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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A125734 Primes of the form 4*3^k + 1.

Original entry on oeis.org

5, 13, 37, 109, 2917, 19131877, 57395629, 16210220612075905069, 3187367866510497232065375864429355521950801431840733951694899540869109890815626195932616388528013, 254244997489062154119688681828370010268347235132197783249391539881181660045297550875174703528321187968562717038040968333
Offset: 1

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Author

David Eppstein, Feb 06 2007, Feb 07 2007

Keywords

Comments

Venkataraman showed that, for every p of this form, 3p is a perfect totient number (cf. A082897).

Examples

			37 = 4*3^2 + 1 is a prime of this form. 973 = 4*3^5 + 1 = 7*139 is not a prime, so is not included in this sequence.

References

  • T. Venkataraman, Perfect totient number, The Mathematics Student, Vol. 43 (1975), p. 178. MR0447089.

Links

Crossrefs

Cf. A005537, A082897.

Programs

  • Mathematica
    Do[p = 4*3^i + 1; If[PrimeQ@p, Print@p], {i, 0, 300}] (* Robert G. Wilson v, Feb 20 2007 *)

Formula

4*3^k + 1 where k belongs to A005537.

Extensions

2 more terms from Robert G. Wilson v, Feb 20 2007

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 *)

A385746 Numbers that are equal to the sum of their iterated infinitary analog of the totient function A384247.

Original entry on oeis.org

3, 10, 18, 21, 48, 160, 288, 3252, 9304, 13965, 68526, 719631, 1531101, 1954782, 28900572, 39189195, 14708055957
Offset: 1

Views

Author

Amiram Eldar, Jul 08 2025

Keywords

Comments

Numbers k such that A385745(k) = k.

Examples

			  n | a(n) | iterations                        | sum
  --+------+-----------------------------------+----------------------------
  1 |    3 | 3 -> 2 -> 1                       | 2 + 1 = 3
  2 |   10 | 10 -> 4 -> 3 -> 2 -> 1            | 4 + 3 + 2 + 1 = 10
  3 |   18 | 18 -> 8 -> 4 -> 3 -> 2 -> 1       | 8 + 4 + 3 + 2 + 1 = 18
  4 |   21 | 21 -> 12 -> 6 -> 2 -> 1           | 12 + 6 + 2 + 1 = 21
  5 |   48 | 48 -> 30 -> 8 -> 4 -> 3 -> 2 -> 1 | 30 + 8 + 4 + 3 + 2 + 1 = 48

Crossrefs

Cf. A384247, A385744, A385745, A385747.
Similar sequences: A082897, A286067, A330273.

Programs

  • Mathematica
    f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n];
infPerfTotQ[n_] := Plus @@ FixedPointList[iphi, n] == 2*n + 1; infPerfTotQ[1] = False; Select[Range[10^5], infPerfTotQ]
  • PARI
    iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2)))); }
s(n) = if(n == 1, 0, my(i = iphi(n)); i + s(i));
isok(k) = s(k) == k;

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 *)

A349306 Numbers k that divide A092694(k).

Original entry on oeis.org

1, 8, 16, 27, 32, 54, 64, 81, 108, 125, 128, 162, 216, 243, 250, 256, 324, 343, 432, 486, 500, 512, 625, 648, 686, 729, 864, 972, 1000, 1024, 1029, 1250, 1296, 1331, 1372, 1458, 1728, 1944, 2000, 2048, 2058, 2187, 2197, 2401, 2500, 2592, 2662, 2744, 2916, 3087
Offset: 1

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Author

Amiram Eldar, Nov 14 2021

Keywords

Comments

Includes all the numbers of the form 2^k * m, where k >= 0 and m is an odd cubefull number, except for 2 and 4. In particular, includes all the cubefull numbers (A036966).
Terms whose odd part is not cubefull are 1029, 2058, 3087, 4116, 6174, 6591, ...

Examples

			8 is a term since A092694(8) = 8 is divisible by 8.
16 is a term since A092694(16) = 64 is divisible by 16.

Links

Crossrefs

Cf. A000265, A082897, A092694.
A036966 is a subsequence.

Programs

  • Mathematica
    Select[Range[3000], Divisible[Times @@ FixedPointList[EulerPhi, #]/#, #] &]

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

A286233 Amicable totient numbers: pairs of numbers (m, n) such that n = A092693(m) and m = A092693(n).

Original entry on oeis.org

579, 639, 14911, 18207, 38575, 47223, 310399, 492855, 16632919, 20238207, 34696495, 37400607, 37852351, 52463103, 84250111, 122992023
Offset: 1

Views

Author

Amiram Eldar, May 04 2017

Keywords

Comments

Analogous to amicable pairs (A063990) as perfect totient numbers (A082897) are analogous to perfect numbers (A000396).
The sequence lists the numbers in increasing order. The first 8 pairs (m, n) are adjacent to each other in the list.
No other terms below 10^9.

Examples

			A092693(579) = phi(579) + phi(phi(579)) + ... = 384 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 639, and A092693(639) = phi(639) + phi(phi(639)) + ... = 420 + 96 + 32 + 16 + 8 + 4 + 2 + 1 = 579.

Crossrefs

Cf. A092693, A082897, A091847.

Programs

  • Mathematica
    totSum[n_] := Plus @@ FixedPointList[EulerPhi@# &, n] - n - 1; amicableTotQ[n_] := If[Nest[totSum, n, 2] == n && totSum[n] != n, True, False]; Select[Range[10^9], amicableTotQ[#] &]

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]

A331017 Numbers that equal to the sum of their iterated absolute alternating sum-of-divisors function (A206369).

Original entry on oeis.org

3, 10, 21, 63, 104, 152, 170, 358, 567, 651, 3826, 4664, 5583, 5943, 39248, 943228, 1906503, 9166540, 35702868, 701792828, 825415941, 2275142000, 5805372939
Offset: 1

Views

Author

Amiram Eldar, Jan 06 2020

Keywords

Examples

			10 is a term since the iterations of A206369 with a starting value of 10 give A206369(10) = 4, A206369(4) = 3, A206369(3) = 2, and A206369(2) = 1, whose sum is 4 + 3 + 2 + 1 = 10.

Crossrefs

Cf. A082897, A206369, A330786 (number of iterations).

Programs

  • Mathematica
    f[p_, e_] := Sum[(-1)^(e - k)*p^k, {k, 0, e}]; s[1] = 1; s[n_] := s[n] = Times @@ (f @@@ FactorInteger[n]); Select[Range[1000], Plus @@ FixedPointList[s, #] == 2 # + 1 &]

A333103 Quasiperfect cototient numbers: numbers k such that the sum of the iterated cototient function of k is equal to k+1.

Original entry on oeis.org

6, 14, 62, 254, 16382, 78585, 87465, 262142, 1048574
Offset: 1

Views

Author

Amiram Eldar, Mar 07 2020

Keywords

Comments

If m is in A050475 then 2^m - 2 is a term.
3*10^8 < a(10) <= 4294967292.

Examples

			6 is a term since A051953(6) = 4, A051953(4) = 2, A051953(2) = 1, and 4 + 2 + 1 = 7 = 6 + 1.

Links

Crossrefs

Cf. A050475, A051953, A082897, A286067, A330273.

Programs

  • Mathematica
    cot[n_] := n - EulerPhi[n]; s[n_] := Plus @@ FixedPointList[cot, n]; Select[Range[10^5], s[#] == 2*# + 1 &]
Previous Showing 11-20 of 27 results. Next

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