A063507 -id:A063507 - cp's OEIS frontend

A016097 Erroneous version of A063507.

1, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36

Offset: 1

dead

A063740 Number of integers k such that cototient(k) = n.

1, 1, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 2, 3, 3, 1, 3, 1, 3, 1, 4, 4, 3, 0, 4, 1, 4, 3, 3, 4, 3, 0, 5, 2, 2, 1, 4, 1, 5, 1, 4, 2, 4, 2, 6, 5, 5, 0, 3, 0, 6, 2, 4, 2, 5, 0, 7, 4, 3, 1, 8, 4, 6, 1, 3, 1, 5, 2, 7, 3, 5, 1, 7, 1, 8, 1, 5, 2, 6, 1, 9, 2, 6, 0, 4, 2, 10, 2, 4, 2, 5, 2, 7, 5, 4, 1, 8, 0, 9, 1, 6, 1, 7

Offset: 2

Labos Elemer, Aug 13 2001

nonn

Note that a(0) is also well-defined to be 1 because the only solution to x - phi(x) = 0 is x = 1. - Jianing Song, Dec 25 2018

			Cototient(x) = 101 for x in {485, 1157, 1577, 1817, 2117, 2201, 2501, 2537, 10201}, with a(101) = 8 terms; e.g. 485 - phi(485) = 485 - 384 = 101. Cototient(x) = 102 only for x = 202 so a(102) = 1.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A000010, A051953, A063507.
Cf. A063748 (greatest solution to x-phi(x)=n).
Cf. A005278, A063741, A131825.

Table[Count[Range[n^2], k_ /; k - EulerPhi@ k == n], {n, 2, 105}] (* Michael De Vlieger, Mar 17 2017 *)

first(n)=my(v=vector(n),t); forcomposite(k=4,n^2, t=k-eulerphi(k); if(t<=n, v[t]++)); v[2..n] \\ Charles R Greathouse IV, Mar 17 2017

From Amiram Eldar, Apr 08 2023 (Start)
a(A005278(n)) = 0.
a(A131825(n)) = 1.
a(A063741(n)) = n. (End)

Name edited by Charles R Greathouse IV, Mar 17 2017

A063741 Smallest number whose inverse cototient set has n elements.

Original entry on oeis.org

10, 0, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 143, 119, 197, 167, 279, 233, 281, 209, 269, 323, 299, 359, 497, 329, 455, 605, 389, 461, 479, 419, 539, 599, 509, 755, 791, 713, 875, 797, 719, 629, 659, 1025, 1163, 929, 779, 1193, 1121, 899, 1133, 1091, 839

Offset: 0

Labos Elemer, Aug 13 2001

nonn

Note that 1 is the only number that has infinitely many cototient-inverses, namely, all the primes.

			For n = 1, 2, 3, 4, 5, ..., the corresponding inverse sets are as follows: {}, {4}, {6, 8}, {12, 14, 16}, {95, 119, 143, 529}, {75, 155, 203, 299, 323}, ..., {455, 815, 1727, 2567, 2831, 4031, 4247, 4847, 5207, 6431, 6527, 6767, 6887, 7031, 27889}, including 0, 1, 2, 3, 4, 5, ..., 15 numbers.

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A000010, A051953 (cototient: n - phi(n)), A063507.

Cf. A063740 (number of k such that cototient(k) = n).

With[{s = Array[Count[Range[#^2], k_ /; k - EulerPhi@ k == #] &, 300, 2]}, ReplacePart[TakeWhile[First@ FirstPosition[s, #] + 1 & /@ Range[0, Max@ s], IntegerQ], 2 -> 0]] (* Michael De Vlieger, Jan 11 2018 *)

a(n) = min {x: |InvCot(x)| = n}.

a(n) = min { k | A063740(k) = n }. - M. F. Hasler, Jan 11 2018

More terms from David Wasserman, Jul 11 2002

A063748 Greatest x that is a solution to x-phi(x)=n or zero if there is no solution, where phi(x) is Euler's totient function.

Original entry on oeis.org

4, 9, 8, 25, 10, 49, 16, 27, 0, 121, 22, 169, 26, 55, 32, 289, 34, 361, 38, 85, 30, 529, 46, 133, 0, 187, 52, 841, 58, 961, 64, 253, 0, 323, 68, 1369, 74, 391, 76, 1681, 82, 1849, 86, 493, 70, 2209, 94, 589, 0, 667, 0, 2809, 106, 703, 104, 697, 0, 3481, 118, 3721, 122

Offset: 2

Labos Elemer, Aug 13 2001

nonn

See A051953 for x-phi(x), the cototient function. Note that a(n)=0 for n in A005278. Also note that n=1 has an infinite number of solutions. If n is prime, then a(n)=n^2. If n is even, then a(n)<=2n. In particular, if n=p+1 for a prime p, then a(n)=2n-2. Also, if n=2^k, then a(n)=2n. If n>9 is odd and composite, then a(n)=pq, with p>q odd primes with p+q=n+1 and p-q minimal. We can take p=A078496((n+1)/2) and q=A078587((n+1)/2).

