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Search: a063507 -id:a063507
Displaying 1-8 of 8 results found. page 1
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A016097 Erroneous version of A063507. +20
0
1, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..24.
KEYWORD
dead
STATUS
approved
A063740 Number of integers k such that cototient(k) = n. +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 2..10000
FORMULA
From Amiram Eldar, Apr 08 2023 (Start)
a(A005278(n)) = 0.
a(A131825(n)) = 1.
a(A063741(n)) = n. (End)
EXAMPLE
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.
MATHEMATICA
Table[Count[Range[n^2], k_ /; k - EulerPhi@ k == n], {n, 2, 105}] (* Michael De Vlieger, Mar 17 2017 *)
PROG
(PARI) 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
CROSSREFS
Cf. A000010, A051953, A063507.
Cf. A063748 (greatest solution to x-phi(x)=n).
Cf. A005278, A063741, A131825.
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 13 2001
EXTENSIONS
Name edited by Charles R Greathouse IV, Mar 17 2017
STATUS
approved
A063741 Smallest number whose inverse cototient set has n elements. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Note that 1 is the only number that has infinitely many cototient-inverses, namely, all the primes.
LINKS
Donovan Johnson, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = min {x: |InvCot(x)| = n}.
a(n) = min { k | A063740(k) = n }. - M. F. Hasler, Jan 11 2018
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
Cf. A000010, A051953 (cototient: n - phi(n)), A063507.
Cf. A063740 (number of k such that cototient(k) = n).
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 13 2001
EXTENSIONS
More terms from David Wasserman, Jul 11 2002
STATUS
approved
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. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
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).
LINKS
T. D. Noe, Table of n, a(n) for n=2..1000
FORMULA
a(n)=Max{x : A051953(x)=n} if the inverse set is not empty; a(n)=0 if no inverse exists.
EXAMPLE
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.
MATHEMATICA
nn=10^4; lim=Floor[Sqrt[nn]]; mx=Table[0, {lim}]; Do[c=n-EulerPhi[n]; If[0<c<=lim, mx[[c]]=n], {n, nn}]; Rest[mx] (* T. 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 *)
CROSSREFS
Cf. A000010, A051953.
Cf. A063507 (least solution to x-phi(x)=n), A063740 (number of solutions to x-phi(x)=n).
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 13 2001
EXTENSIONS
Corrected and edited by T. D. Noe, Oct 30 2006
STATUS
approved
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. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Is there any n for which a(n) = -1?
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..1292
FORMULA
A362181(a(n)) = n.
MATHEMATICA
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]]]
CROSSREFS
The unitary version of A063507.
Cf. A323410, A362181.
Similar sequences: A007374, A361970.
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 10 2023
STATUS
approved
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. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 2..9674 (rows 2..1000)
EXAMPLE
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;
MATHEMATICA
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]
CROSSREFS
Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.
Similar sequences: A032447, A361966, A362180.
KEYWORD
nonn,tabf
AUTHOR
Amiram Eldar, Apr 11 2023
STATUS
approved
A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
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
EXAMPLE
The 31st term is 255 since 255 - phi(255) = 127, the 31st prime, and no number less than 255 has this property.
MATHEMATICA
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 *)
PROG
(PARI) a(n) = {my(k = 1); while(k - eulerphi(k) != prime(n), k++); k; } \\ Michel Marcus, Feb 02 2015
CROSSREFS
Cf. A050530, A051953.
KEYWORD
nonn
AUTHOR
Labos Elemer, Jan 05 2000
STATUS
approved
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. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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?
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..300
MATHEMATICA
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 *)
CROSSREFS
Cf. A091732, A362484, A362485.
Similar sequences: A007374, A063507, A361970, A362186.
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 22 2023
STATUS
approved
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Last modified July 21 00:28 EDT 2024. Contains 374461 sequences. (Running on oeis4.)