, 1, 1, 10, 1, 8, 9, 12, 1, 18, 1, 14, 11, 10, 1, 22, 1, 17, 13, 18, 11, 12, 1, 20, 15, 28, 1, 30, 1, 14, 13, 24, 1, 18, 1, 26, 19, 16, 1, 38, 15, 38, 21, 30, 1, 36, 1, 32, 15, 19, 17, 46, 1, 20, 25, 46, 1, 48, 1, 38, 27, 22, 17, 54, 1, 20, 1, 42, 1, 48, 21, 44, 31, 58, 1, 58, 19

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..21000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A037445, A049417, A050376, A091732, A327575.

Cf. also A051953, A126168, A323410.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 0; a[n_] := n - Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); Array[a, 100] (* Amiram Eldar, Jan 09 2021 *)

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A323413(n) = (n-A091732(n));

a(n) = n - A091732(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - A327575 = 0.171064... . - Amiram Eldar, Dec 15 2023

A333104 Unitary quasiperfect cototient numbers: numbers k such that the sum of the iterated unitary cototient function of k is equal to k+1.

Original entry on oeis.org

10, 22, 98, 118, 230, 266, 1452, 88894, 114214, 1274198, 51675986, 61177358, 82986118

Offset: 1

Amiram Eldar, Mar 07 2020

a(14) > 10^9.

			10 is a term since A323410(10) = 6, A323410(6) = 4, A323410(4) = 1 and 6 + 4 + 1 = 11 = 10 + 1.

Cf. A082897, A286067, A323410, A330273, A333103.

uphi[0] = 0; uphi[1] = 1; uphi[n_] := (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]; ucot[n_] := n - uphi[n]; Select[Range[10^4], Plus @@ FixedPointList[ucot, #] == 2*# + 1 &]

cp's OEIS Frontend

A362181 Number of numbers k such that A323410(k) = n.

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A362180 Irregular table read by rows in which the n-th row consists of all the numbers m such that A323410(m) = n.

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A362183 Unitary highly cototient numbers: numbers k that have more solutions x to the equation A323410(x) = k than any smaller k.

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A362185 Numbers k with a single solution x to the equation A323410(x) = k.

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A362182 Unitary noncototient numbers: numbers k such that A323410(x) = k has no solution.

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A362186 a(n) is the least number k such that the equation A323410(x) = k has exactly n solutions, or -1 if no such k exists.

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A362211 a(n) is the unique solution to A323410(x) = A362185(n).

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A323409 Greatest common divisor of Product (p_i^e_i)-1 and n, when n = Product (p_i^e_i); a(n) = gcd(n, A047994(n)).

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A323413 Infinitary analog of cototient function A051953: a(n) = n - A091732(n).

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