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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A096990 Initial values for f(x)=sigma(phi(x))=A062402(x) such that iteration of f ends in cycle of length=3.

Original entry on oeis.org

17, 19, 27, 29, 31, 32, 34, 35, 38, 39, 40, 41, 45, 47, 48, 52, 54, 55, 56, 58, 59, 60, 62, 69, 70, 72, 75, 78, 82, 84, 88, 90, 92, 94, 100, 110, 118, 132, 138, 150, 1057, 1117, 1153, 1201, 1237, 1241, 1261, 1271, 1301, 1313, 1321, 1333, 1349, 1351, 1359, 1381
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Links

Crossrefs

Cf. A062402, A095953.

Programs

  • Maple
    kT:= {}: kF:= {}:
f:= proc(t) uses numtheory; local S,R,i,val,s; global kT, kF;
  if member(t,kT) then return true elif member(t,kF) then return false fi;
  R[0]:= t;
  S:= {t};
  for i from 1 do
    R[i]:= sigma(phi(R[i-1]));
    if member(R[i], kT) then val:= true
    elif member(R[i], kF) then val:= false
    elif member(R[i],S) then
      val:= evalb(R[i-3] = R[i]) and not member(R[i],[R[i-1],R[i-2]])
    else val:= fail; S:= S union {R[i]}
    fi;
    if val = true then kT:= kT union {R[i]} union S; return true
    elif val = false then kF:= kF union {R[i]} union S; return false
    fi
  od;
end proc:
select(f, [$1..3000]); # Robert Israel, Jun 09 2024
  • Mathematica
    f[n_] := DivisorSigma[1, EulerPhi[n]]; g[n_] := Block[{l = NestWhileList[f, n, UnsameQ, All]}, -Subtract @@ Flatten[ Position[l, l[[ -1]]]]]; Select[ Range[ 1396], g[ # ] == 3 &] (* Robert G. Wilson v, Jul 23 2004 *)

Extensions

Edited and extended by Robert G. Wilson v, Jul 23 2004

A096996 a(n) is the smallest initial value if function f(x)=sigma(phi(x))=A062402(x) is iterated and the iteration ends in a cycle of length n.

Original entry on oeis.org

1, 5, 17, 401
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Comments

a(5) exists and <= 2^80.
a(6) = 883, a(11) = 88897, a(15) = 470137, a(18) = 448181;
a(9) = 24877841, a(12) = 16295171, a(21) <= 726569551. 254808457 does not reach a cycle after 14000 iterations. - Hiroaki Yamanouchi, Sep 06 2014
a(5) <= 9248288975491. - Hiroaki Yamanouchi, Sep 10 2014
a(5) > 8.5*10^7. - Tyler Busby, Mar 29 2024

Examples

			a(4) = 401. 401 -> 961 -> 2304 -> 2044 -> 2520 -> 1651 -> 4800 -> 3066 -> 2520 (cycle length = 4). - _Hiroaki Yamanouchi_, Sep 06 2014

Crossrefs

Cf. A062402, A095956.

A096998 Consider iteration of the function f(x) = sigma(phi(x)) = A062402(x). Sequence lists the numbers k such that the trajectory of k returns to k.

Original entry on oeis.org

1, 3, 7, 12, 15, 28, 31, 60, 72, 124, 168, 195, 252, 255, 744, 1240, 1512, 1651, 2418, 2520, 3066, 3844, 4092, 4800, 5080, 5376, 6045, 6138, 6552, 9906, 9920, 10200, 12264, 20440, 30855, 36792, 46228, 58968, 60984, 65535, 67963, 81880, 122640
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Examples

			96 => 63 => 91 => 195 => 252 => 195 => ..., therefore 195 and 252 are in the sequence.

Crossrefs

Cf. A062402, A096850.

Programs

  • Mathematica
    a = {}; f[n_] := DivisorSigma[1, EulerPhi[ n]]; Do[ AppendTo[ a, NestWhileList[f, n, UnsameQ, All][[ -1]]]; a = Union[a], {n, 10^6}]; Take[ a, 43] (* Robert G. Wilson v, Jul 21 2004 *)

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jul 21 2004

A097004 Function A062402(x)=phi(sigma(x)) is iterated. Starting with 2^n, the n-th power of 2, a(n) is the count of distinct terms arising in trajectory; a(n)=t(n)+c(n)=t+c, where t is the number of transient terms, c is the number of recurrent terms [in the terminal cycle].

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 5, 2, 4, 5, 11, 4, 4, 12, 17, 2, 8, 11, 14, 26, 11, 6, 80, 59, 100, 101, 95, 93, 60, 38, 55, 2
Offset: 0

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=13: 2^n=8192, trajectory ={8192,8191,26208,[20440],.. }, a[13]=3+1=4 with 3 transients and one recurrent term.

Crossrefs

Cf. A000010, A000203, A062402, A096852, A096857.

Programs

  • Mathematica
    EulerPhi[DivisorSigma[1, x]] itef[x_, len_] :=NestList[fs, x, len] Table[Length[Union[itef[2^w, 1000]]], {w, 1, 256}]

A281627 a(n) is the smallest number k such that sigma(phi(k)) = A062402(k) is the n-th Mersenne prime (A000668(n)), or 0 if no such k exists.

