Search: a063748 -id:a063748
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A063740
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Number of integers k such that cototient(k) = n.
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+10
13
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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
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OFFSET
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2,3
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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Table[Count[Range[n^2], k_ /; k - EulerPhi@ k == n], {n, 2, 105}] (* Michael De Vlieger, Mar 17 2017 *)
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PROG
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(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
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CROSSREFS
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Cf. A063748 (greatest solution to x-phi(x)=n).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A063507
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Least k such that k - phi(k) = n, or 0 if no such k exists.
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+10
9
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2, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36, 69, 0, 63, 52, 161, 42, 87, 48, 93, 0, 75, 54, 217, 74, 99, 76, 185, 82, 123, 60, 117, 66, 215, 72, 141, 0, 235, 0, 329, 78, 159, 98, 105, 0, 371, 84, 177, 122, 135, 96, 305, 90, 427
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OFFSET
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1,1
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COMMENTS
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Inverse cototient (A051953) sets represented by their minimum, as in A002181 for totient function. Impossible values (A005278) are replaced by zero.
If a(n) > 0, then it appears that a(n) > 1.26n. - T. D. Noe, Dec 06 2006
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LINKS
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FORMULA
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a(n)-A051953(a(n)) = n if possible and a(n)=0 if n belongs to A005278.
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EXAMPLE
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x = InvCototient[24] = {36, 40, 44, 46}; Phi[x] = Phi[{36, 40, 44, 46}] = {12, 16, 20, 22}; x-Phi[x] = {24, 24, 24, 24}, so a(24) = Min[InvCototient[24]]; a(10) = 0 because 10 is in A005278.
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MATHEMATICA
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Table[SelectFirst[Range[n^2 + 1], # - EulerPhi[#] == n &] /. k_ /; ! IntegerQ@ k -> 0, {n, 67}] (* Michael De Vlieger, Jan 11 2018 *)
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CROSSREFS
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Cf. A063748 (greatest solution to x-phi(x)=n).
Cf. A063740 (number of k such that cototient(k) = n).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A362213
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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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+10
3
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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
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OFFSET
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2,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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;
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A005278, A051953, A063507, A063740 (row lengths), A063741, A063742, A063748, A100827, A101373, A131825, A131826.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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