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 10 results.

A293184 Numbers k such that bphi(k) = bphi(k+1), where bphi(k) is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

1, 14, 20, 57, 187, 188, 916, 1603, 93928, 142891, 432976, 549815, 692259, 773887, 872191, 4297168, 9478088, 127162432, 127991488, 129015616, 132527167
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2017

Keywords

Comments

187 is the first solution to bphi(k) = bphi(k+1) = bphi(k+2).
a(22) > 1.6*10^9, if it exists. - Amiram Eldar, Jul 16 2022

Examples

			14 is in the sequence since bphi(14) = bphi(15) = 9.

Crossrefs

Cf. A001274, A116550, A287055, A294030.

Programs

  • Mathematica
    bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k];If[b1 == b2, a = AppendTo[a, k - 1]]; b1 = b2, {k, 1, 10^3}]; a (* after Jean-François Alcover at A116550 *)
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biuphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
isok(n) = biuphi(n) == biuphi(n+1);
lista(nn) = {x = biuphi(1); for (n=2, nn, y = biuphi(n); if (x==y, print1(n-1, ", ")); x = y;);} \\ Michel Marcus, Nov 09 2017

Extensions

a(10) from Michel Marcus, Nov 11 2017
a(11) from Michel Marcus, Nov 12 2017
a(12)-a(21) from Amiram Eldar, Jul 16 2022

A326403 Numbers k such that iphi(k) = iphi(k+1), where iphi(k) is an infinitary analog to the Euler totient function (A091732).

Original entry on oeis.org

1, 20, 35, 143, 194, 208, 740, 1220, 1299, 1419, 1803, 1892, 3705, 3716, 3843, 5186, 5635, 7868, 10659, 13634, 13905, 17948, 18507, 18914, 18980, 21007, 25388, 25545, 30380, 31599, 32304, 34595, 37820, 47067, 70394, 73059, 78064, 87856, 94874, 105908, 116963
Offset: 1

Views

Author

Amiram Eldar, Sep 14 2019

Keywords

Examples

			20 is in the sequence since iphi(20) = iphi(21) = 12.

Links

Crossrefs

Cf. A001274, A091732, A287055, A293184, A301866.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Times @@ (Flatten @(f @@@ FactorInteger[n]) - 1); a1 = 1; s = {}; Do[a2 = a[n]; If[a1 == a2, AppendTo[s, n - 1]]; a1 = a2, {n, 2, 10^5}]; s

A349307 Numbers k such that A072911(k) = A072911(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 567, 624, 728, 783, 944, 1160, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3087, 3320, 3429, 3536, 3591, 3624, 3752, 3992, 4023, 4184, 4239, 4375, 4400, 4455, 4624, 4671, 4887, 4912, 4913, 5048, 5103
Offset: 1

Views

Author

Amiram Eldar, Nov 14 2021

Keywords

Comments

Without the restriction that A072911(k) > 1, all the terms of A340152 would be in this sequence.
In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 4912, 5750, 6858, ...

Examples

			80 is a term since A072911(80) = A072911(81) = 2.

Links

Crossrefs

Cf. A072911, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349308.

Programs

  • Mathematica
    f[p_, e_] := EulerPhi[e]; ephi[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[5000], ephi[#] == ephi[# + 1] > 1 &]

A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 624, 728, 1160, 1431, 1592, 1624, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4671, 4887, 4913, 5048, 5144, 5264, 5319, 5480, 5696, 6183, 6344, 6375, 6591, 6615, 6776, 6858, 6859, 7479, 7624, 7640
Offset: 1

Views

Author

Amiram Eldar, Nov 14 2021

Keywords

Comments

Without the restriction that A321167(k) > 1, all the terms of A340152 would be in this sequence.
In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 6858, 13375, 22625, ...

Examples

			80 is a term since A321167(80) = A321167(81) = 3.

Links

Crossrefs

Cf. A321167, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349307.