Original entry on oeis.org

3, 5, 17, 85, 4369, 65537, 327685, 1431655765, 2305843009213693952, 618970019642690137449562112, 162259276829213363391578010288128, 170141183460469231731687303715884105728
Offset: 1

Views

Author

Jaroslav Krizek, Feb 11 2017

Keywords

Comments

Conjecture 1: If A062402(n) = A000203(A000010(n)) = sigma(phi(n)) is a prime p for some n, then p is Mersenne prime (A000668); a(n) > 0 for all n.
Conjecture 2: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 divisors; see A276044.
Conjecture 3: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 prime factors (counted with multiplicity); see A275969.
a(n) <= A000668(n) for n = 1-18; conjecture: a(n) <= A000668(n) for all n.
Equals A002181 (index in A002202 of (intersection of A023194 and A002202)). - Michel Marcus, Feb 12 2017

Links

Crossrefs

Cf. A000010, A000043, A000203, A000668, A062402, A062514, A275969, A276044.
Cf. A002181, A002202, A023194.
Cf. A053576 (includes the first 13 known terms of this sequence).

Programs

  • Magma
    A281627:=func; Set(Sort([A281627(n): n in [SumOfDivisors(EulerPhi(n)): n in[1..1000000] | IsPrime(SumOfDivisors(EulerPhi(n)))]]));
  • PARI
    terms() = {v = readvec("b023194.txt"); for(i=1, #v, if (istotient(v[i], &n), print1(n/2, ", ")););} \\ Michel Marcus, Feb 12 2017
  • PARI
    f(p) = {my(s = invsigma(p), t, minv = 0); for(i = 1 ,#s, t = invphi(s[i]); for(j = 1, #t, if(minv == 0, minv = t[j]); if(t[j] < minv, minv = t[j]))); minv;} \\ using Max Alekseyev's invphi.gp
list(pmax) = forprime(p = 1, pmax, if(isprime(2^p-1), print1(f(2^p-1), ", "))); \\ Amiram Eldar, Dec 23 2024

Extensions

a(8) from Michel Marcus, Feb 12 2017
a(9)-a(12) from Amiram Eldar, Dec 23 2024

A096989 Initial values for f(x)=sigma(phi[x])=A062402[x] such that iteration of f ends in cycle of length=2.

Original entry on oeis.org

5, 7, 8, 9, 10, 11, 12, 14, 18, 22, 37, 43, 49, 51, 53, 57, 61, 63, 64, 65, 67, 68, 71, 73, 74, 76, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 93, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122, 123, 124
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Crossrefs

Cf. A062402, A095953.

A096991 Initial values for f(x)=sigma(phi[x])=A062402[x] such that iteration of f ends in cycle of length=4.

Original entry on oeis.org

401, 451, 489, 491, 505, 513, 567, 577, 629, 631, 652, 661, 673, 679, 685, 691, 713, 731, 737, 751, 757, 769, 773, 791, 802, 808, 811, 817, 825, 829, 833, 847, 851, 853, 859, 867, 869, 871, 873, 877, 889, 893, 899, 901, 902, 907, 911, 917, 919, 923, 931
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Crossrefs

Cf. A062402, A096926.

A096992 Initial values for f(x)=sigma(phi[x])=A062402[x] such that iteration of f ends in cycle of length=6.

Original entry on oeis.org

883, 985, 1009, 1025, 1037, 1051, 1067, 1073, 1141, 1145, 1147, 1159, 1199, 1203, 1205, 1217, 1249, 1267, 1285, 1291, 1297, 1303, 1309, 1343, 1345, 1353, 1355, 1361, 1369, 1371, 1373, 1385, 1387, 1417, 1435, 1461, 1463, 1465, 1467, 1471, 1493, 1505
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Crossrefs

Cf. A062402, A095954.

A096997 If the function f(x) = sigma(phi(x)) = A062402(x) is iterated starting from these listed values, then the starting value never returns. These are the transient terms of this iteration; they never occur in terminal cycles.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Labos Elemer, Jul 19 2004

Keywords

Crossrefs

Cf. A062402, A096849.

A097001 A062402(x)=sigma(phi[x]) function is iterated; initial value=2^n; a(n)=largest term of trajectory.

Original entry on oeis.org

2, 4, 12, 16, 72, 252, 312, 256, 1512, 1860, 12264, 6552, 26208, 34200, 93600, 65536, 833280, 1116024, 2239920, 4464096, 9865440, 8124480, 569540160, 569540160, 1100946774480, 1100946774480, 1100946774480, 1100946774480, 34696672920
Offset: 1

Views

Author

Labos Elemer, Jul 21 2004

Keywords

Examples

			n=13: 2^n=8192, list={8192,8191,26208,[20440],20440,.. a[13]=26208 arose in transient.

Crossrefs

Cf. A000010, A000203, A062402, A096852.

Programs

  • Mathematica
    gf[x_] :=DivisorSigma[1, EulerPhi[x]] gite[x_, hos_] :=NestList[gf, x, hos] Table[Max[gite[2^w, 200]], {w, 1, 30}]
Previous Showing 11-20 of 76 results. Next

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