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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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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
EXAMPLE
20 is in the sequence since iphi(20) = iphi(21) = 12.
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
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
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, ...
MATHEMATICA
f[p_, e_] := EulerPhi[e]; ephi[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[5000], ephi[#] == ephi[# + 1] > 1 &]
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
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, ...
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 &]
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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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
COMMENTS
The corresponding common values A103224(k) = A103224(k+1) are 8, 8, 288, 640, 7200, 139392, 3744000, 5760000, ...
EXAMPLE
4 is a term since norm(phi(4)) = norm(phi(5)) = 8.
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 *)
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
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] &]
PROG
(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
Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function ( A116550).
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1, 9, 14, 42, 161, 161, 798, 1400, 86156, 123656, 419430, 387868, 508797, 772121, 870233, 4162866, 8754569, 126168912, 126991491, 128007618, 131491736
COMMENTS
The bi-unitary totient function of numbers k such that k and k+1 have the same function value ( A293184).
EXAMPLE
9 is in the sequence since 9 = bphi(14) = bphi(15).
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, b1]]; b1 = b2, {k, 1, 10^2}]; a (* after Jean-François Alcover at A116550 *)
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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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
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
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