Programs

  • Mathematica
    f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; fe[p_, e_] := uphi[e]; euphi[n_] := Times @@ fe @@@ FactorInteger[n]; Select[Range[8000], euphi[#] == euphi[# + 1] > 1 &]

A332316 Numbers k such that k and k + 1 have the same value of the norm of the totient function for Gaussian integers (A103224).

Original entry on oeis.org

4, 5, 35, 51, 154, 804, 3596, 6200, 7595, 916538, 1638039, 2794805, 6804035, 24724472, 40128444, 52424787, 69918849, 82954611, 124077316, 160245605, 204215514, 361275551, 371254235, 661831521, 1314759754, 1695554762, 2246110022, 2378746131, 2889320799, 4181707719
Offset: 1

Views

Author

Amiram Eldar, Feb 09 2020

Keywords

Comments

The corresponding common values A103224(k) = A103224(k+1) are 8, 8, 288, 640, 7200, 139392, 3744000, 5760000, ...

Examples

			4 is a term since norm(phi(4)) = norm(phi(5)) = 8.

Links

Crossrefs

Cf. A001274, A103222, A103223, A103224, A287055, A293184, A326403.

Programs

  • Mathematica
    phi[z_] := Module[{f, k, prod}, If[Abs[z]==1, z, f = FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, k=2; prod=f[[1, 1]], k=1; prod=1]; Do[prod=prod*(f[[i, 1]]-1)f[[i, 1]]^(f[[i, 2]]-1), {i, k, Length[f]}]; prod]]; Select[Range[10^3], Abs[phi[#]]^2 == Abs[phi[# + 1]]^2 &] (* after T. D. Noe at A103222 *)

A349309 Numbers k such that A254926(k) = A254926(k+1).

Original entry on oeis.org

7, 26, 124, 342, 1330, 2196, 12166, 24388, 29790, 79506, 103822, 148876, 205378, 226980, 300762, 357910, 493038, 571786, 1030300, 1092726, 1225042, 2248090, 2685618, 3307948, 3442950, 3869892, 4657462, 5177716, 5735338, 6967870, 7645372, 9393930, 11089566, 11697082
Offset: 1

Views

Author

Amiram Eldar, Nov 14 2021

Keywords

Examples

			7 is a term since A254926(7) = A254926(8) = 7.

Links

Crossrefs

Cf. A254926.
Similar sequences: A001274, A287055, A293184, A326403, A336673, A349307, A349308.

Programs

  • Mathematica
    f[p_, e_] := If[e < 3, p^e, p^e - p^(e - 3)]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^6], s[#] == s[# + 1] &]
  • Python
    from math import prod
from itertools import count, islice
from sympy import factorint
def A349309_gen(startvalue=1): # generator of terms >= startvalue
    a = prod(p**e - (p**(e-3) if e >= 3 else 0) for p, e in factorint(max(startvalue,1)).items())
    for k in count(max(startvalue,1)):
        b = prod(p**e - (p**(e-3) if e >= 3 else 0) for p, e in factorint(k+1).items())
        if a == b:
            yield k
        a = b
A349309_list = list(islice(A349309_gen(),10)) # Chai Wah Wu, Jan 24 2022

A290304 Values of uphi(k) = uphi(k+1).

Original entry on oeis.org

1, 12, 24, 120, 96, 180, 432, 744, 720, 864, 840, 1200, 1260, 1680, 2520, 1728, 2784, 2880, 3744, 4032, 5040, 2592, 4224, 5040, 5760, 11520, 11880, 9216, 18000, 20160, 17280, 12480, 17280, 20160, 28080, 20160, 23040, 21600, 32256, 30240, 52080, 34560, 57600
Offset: 1

Views

Author

Amiram Eldar, Jul 26 2017

Keywords

Comments

The values of unitary totient function of numbers such that k and k+1 have the same value.
The unitary version of A003275.

Links

Crossrefs

Cf. A001274, A003275, A047994, A287055.

