A293184 -id:A293184 - cp's OEIS frontend

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

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

Amiram Eldar, Sep 14 2019

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

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

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

Amiram Eldar, Nov 14 2021

nonn

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, ...

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Amiram Eldar, Nov 14 2021

nonn

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, ...

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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

Amiram Eldar, Feb 09 2020

nonn

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

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

Amiram Eldar, Table of n, a(n) for n = 1..37

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

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

Amiram Eldar, Nov 14 2021

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A254926.

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

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] &]

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

A294030 Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

1, 9, 14, 42, 161, 161, 798, 1400, 86156, 123656, 419430, 387868, 508797, 772121, 870233, 4162866, 8754569, 126168912, 126991491, 128007618, 131491736

Offset: 1

Amiram Eldar, Oct 22 2017

The bi-unitary totient function of numbers k such that k and k+1 have the same function value (A293184).

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

The bi-unitary version of A003275.

Cf. A116550, A293184.

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, b1]]; b1 = b2, {k, 1, 10^2}]; a (* after Jean-François Alcover at A116550 *)

a(n) = A116550(A293184(n)).

a(10)-a(11) from Michel Marcus, Nov 14 2017

a(12)-a(21) from Amiram Eldar, Jul 16 2022

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

Amiram Eldar, Feb 15 2020

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..152

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

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

Amiram Eldar, Jul 08 2025

nonn

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?

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

Amiram Eldar, Table of n, a(n) for n = 1..4006 (terms below 10^11)

Cf. A384247.

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

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] &]

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);}

cp's OEIS Frontend

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

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

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

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

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

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A294030 Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function (A116550).

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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.

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

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