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.

Previous Showing 11-16 of 16 results.

A329153 Sum of the iterated unitary totient function (A047994).

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 16, 24, 10, 20, 9, 21, 9, 24, 39, 55, 24, 42, 21, 21, 20, 42, 23, 47, 21, 47, 42, 70, 24, 54, 85, 41, 55, 47, 47, 83, 42, 47, 70, 110, 21, 63, 54, 117, 42, 88, 54, 102, 47, 117, 83, 135, 47, 110, 63, 83, 70, 128, 47, 107, 54, 102, 165, 102
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2020

Keywords

Comments

Analogous to A092693 with the unitary totient function uphi instead of the Euler totient function phi (A000010).

Examples

			a(4) = uphi(4) + uphi(uphi(4)) + uphi(uphi(uphi(4))) = 3 + 2 + 1 = 6.

Links

Crossrefs

Cf. A003271, A047994, A049865, A092693, A225172, A225173, A286067.

Programs

  • Mathematica
    uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); Table[Plus @@ FixedPointList[uphi, n] - n - 1, {n, 1, 100}]

Formula

a(n) = n for n in A286067.

A225175 Largest number which requires n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 12, 18, 30, 42, 78, 106, 210, 366, 550, 603, 750, 1290, 2562, 4398, 4305, 7470, 9090, 14322, 24558, 35382, 55482, 78020, 141190, 207519, 301642, 429870, 552693, 684846, 1060710, 1391390, 2385246, 3454044
Offset: 0

Views

Author

N. J. A. Sloane, May 01 2013

Keywords

Comments

a(26) >= 55482. a(27) >= 78020. - R. J. Mathar, May 05 2013

References

  • M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Crossrefs

Cf. A116550, A005424, A225176, A225174, A047994, A003271, A225172, A225173.

Extensions

a(26)-a(37) from Donovan Johnson, Dec 07 2013

A225176 Number of numbers which require n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 2, 4, 7, 6, 13, 12, 16, 24, 31, 51, 66, 87, 126, 139, 187, 260, 331, 412, 551, 693
Offset: 1

Views

Author

N. J. A. Sloane, May 01 2013

Keywords

References

  • M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Crossrefs

Cf. A116550, A005424, A225175, A225174, A047994, A003271, A225172, A225173.

A385747 Least number that reaches 1 after exactly n iterations of the infinitary analog of the totient function A384247.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 16, 17, 41, 73, 101, 197, 467, 829, 1109, 2761, 4849, 7831, 12401, 26189, 52379, 85853, 139589, 237007, 395533, 947043, 1967027, 3446033, 5396427, 9510437, 17502533, 35005067, 71202449, 90187609, 164664701, 395199461, 705113873, 1265735729, 1803553457
Offset: 0

Views

Author

Amiram Eldar, Jul 08 2025

Keywords

Comments

a(n) is the least number k such that A385744(k) = n.
Also, indices of records of A385744.

Examples

			  n | a(n) | iterations
  --+------+---------------------------
  1 |    2 | 2 -> 1
  2 |    3 | 3 -> 2 -> 1
  3 |    4 | 4 -> 3 -> 2 -> 1
  4 |    5 | 5 -> 4 -> 3 -> 2 -> 1
  5 |    9 | 9 -> 8 -> 4 -> 3 -> 2 -> 1

Crossrefs

Cf. A384247, A385744, A385745, A385746.
Similar sequences: A003271, A005424, A007755, A333610.

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];
numiter[n_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1;
seq[len_] := Module[{s = {}, k = 0, i = 0}, While[Length[s] < len, k++; If[numiter[k] == i, AppendTo[s, k]; i++]]; s]; seq[25]
  • PARI
    iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2))));}
numiter(n) = if(n ==  1, 0, 1 + numiter(iphi(n)));
list(len) = {my(k = 0, i = 0, c = 0); while(c < len, k++; if(numiter(k) == i, c++; print1(k, ", "); i++));}

A333610 Least number that reaches 1 after n iterations of the infinitary totient function A091732.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 17, 47, 85, 227, 257, 919, 1229, 2459, 4369, 9839, 30865, 101503, 148157, 438499, 828297, 2201671, 3316617, 11055391, 35354993, 140810491, 188991053, 377982107, 848170377, 1704741139, 6933926513
Offset: 0

Views

Author

Amiram Eldar, Mar 28 2020

Keywords

Crossrefs

Cf. A003271, A007755, A091732, A333609.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); iphi[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); numiter[n_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1; n = 0; seq = {}; Do[If[numiter[k] == n, AppendTo[seq, k]; n++], {k, 1, 1000}]; seq

Formula

A333609(a(n)) = n.

A362025 a(n) is the least number that reaches 1 after n iterations of the infinitary totient function A064380.

Original entry on oeis.org

2, 3, 4, 5, 9, 11, 16, 17, 28, 29, 46, 47, 99, 145, 167, 205, 314, 397, 437, 793, 851, 1137, 1693, 2453, 2771, 2989, 3701, 5099, 6801, 9299, 12031, 15811, 16816, 21520, 21521, 29547, 39685, 62077, 83191, 103473, 112117, 149535, 157159, 196049, 200267, 303022
Offset: 1

Views

Author

Amiram Eldar, Apr 05 2023

Keywords

Crossrefs

Cf. A064380.
Indices of records of A362024.
Similar sequences: A003271, A007755, A333610.

Programs

  • Mathematica
    infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
iphi[n_] := Sum[Boole[infCoprimeQ[j, n]], {j, 1, n - 1}];
numiter[n_] := Length@ NestWhileList[iphi, n, # > 1 &] - 1;
seq[kmax_] := Module[{v = {}, n = 1}, Do[If[numiter[k] == n, AppendTo[v, k]; n++], {k, 2, kmax}]; v]; seq[1000]
  • PARI
    isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
iphi(n) = sum(j = 1, n-1, isinfcoprime(j, n));
numiter(n) = if(n==2, 1, numiter(iphi(n)) + 1);
lista(kmax) = {my(n = 1); for(k = 2, kmax, if(numiter(k) == n, print1(k, ", "); n++)); }

Formula

A362024(a(n)) = n, and A362024(k) < n for all k < a(n).
Previous Showing 11-16 of 16 results.

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