Programs

  • Mathematica
    uphi[n_] := If[n==1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]]; a={}; u1=0; For[k=0, k<10^5, k++; u2=uphi[k]; If[u1==u2, a = AppendTo[a, u1]]; u1=u2]; a

Formula

a(n) = A047994(A287055(n)).

A332530 Numbers k such that k and k + 1 has the same value of A319445, the equivalent of the Euler totient function in the ring of Eisenstein integers.

Original entry on oeis.org

34, 51, 152, 679, 1065, 1845, 6525, 12122, 12970, 15656, 38607, 48398, 175473, 272935, 401505, 953342, 1035895, 1210054, 1222988, 1406665, 1589245, 1607095, 2108186, 2116975, 2272425, 2500615, 2751160, 3399591, 4542225, 5298559, 5412986, 6813585, 6898736, 7115553
Offset: 1

Views

Author

Amiram Eldar, Feb 15 2020

Keywords

Examples

			34 is a term since A319445(34) = A319445(35) = 864.

Links

Crossrefs

Cf. A001274, A287055, A293184, A319445, A326403, A332316.

Programs

  • Mathematica
    f[p_, e_] := If[p == 3, 2*3^(2*e - 1), Switch[Mod[p, 3], 1, (p - 1)^2*p^(2*e - 2), 2, (p^2 - 1)*p^(2*e - 2)]]; eisPhi[1] = 1; eisPhi[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; e1 = eisPhi[1]; Do[e2 = eisPhi[n]; If[e1 == e2, AppendTo[seq, n - 1]]; e1 = e2, {n, 2, 10^6}]; seq

A385743 Numbers k such that A384247(k) = A384247(k+1).

Original entry on oeis.org

1, 20, 27, 35, 63, 64, 104, 143, 194, 208, 740, 836, 1220, 1299, 1419, 1803, 1892, 2625, 3255, 3705, 3716, 3843, 4096, 5184, 5186, 5635, 5695, 7868, 10659, 13365, 16904, 17948, 18507, 18914, 21007, 22935, 25388, 25545, 27675, 30380, 31599, 32304, 32864, 34595
Offset: 1

Views

Author

Amiram Eldar, Jul 08 2025

Keywords

Comments

63 is the only number k below 10^11 such that A384247(k) = A384247(k+1) = A384247(k+2). Are there any other such terms?

Examples

			1 is a term since A384247(1) = A384247(2) = 1.
20 is a term since A384247(20) = A384247(21) = 12.

Links

Crossrefs

Cf. A384247.
Similar sequences: A001274, A287055, A293184, A301866, A326403, A349307.

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]; Select[Range[35000], iphi[#] == iphi[# + 1] &]
  • PARI
    iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2)))); }
list(lim) = {my(s1 = iphi(1), s2); for(k = 2, lim, s2 = iphi(k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}

A291530 a(n) is the smallest k such that uphi(k*n) = uphi(k*n+1), or 0 if no such k exists.

Original entry on oeis.org

1, 10, 373, 5, 4, 372, 5, 26, 248, 2, 13, 186, 11, 562, 247, 13, 627, 124, 195, 1, 183, 86, 245, 93, 5184, 8, 185, 281, 1623, 4320, 72, 738, 43, 2296, 1, 62, 20, 2312, 95, 3240, 576, 732, 33, 43, 111, 4600, 540100, 492, 115, 2592, 209, 4, 25383, 2388, 629, 549, 65, 1732, 64476, 2160, 20, 36, 61
Offset: 1

Views

Author

Altug Alkan, Aug 25 2017

Keywords

Examples

			a(3) = 373 because uphi(373*3) = uphi(373*3+1) and 373 is the smallest number with this property.

Crossrefs

Cf. A047994, A275412, A287055.

Programs

  • PARI
    uphi(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[1, 2]-1);
a(n) = {my(k = 1); while (uphi(k*n) != uphi(k*n+1), k++); k; }
Showing 1-10 of 10 results.

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