This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
n = 12, phi(12) = 4 = |{1, 5, 7, 11}|, a(12) = 12 - phi(12) = 8, numbers not exceeding 12 and not coprime to 12: {2, 3, 4, 6, 8, 9, 10, 12}.
a051953 n = n - a000010 n -- Reinhard Zumkeller, Jan 21 2014
with(numtheory); A051953 := n->n-phi(n);
Table[n - EulerPhi[n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)
A051953(n) = n - eulerphi(n); \\ Michael B. Porter, Jan 28 2010
from sympy.ntheory import totient print([i - totient(i) for i in range(1, 101)]) # Indranil Ghosh, Mar 17 2017
nmax = 520; cototientQ[n_?EvenQ] := (x = n; While[test = x - EulerPhi[x] == n ; Not[test || x > 2*nmax], x++]; test); cototientQ[n_?OddQ] = True; Select[Range[nmax], !cototientQ[#]&] (* Jean-François Alcover, Jul 20 2011 *)
lista(nn)=v = vecsort(vector(nn^2, n, n - eulerphi(n)), ,8); for (n=1, nn, if (! vecsearch(v, n), print1(n, ", "))); \\ Michel Marcus, Oct 03 2016
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.
Table[SelectFirst[Range[n^2 + 1], # - EulerPhi[#] == n &] /. k_ /; ! IntegerQ@ k -> 0, {n, 67}] (* Michael De Vlieger, Jan 11 2018 *)
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[n_] := Length[invIPhi[n]]; Array[a, 100] (* using the function invIPhi from A362484 *)
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.
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 *)
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.
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 *)
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;
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]
Neither 50 nor 52 have cototient-inverses and since 50 is the first of the two and the least number with this property, a(2) = 50.
a = Table[0, {5*10^7}]; Do[b = n - EulerPhi[n]; If[ b < 5*10^7 + 1, a[[b/2]]++ ], {n, 2, 615437100}] (* used to find a(7) *) Do[ If[ a[[n]] == a[[n + 1]] == a[[n + 2]] == a[[n + 3]] == a[[n + 4]] == a[[n + 5]] == a[[n + 6]] == 0, Print[n]], {n, 1, 10^6}]
Comments