cp's OEIS Frontend

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.

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A241668 Sum of iterates of A241663 up to and including either 0 or 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 10, 0, 9, 0, 0, 0, 22, 0, 15, 0, 0, 0, 34, 0, 6, 0, 0, 0, 31, 0, 27, 0, 0, 0, 3, 0, 33, 0, 0, 0, 70, 0, 39, 0, 0, 0, 82, 0, 21, 0, 0, 0, 70, 0, 10, 0, 0, 0, 65, 0, 57, 0, 0, 0, 9, 0, 63, 0, 0, 0, 130, 0, 69, 0, 0, 0, 21, 0, 75, 0
Offset: 1

Views

Author

Colin Defant, Apr 26 2014

Keywords

Comments

This sequence is similar to the totient summatory function A092693, but uses the Schemmel totient function A241663 instead of the phi function.

Examples

			A241663(11)=7, A241663(7)=3, A241663(3)=0, so a(11)=7+3+0=10.
A241663(9)=0, so a(9)=0.

Links

Crossrefs

Cf. A092693, A241663, A241665.

Programs

  • Mathematica
    L[n_, m_] :=
If[Min[Select[Divisors[n], PrimeQ]] <= m, 0,
  n*Times @@ (1 - m/(Select[Divisors[n], PrimeQ]))]
a[0]:=0
a[5]:=1
a[n_]:=L[n, 4]+a[L[n, 4]]
  • PARI
    A241663(n) = {my(f = factor(n)); prod(i=1, #f~, if ((f[i, 1] == 2) || (f[i, 1] == 3), 0, f[i, 1]^(f[i, 2]-1)*(f[i, 1]-4))); } \\ From A241663
A241668(n) = { my(s=(1==n)); while(n>1, n = A241663(n); s += n); (s); }; \\ Antti Karttunen, Oct 01 2018

A348215 a(n) is the sum of the iterated A348158 starting from n until a fixed point is reached.

Original entry on oeis.org

0, 1, 0, 3, 0, 3, 0, 7, 0, 5, 0, 7, 0, 7, 0, 15, 0, 9, 0, 15, 0, 11, 0, 15, 0, 13, 0, 21, 0, 15, 0, 31, 0, 17, 0, 25, 0, 19, 0, 31, 0, 21, 0, 33, 0, 23, 0, 31, 0, 25, 0, 39, 0, 27, 0, 49, 0, 29, 0, 31, 0, 31, 57, 120, 0, 33, 0, 51, 0, 35, 0, 57, 0, 37, 0, 57
Offset: 1

Views

Author

Amiram Eldar, Oct 07 2021

Keywords

Comments

The first odd number k with a(k) > 0 is k = 63.

Examples

			a(4) = 3 since the iterations of the map x -> A348158(x) starting from 4 are 4 -> 3.
a(64) = 120 since the iterations of the map x -> A348158(x) starting from 64 are 64 -> 63 -> 57, and 63 + 57 = 120.

Links

Crossrefs

Cf. A092693, A326835, A348158, A348213, A348214.

Programs

  • Mathematica
    f[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; a[n_] := Plus @@ Most @ FixedPointList[f, n] - n; Array[a, 100]

Formula

a(n) = 0 if and only if n is in A326835.
a(2*n) > 0 for all n.

A113808 Positive integers n such that S(n) divides n, where S(n) is the sum of the iterates of the Euler phi-function of n, that is, S(n) = phi(n)+phi(phi(n))+....+ 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 15, 18, 27, 30, 39, 54, 78, 81, 111, 162, 183, 222, 243, 255, 327, 363, 366, 471, 486, 510, 654, 726, 729, 942, 1458, 2187, 2199, 3063, 4359, 4374, 4375, 4398, 5571, 6126, 6561, 8718, 8750, 8751, 11142, 13122, 15723, 17502, 19683, 31446, 36759
Offset: 1

Views

Author

Jeffrey Shallit, Jan 22 2006

Keywords

Examples

			18 is in the sequence because phi(18)+phi(phi(18))+phi(phi(phi(18))) = 6 + 2 + 1 = 9, which divides 18.

Links

Crossrefs

Cf. A092693, A082897.

Programs

  • Mathematica
    s[1]=1; s[n_] := Total@NestWhileList[EulerPhi, n, #>1 &] - n; Select[Range@ 1000, Mod[#, s@#] == 0 &] (* Giovanni Resta, May 25 2013 *)
  • PARI
    lista(nn) = {for (n=1, nn, s = 0; m = n; until (m == 1, m = eulerphi(m); s += m;); if ((n % s == 0), print1(n, ", ")););} \\ Michel Marcus, May 25 2013

A288453 Weird totient numbers: totient abundant numbers (A286265) that are not pseudoperfect totient numbers (A288452).

