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 1-10 of 27 results. Next

A181659 Numbers at which the sum of the iterated totient function (A092693) attains a record.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 23, 29, 37, 41, 47, 53, 59, 71, 73, 79, 83, 89, 101, 107, 131, 137, 167, 179, 227, 233, 257, 289, 317, 347, 359, 389, 443, 449, 461, 467, 503, 557, 563, 569, 587, 641, 719, 809, 839, 857, 929, 977, 983, 1013, 1019, 1097, 1187, 1193, 1283
Offset: 1

Views

Author

T. D. Noe, Nov 04 2010

Keywords

Comments

Most of these numbers are prime. The first four composites are 289, 2329, 4369, and 4913.

Links

Crossrefs

Cf. A181660
Cf. A126106 (record values)

Programs

  • Mathematica
    kMax=2*3*5*7*11*13; t=Table[0,{kMax}]; Do[e=EulerPhi[k]; t[[k]]=e+t[[e]], {k,2,kMax}]; mx=-1; Reap[Do[If[t[[k]]>mx, mx=t[[k]]; Sow[k]], {k,kMax}]][[2,1]]

A286265 Totient abundant numbers: numbers k such that A092693(k) > k.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 123, 125, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153, 155, 157
Offset: 1

Views

Author

Amiram Eldar, May 05 2017

Keywords

Examples

			19 is a totient abundant number since A092693(19) = phi(19) + phi(phi(19)) + ... = 18 + 6 + 2 + 1 = 27 > 19.

Links

Crossrefs

Cf. A000010, A082897, A092693.

Programs

  • Mathematica
    totAbundantQ[n_] := Plus @@ FixedPointList[ EulerPhi@ # &, n] > 2*n+1; Select[Range[1000],totAbundantQ]
  • Python
    from sympy import totient
def a092693(n): return 0 if n==1 else totient(n) + a092693(totient(n))
print([n for n in range(1, 201) if a092693(n) > n]) # Indranil Ghosh, May 05 2017

A126106 Record values in A092693 = sum of iterated phi(n).

Original entry on oeis.org

0, 1, 3, 7, 9, 17, 19, 31, 39, 47, 55, 71, 85, 91, 105, 109, 111, 117, 153, 159, 171, 197, 209, 263, 319, 337, 417, 423, 511, 527, 551, 641, 695, 707, 761, 767, 779, 889, 923, 991, 1001, 1007, 1101, 1151, 1413, 1495, 1515, 1647, 1695, 1711, 1719, 1739, 1889
Offset: 1

Views

Author

Robert G. Wilson v, Feb 20 2007

Keywords

Links

Crossrefs

Cf. A092693.

Programs

  • Mathematica
    lst = {}; mim = -1; Do[a = Plus @@ FixedPointList[EulerPhi, n] - n - 1; If[a > mim, mim = a; AppendTo[lst, a]], {n, 1100}]; lst

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 *)

A181660 Numbers at which the sum of the iterated totient function (A092693) attains a minimum.

Original entry on oeis.org

1, 2, 6, 12, 18, 30, 42, 54, 60, 66, 90, 126, 150, 210, 270, 294, 330, 420, 462, 630, 660, 840, 882, 1050, 1260, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3234, 3570, 3990, 4410, 4620, 4830, 5250, 5460, 5670, 6090, 6930, 7350, 8190, 9030, 9240, 9450, 9660
Offset: 1

Views

Author

T. D. Noe, Nov 04 2010

Keywords

Comments

That is, for each n in this sequence, A092693(n) < A092693(m) for m > n. Do all primorials appear here?

Links

Crossrefs

Cf. A181659

Programs

  • Mathematica
    kMax=2*3*5*7*11*13; t=Table[0,{kMax}]; Do[e=EulerPhi[k]; t[[k]]=e+t[[e]], {k,2,kMax}]; mn=Infinity; Reverse[Reap[Do[If[t[[ -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 *)

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[#] &]

A291173 Totient superdeficient numbers: numbers n > 1 such that s(n)/n < s(m)/m for all m < n, where s(n) is the sum of iterated phi(n) (A092693).

Original entry on oeis.org

2, 42, 90, 210, 630, 1050, 1470, 2310, 6930, 16170, 30030, 90090, 150150, 210210, 570570, 690690, 870870, 2072070, 3573570, 3993990, 4834830
Offset: 1

Views

Author

Amiram Eldar, Aug 19 2017

Keywords

Comments

A variation of A286268 (Totient superabundant numbers).

Examples

			The values of A092693(n)/n for n = 2, 42, 90 are 0.5, 0.452..., 0.433..., which are records of lowest values.

Crossrefs

Cf. A092693, A286268, A291174.

Programs

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

A291177 Numbers k such that s(k) = s(k+1) but phi(k) != phi(k+1), where s(k) = phi(k) + phi(phi(k)) + ... + 1 is the sum of iterated phi (A092693).

Original entry on oeis.org

45, 297, 356, 375, 1335, 1935, 3915, 4743, 5271, 6015, 6375, 6903, 20894, 22311, 25347, 28118, 31664, 32384, 39632, 49155, 50954, 55935, 59984, 64514, 70275, 119324, 125054, 162944, 209715, 334304, 342975, 472718, 767584, 798567, 862802, 908775, 1280096
Offset: 1

Views

Author

Amiram Eldar, Aug 19 2017

Keywords

Comments

The restriction phi(k) != phi(k+1) is intended to exclude all the (trivial) terms of A001274.

Links

Crossrefs

Cf. A001274, A092693.

Programs

  • Mathematica
    s[n_]:=Plus @@ FixedPointList[EulerPhi, n] - (n + 1);seqQ[n_]:=(s[n+1]==s[n])&&(EulerPhi[n+1]!=EulerPhi[n]);Select[Range[10^5],seqQ]

A290149 Totient sublime numbers: numbers k such that the number of terms in the iterations of phi(k) from k to 1, A032358(k)+2, and their sum, A092693(k) are both perfect totient numbers (A082897).

Original entry on oeis.org

6, 2916, 4374, 109100, 113708, 3188646
Offset: 1

Views

Author

Amiram Eldar, Jul 21 2017

Keywords

Comments

Analogous to A081357 (sublime numbers), as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).
No other terms below 10^8.

Examples

			There are 9 terms in the iterations of phi(k) for 2916: 2916, 972, 324, 108, 36, 12, 4, 2, 1. Their sum is 4375. Both 9 and 4375 are perfect totient numbers (A082897).

Crossrefs

Cf. A032358, A081357, A082897, A092693.

Programs

  • Mathematica
    iterList [n_] := FixedPointList[EulerPhi@# &, n]; sumIter [n_] := Plus @@ iterList[n] - 1; numIter[n_] := Length[iterList[n]] - 1; perfTotQ[n_] := sumIter[n] == 2 n; totSublimeQ[n_] := perfTotQ[numIter[n]] && perfTotQ[sumIter[n]]; Select[Range [10^8], totSublimeQ]
Showing 1-10 of 27 results. Next

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