			For n=15, the solutions are x=39 and x=55, so a(15)=55. Note that 55=5*11 and 5+11=n+1.

T. D. Noe, Table of n, a(n) for n=2..1000

Cf. A000010, A051953.

Cf. A063507 (least solution to x-phi(x)=n), A063740 (number of solutions to x-phi(x)=n).

nn=10^4; lim=Floor[Sqrt[nn]]; mx=Table[0,{lim}]; Do[c=n-EulerPhi[n]; If[0T. D. Noe *)
Table[Module[{k = n^2}, While[And[k - EulerPhi@ k != n, k > 0], k--];
k], {n, 2, 62}] (* Michael De Vlieger, Mar 17 2017 *)

a(n)=Max{x : A051953(x)=n} if the inverse set is not empty; a(n)=0 if no inverse exists.

Corrected and edited by T. D. Noe, Oct 30 2006

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.

A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

Original entry on oeis.org

4, 9, 6, 8, 25, 10, 15, 49, 12, 14, 16, 21, 27, 35, 121, 18, 20, 22, 33, 169, 26, 39, 55, 24, 28, 32, 65, 77, 289, 34, 51, 91, 361, 38, 45, 57, 85, 30, 95, 119, 143, 529, 36, 40, 44, 46, 69, 125, 133, 63, 81, 115, 187, 52, 161, 209, 221, 841, 42, 50, 58, 87, 247, 961

Offset: 2

Amiram Eldar, Apr 11 2023

The offset is 2 since cototient(p) = 1 for all primes p.

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

			The table begins:
  n   n-th row
  --  -----------
   2  4;
   3  9;
   4  6, 8;
   5  25;
   6  10;
   7  15, 49;
   8  12, 14, 16;
   9  21, 27;
  10
  11  35, 121;
  12  18, 20, 22;

Amiram Eldar, Table of n, a(n) for n = 2..9674 (rows 2..1000)

Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.

Similar sequences: A032447, A361966, A362180.

With[{max = 50}, cot = Table[n - EulerPhi[n], {n, 1, max^2}]; row[n_] := Position[cot, n] // Flatten; Table[row[n], {n, 2, max}] // Flatten]

A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime.

Original entry on oeis.org

4, 9, 25, 15, 35, 33, 65, 51, 95, 161, 87, 217, 185, 123, 215, 329, 371, 177, 427, 335, 213, 511, 395, 581, 1501, 485, 303, 515, 321, 545, 255, 635, 917, 411, 1529, 447, 1057, 1099, 455, 1169, 1211, 537, 1991, 573, 965, 591, 435, 2743, 1115, 681, 665

Offset: 1

Labos Elemer, Jan 05 2000

nonn

			The 31st term is 255 since 255 - phi(255) = 127, the 31st prime, and no number less than 255 has this property.

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

Cf. A050530, A051953.

With[{c=Table[n-EulerPhi[n],{n,4000}]},Table[Position[c,p,1,1],{p,Prime[ Range[ 60]]}]]//Flatten (* Harvey P. Dale, Sep 14 2020 *)

a(n) = {my(k = 1); while(k - eulerphi(k) != prime(n), k++); k;} \\ Michel Marcus, Feb 02 2015

A050530(a(n)) = prime(n) and a(n) is the least number with this property.

a(n) = A063507(A000040(n)). - Michel Marcus, Feb 02 2015

A362489 a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

5, 1, 6, 12, 36, 24, 396, 48, 216, 96, 528, 144, 384, 2784, 432, 240, 1296, 288, 1584, 1800, 480, 1680, 1080, 864, 576, 3240, 2016, 960, 6624, 720, 1152, 7776, 12000, 8448, 5280, 1728, 10752, 2304, 4032, 4800, 6048, 3840, 2160, 5184, 4608, 6336, 1440, 10560, 29568

Offset: 0

Amiram Eldar, Apr 22 2023

nonn

a(n) is the least number k such that A362485(k) = 2*n. Odd values of A362485 are impossible.

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

Amiram Eldar, Table of n, a(n) for n = 0..300

Cf. A091732, A362484, A362485.

Similar sequences: A007374, A063507, A361970, A362186.

solnum[n_] := Length[invIPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k]/2 + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invIPhi from A362484 *)

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A016097 Erroneous version of A063507.

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A063740 Number of integers k such that cototient(k) = n.

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A063741 Smallest number whose inverse cototient set has n elements.

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A063748 Greatest x that is a solution to x-phi(x)=n or zero if there is no solution, where phi(x) is Euler's totient function.

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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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A362213 Irregular table read by rows in which the n-th row consists of all the numbers m such that cototient(m) = n, where cototient is A051953.

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A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime.

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A362489 a(n) is the least number k such that the equation iphi(x) = k has exactly 2*n solutions, or -1 if no such k exists, where iphi is the infinitary totient function A091732.

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