Original entry on oeis.org

91, 95, 133, 145, 185, 203, 215, 217, 259, 275, 301, 335, 343, 355, 365, 395, 427, 469, 497, 545, 551, 553, 575, 635, 637, 649, 655, 703, 725, 755, 763, 767, 785, 815, 817, 833, 865, 889, 893, 905, 917, 931, 949, 955, 973, 985, 995, 1007, 1027, 1057, 1073
Offset: 1

Views

Author

Amiram Eldar, Jun 09 2017

Keywords

Comments

Analogous to A006037 (weird numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

Examples

			The set of iterated phi of 91 is {72, 24, 8, 4, 2, 1} and none of its subsets sums to 91.

Links

Crossrefs

Cf. A000010, A000396, A006037, A092693, A082897, A286265.

Programs

  • Mathematica
    pseudoPerfectTotQ[n_] := Module[{tots = Most[Rest[FixedPointList[EulerPhi@# &, n]]]}, MemberQ[Total /@ Subsets[tots, Length[tots]], n]];
totAbundantQ[n_] := Plus @@ FixedPointList[EulerPhi@# &, n] > 2*n + 1;
weirdTotient[n_] := totAbundantQ[n] && ! pseudoPerfectTotQ[n];
Select[Range[1100], weirdTotient]

A291174 Unitary totient superdeficient numbers: numbers n > 1 such that s(n)/n < s(m)/m for all m < n, where s is the sum of iterated uphi (A047994).

Original entry on oeis.org

2, 3, 6, 12, 14, 26, 30, 42, 186, 210, 2310, 66990, 903210, 1037190, 1138830
Offset: 1

Views

Author

Amiram Eldar, Aug 19 2017

Keywords

Comments

The unitary version of A291173.

Crossrefs

Cf. A047994, A291173.

Programs

  • Mathematica
    uphi[n_] :=  If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@  FactorInteger[n]))[[1]]]; Function[s, Flatten[First@Position[s, #] & /@ Union@Rest@FoldList[Max, 0, s]]]@ Table[n/(Total@FixedPointList[uphi, n] - (n-5)), {n, 2, 10^4}]+1 (* after Michael De Vlieger at A286268 and Alonso del Arte at A092693 *)

A286266 Number of totient abundant numbers <= 10^n.

Original entry on oeis.org

2, 36, 383, 3708, 35731, 347505, 3407290, 33579303, 332026623, 3290205509
Offset: 1

Views

Author

Amiram Eldar, May 05 2017

Keywords

Comments

Totient abundant numbers are defined in A286265.
a(3)-a(8) were calculated by Loomis & Luca (2008).

Examples

			There are 2 totient abundant numbers <= 10^1 (5 and 7), thus a(1)=2.

Links

Crossrefs

Cf. A000010, A082897, A092693, A286265.

Programs

  • Mathematica
    Accumulate@ Table[Count[Select[Range[10^(n - 1) + 1, 10^n], (Total@ FixedPointList[EulerPhi, #] - (# + 1)) > # &], k_ /; k <= 10^n], {n, 6}] (* Michael De Vlieger, May 06 2017, after Alonso del Arte at A092693 *)
  • PARI
    s(n) = {n=eulerphi(n); if(n==1, 1, n+s(n));}
lista(nmax) = {my(c = 0, r = 10); for(k = 1, 10^nmax, if(s(k) > k, c++); if(k == r, print1(c, ", "); r *= 10));} \\ Amiram Eldar, Mar 26 2023

Extensions

a(9)-a(10) from Amiram Eldar, Mar 26 2023

A332737 Composite terms of A181659, where the sum of the iterated totient function attains a record.

Original entry on oeis.org

289, 2329, 4369, 4913, 18769, 21331, 35209, 66049, 128881, 197143, 258121, 281929, 516961, 739903, 971203, 1762249, 1942663, 2070721, 2898703, 2952673, 3820819, 4142881, 8288641, 16773619, 16843009, 16974593, 20229241, 21762361, 32472241, 132575071, 187903693
Offset: 1

Views

Author

Amiram Eldar, Feb 21 2020

Keywords

Comments

Most of the terms of A181659 are primes. Out of the first 10^4 terms of A181659 only 28 are composites.
The indices of the terms of this sequence in A181659 are 30, 73, 93, 99, 154, 161, 191, 236, 286, 316, ...
The corresponding record values (terms of A126106) are 527, 4223, 8191, 8847, 35527, 39423, 67583, 131327, 246869, 376559, 493739, 550911, 1009981, 1466879, 1884671, 3442687, 3819519, 4089245, 5707263, 5791743, 7444991, 8178491, 16464253, 33260031, 33554431, 33718527, 39989247, 42809067, 63932219, 263382015, 372697723.

Crossrefs

Cf. A092693, A126106, A181659.

Programs

  • Mathematica
    s[n_] := Plus @@ FixedPointList[EulerPhi, n] - n - 1; seq={}; smax = 1; Do[s1 =s[n];  If[s1 >smax, smax = s1; If[CompositeQ[n], AppendTo[seq, n]]], {n, 1,  5000}]; seq
Previous Showing 21-27 of 27 results.

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