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.

Showing 1-6 of 6 results.

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]

A288452 Pseudoperfect totient numbers: numbers n such that equal the sum of a subset of their iterated phi(n).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 131, 137, 139, 141, 143, 149, 151, 153, 155
Offset: 1

Views

Author

Amiram Eldar, Jun 09 2017

Keywords

Comments

Analogous to A005835 (pseudoperfect numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).
All the odd primes are in this sequence.
Number of terms < 10^k: 4, 40, 350, 2956, 24842, etc. - Robert G. Wilson v, Jun 17 2017
All terms are odd. If n is even, phi(n) <= n/2, and except for n = 2, we will have phi(n) also even. So the sum of the phi sequence < n*(1/2 + 1/4 + ...) = n. - Franklin T. Adams-Watters, Jun 25 2017

Examples

			The iterated phi of 25 are 20, 8, 4, 2, 1 and 25 = 20 + 4 + 1.

Links

Crossrefs

Supersequence of A082897. Subsequence of A286265.
Cf. A000010, A000396, A005835, A053478, A092693.

Programs

  • Mathematica
    pseudoPerfectTotQ[n_]:= Module[{tots = Most[Rest[FixedPointList[EulerPhi@# &, n]]]}, MemberQ[Total /@ Subsets[tots, Length[tots]], n]]; Select[Range[155], pseudoPerfectTotQ]
  • PARI
    subsetSum(v, target)=if(setsearch(v,target), return(1)); if(#v<2, return(target==0)); my(u=v[1..#v-1]); if(target>v[#v] && subsetSum(u, target-v[#v]), return(1)); subsetSum(u,target);
is(n)=if(isprime(n), return(n>2)); my(v=List(),k=n); while(k>1, listput(v,k=eulerphi(k))); subsetSum(Set(v),n) \\ Charles R Greathouse IV, Jun 25 2017

A286267 Totient highly abundant numbers: numbers n such that A092693(n)+n > A092693(m)+m for all m < n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 23, 29, 37, 41, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 107, 127, 131, 137, 149, 167, 179, 221, 223, 227, 233, 251, 257, 289, 317, 347, 353, 359, 383, 389, 431, 443, 449, 461, 467, 503, 557, 563, 569, 587, 641, 677, 697
Offset: 1

Views

Author

Amiram Eldar, May 05 2017

Keywords

Comments

Analogous to A002093 (highly abundant numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

Links

Crossrefs

Cf. A000396, A002093, A082897, A092693, A286265.

Programs

  • Mathematica
    Function[s, Flatten[First@ Position[s, #] & /@ Union@ Rest@ FoldList[Max, 0, s]]]@ Table[(Total@ FixedPointList[EulerPhi, n] - 1), {n, 10^3}] (* Michael De Vlieger, May 06 2017 *)

A286268 Totient superabundant numbers: numbers n such that A092693(n)/n > A092693(m)/m for all m < n.

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 83, 137, 257, 2879, 46049, 65537
Offset: 1

Views

Author

Amiram Eldar, May 05 2017

Keywords

Comments

Analogous to A004394 (superabundant numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).
The first 6 terms of A092506 (primes of the form 2^n + 1) are in this sequence.
a(13) > 1.6*10^10, if it exists. - Giovanni Resta, May 05 2017

Links

Crossrefs

Cf. A000396, A004394, A082897, A092506, A092693, A286265, A286267.

Programs

  • Mathematica
    Function[s, Flatten[First@ Position[s, #] & /@ Union@ Rest@ FoldList[Max, 0, s]]]@ Table[(Total@ FixedPointList[EulerPhi, n] - (n + 1))/n, {n, 10^5}] (* Michael De Vlieger, May 06 2017, after Alonso del Arte at A092693 *)

A335121 Admirable totient numbers: numbers that are equal to the sum of their iterated phi, with one of them taken with a minus sign.

Original entry on oeis.org

5, 7, 33, 35, 55, 87, 95, 175, 201, 215, 219, 245, 531, 747, 927, 939, 1047, 1295, 1463, 1473, 1551, 1855, 2015, 2103, 2421, 2431, 2547, 2619, 2631, 2765, 3535, 4833, 5067, 5215, 7655, 7743, 7851, 10503, 11127, 11307, 13055, 13707, 16247, 16593, 17805, 18471
Offset: 1

Views

Author

Amiram Eldar, May 24 2020

Keywords

Comments

Analogous to A111592 (admirable numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

Examples

			5 is a term since the values of the iterated phi of 5 are 4, 2 and 1 and 5 = 4 + 2 - 1.

Links

Crossrefs

Subsequence of A286265.
Cf. A000010, A053478, A082897, A111592, A288452.

Programs

  • Mathematica
    admTotQ[n_] := Module[{s = Most @ Rest @ FixedPointList[EulerPhi, n]}, (ab = Plus @@ s - n) > 0 && EvenQ[ab] && ab/2 < n && MemberQ[s, ab/2]]; Select[Range[8000], admTotQ]

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
Showing 1-6 of 6 results.

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