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.

A332737 Composite terms of A181659, where the sum of the iterated totient function attains a record.

Original entry on oeis.org

289, 2329, 4369, 4913, 18769, 21331, 35209, 66049, 128881, 197143, 258121, 281929, 516961, 739903, 971203, 1762249, 1942663, 2070721, 2898703, 2952673, 3820819, 4142881, 8288641, 16773619, 16843009, 16974593, 20229241, 21762361, 32472241, 132575071, 187903693
Offset: 1

Views

Author

Amiram Eldar, Feb 21 2020

Keywords

Comments

Most of the terms of A181659 are primes. Out of the first 10^4 terms of A181659 only 28 are composites.
The indices of the terms of this sequence in A181659 are 30, 73, 93, 99, 154, 161, 191, 236, 286, 316, ...
The corresponding record values (terms of A126106) are 527, 4223, 8191, 8847, 35527, 39423, 67583, 131327, 246869, 376559, 493739, 550911, 1009981, 1466879, 1884671, 3442687, 3819519, 4089245, 5707263, 5791743, 7444991, 8178491, 16464253, 33260031, 33554431, 33718527, 39989247, 42809067, 63932219, 263382015, 372697723.

Crossrefs

Cf. A092693, A126106, A181659.

Programs

  • Mathematica
    s[n_] := Plus @@ FixedPointList[EulerPhi, n] - n - 1; seq={}; smax = 1; Do[s1 =s[n];  If[s1 >smax, smax = s1; If[CompositeQ[n], AppendTo[seq, n]]], {n, 1,  5000}]; seq

A330400 Numbers at which the sum of the iterated unitary totient function (A329153) attains a record.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 16, 17, 29, 32, 41, 45, 53, 64, 73, 83, 85, 101, 113, 125, 128, 137, 153, 187, 197, 233, 257, 389, 401, 512, 577, 641, 677, 685, 703, 773, 901, 929, 977, 1153, 1193, 1493, 1537, 1553, 1657, 1697, 2047, 2048, 2313, 2897, 3089, 3137, 3593, 4001
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2020

Keywords

Comments

Analogous to A181659 with the unitary totient function (A047994) instead of the Euler totient function phi (A000010).
The corresponding record values are 0, 1, 3, 6, 10, 16, 24, 39, 55, 70, 85, ... (see the link for more values).

Links

Crossrefs

Cf. A047994, A181659, A329153.

Programs

  • Mathematica
    uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); s[n_] := Plus @@ FixedPointList[uphi, n] - n - 1; seq = {}; smax = -1; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 5000}]; seq

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]]

A331407 Numbers at which the sum of the iterated exponential totient function (A331273) attains a record.

Original entry on oeis.org

1, 2, 8, 32, 128, 864, 3456, 7776, 31104, 279936, 497664, 1990656, 4478976, 17915904, 62208000, 97200000, 559872000, 874800000, 1555200000, 6220800000, 13996800000, 55987200000
Offset: 1

Views

Author

Amiram Eldar, Feb 25 2020

Keywords

Comments

Analogous to A181659 with the exponential totient function (A072911) instead of the Euler totient function phi (A000010).
The corresponding record values are 0, 1, 3, 5, 7, 11, 13, 19, 27, 37, 43, 51, 61, 75, 83, 101, 123, 147, 165, 195, 243, 293, ...

Crossrefs

Cf. A072911, A181659, A331273.

Programs

  • Mathematica
    ephi[n_] := Times @@ EulerPhi[FactorInteger[n][[;; , 2]]]; s[n_] := Plus @@ FixedPointList[ephi, n] - n - 1; seq = {}; smax = -1; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 5000}]; seq

A333612 Numbers at which the sum of the iterated infinitary totient function (A091732) attains a record.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 11, 13, 16, 17, 29, 37, 47, 49, 53, 81, 101, 107, 113, 149, 173, 197, 257, 389, 401, 509, 529, 531, 557, 593, 677, 701, 747, 773, 829, 963, 977, 1109, 1297, 1493, 1675, 1733, 1901, 2417, 2761, 2837, 3089, 3313, 3329, 3413, 3467, 3677, 3803, 3989
Offset: 1

Views

Author

Amiram Eldar, Mar 28 2020

Keywords

Comments

Analogous to A181659 with the infinitary totient function A091732 instead of the Euler totient function phi (A000010).
The corresponding record values are 0, 1, 3, 6, 10, 14, 20, 21, 29, 45, ... (see the link for more values).

Links

Crossrefs

Cf. A091732, A181659, A330400, A331407, A333611.

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); s[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; seq = {}; smax = -1; Do[s1 = s[n];  If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq

A333872 Numbers at which the sum of the iterated absolute Möbius divisor function (A173557) attains a record.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 17, 19, 23, 31, 41, 43, 47, 59, 71, 79, 83, 103, 107, 131, 139, 167, 223, 227, 263, 347, 359, 383, 467, 479, 563, 587, 659, 719, 839, 863, 887, 1019, 1163, 1187, 1223, 1259, 1283, 1307, 1319, 1367, 1439, 1823, 1979, 2027, 2039, 2207, 2447, 2879
Offset: 1

Views

Author

Amiram Eldar, Apr 08 2020

Keywords

Comments

Analogous to A181659 with the absolute Möbius divisor function (A173557) instead of the Euler totient function phi (A000010).
The corresponding record values are 0, 1, 3, 5, 9, 15, 17, 21, 37, 39, 45, ... (see the link for more values).

Links

Crossrefs

Cf. A173557, A181659, A330400, A331407, A333612, A333871.

Programs

  • Mathematica
    f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); s[n_] := Plus @@ FixedPointList[u, n] - n - 1; seq = {}; smax = -1; Do[s1 = s[n];  If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 3000}]; seq
Showing 1-6 of 6 results.

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