A001274 -id:A001274 - cp's OEIS frontend

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

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 *)

A333741 Odd numbers k such that phi(k) = phi(k+2), where phi is the Euler totient function (A000010).

Original entry on oeis.org

7, 635, 1015, 2695, 6497, 10307, 12317, 13445, 46205, 77693, 81303, 133787, 134995, 151823, 162925, 180633, 181427, 220113, 288925, 359905, 392819, 404471, 439097, 453167, 485237, 682649, 739023, 840851, 879303, 910195, 988713, 1392317, 1410119, 1434895, 1503347

Offset: 1

Amiram Eldar, Apr 03 2020

nonn

The odd terms of A001494. These terms are relatively rare: of the first 10000 terms of A001494 only 63 are odd.

			7 is a term since phi(7) = phi(9) = 6.

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

Cf. A000010, A001274.

Select[Range[1, 10^6, 2], EulerPhi[#] == EulerPhi[# + 2] &]
2#-1&/@SequencePosition[EulerPhi[Range[1,151*10^4,2]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 15 2020 *)

A333874 Numbers k such that A173557(k) = A173557(k+1).

Original entry on oeis.org

1, 168, 194, 350, 1368, 1628, 3705, 5186, 5328, 6929, 7475, 25545, 26047, 26864, 28251, 34936, 37248, 56724, 65675, 81732, 82368, 87308, 87367, 88450, 91539, 132308, 164691, 166624, 244215, 265524, 280818, 281897, 388245, 465651, 501024, 577524, 806895, 859901

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

Kim et al. (2019) conjectured that A173557(k) = A173557(k+1) is divisible by 12 for all the terms k > 1.

			1 is a term since A173557(1) = A173557(2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..712 (terms below 2*10^10)
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Certain combinatoric convolution sums arising from Bernoulli and Euler Polynomials, Miskolc Mathematical Notes, No. 20, Vol. 1 (2019): pp. 311-330.

Cf. A001274, A173557, A333875.

f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); s = {}; u1 = 1; Do[u2 = u[n]; If[u1 == u2, AppendTo[s, n-1]]; u1 = u2, {n, 2, 10^5}]; s

A333875 Numbers k such that both k and k+1 are squarefree and phi(k) = phi(k+1), where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 194, 3705, 5186, 25545, 388245, 1659585, 2200694, 2521694, 2619705, 3289934, 3794834, 4002405, 5781434, 6245546, 6372794, 8338394, 12352934, 14144954, 16475414, 22632285, 23553705, 37762394, 40588485, 43834754, 44485454, 59603954, 63298785, 76466985, 81591194

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

Numbers k such that A000010(k) = A000010(k+1) = A173557(k) = A173557(k+1).

			1 is a term since 1 and 2 are both squarefree and phi(1) = phi(2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..1166 (terms below 10^13, calculated from the b-file at A001274)
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Certain combinatoric convolution sums arising from Bernoulli and Euler Polynomials, Miskolc Mathematical Notes, No. 20, Vol. 1 (2019): pp. 311-330.

Subsequence of A001274.

Cf. A000010, A173557, A333874.

s = {}; p1 = 1; Do[p2 = If[SquareFreeQ[n], EulerPhi[n], 0]; If[p2 > 0 && p2 == p1, AppendTo[s, n-1]]; p1 = p2, {n, 2, 10^5}]; s

for(k=1,10^7, if(issquarefree(k), if(issquarefree(k+1), if(eulerphi(k)==eulerphi(k+1),print1(k,", "))))) \\ Hugo Pfoertner, Apr 08 2020

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

A003276 Numbers k such that the multiplicative group of residues prime to k, M_k, is isomorphic to M_{k+1}.

Original entry on oeis.org

1, 3, 15, 104, 495, 975, 22935, 32864, 57584, 131144, 491535, 2539004, 3988424, 6235215, 7378371, 13258575, 17949434, 25637744, 26879684, 29357475, 32235735, 41246864, 48615735, 184611375, 229944855, 257278724, 290849624, 429461864, 550666515, 671054835, 706075095

Offset: 1

N. J. A. Sloane

K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [Cf. Math. Comp., Vol. 27, p. 447, 1973.]
D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 225.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots.
K. Miller, The equation phi(n) = phi(n+1), Unpublished M.S., ND.
K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000, Mathematics of Computation 27 (1973), 47-48. (Annotated scanned copy)

Subsequence of A001274.

{my(z=znstar(1));for(n=1,10^10,my(z1=znstar(n+1)); if(z[1]==z1[1]&&z[2]==z1[2],print1(n,", "));z=z1;); } \\ Joerg Arndt, Mar 17 2016

list(lim)=my(v=List(),old=[1,[]]); forfactored(n=2,lim\1+1, my(cur=znstar(n)[1..2]); if(old==cur, listput(v,n[1]-1)); old=cur); Vec(v) \\ Charles R Greathouse IV, Jul 17 2022

More terms from David W. Wilson

A063739 Squarefree numbers k such that phi(k) = phi(k+1).

Original entry on oeis.org

1, 3, 15, 194, 255, 2834, 3255, 3705, 5186, 5187, 11715, 22935, 25545, 49215, 49335, 65535, 214334, 256274, 388245, 525986, 568815, 589407, 840255, 936494, 1259642, 1574727, 1659585, 1759874, 1788254, 2123583, 2200694, 2521694, 2619705, 3240614, 3289934

Offset: 1

Jason Earls, Aug 13 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3547 (terms below 10^13 calculated from the b-file at A001274; terms 1..75 from Harry J. Smith, terms 76..1000 from Donovan Johnson)

Intersection of A005117 and A001274.

Select[Range[3000000],SquareFreeQ[#]&&EulerPhi[#]==EulerPhi[#+1]&] (* Harvey P. Dale, May 15 2013 *)

for(n=1,10^7, if(issquarefree(n), if(eulerphi(n)==eulerphi(n+1),print(n))))

{ n=0; for (m=1, 10^9, if (eulerphi(m)==eulerphi(m + 1) && issquarefree(m), write("b063739.txt", n++, " ", m); if (n==75, break)) ) } \\ Harry J. Smith, Aug 29 2009

A217773 Numbers n such that tau(n) = tau(n+1) and phi(n) = phi(n+1).

Original entry on oeis.org

104, 3255, 22935, 983775, 1025504, 2200694, 2619705, 4163355, 4447064, 4695704, 6372794, 9718904, 11903775, 23992215, 26879684, 29357475, 37239735, 40588485, 41207144, 48615735, 56424555, 76466985, 81591194, 83864055, 113664135, 118018094, 166758015

Offset: 1

Jayanta Basu, Mar 24 2013

nonn

Intersection of A001274 and A005237. [Alex Ratushnyak, Mar 27 2013]

Donovan Johnson, Table of n, a(n) for n = 1..402 (terms < 10^12)

Cf. A001274, A005237.

Select[Range[100000000],DivisorSigma[0, #] == DivisorSigma[0, # + 1] && EulerPhi[#] == EulerPhi[# + 1] &]
Position[Partition[Table[{DivisorSigma[0,n],EulerPhi[n]},{n,1668*10^5}], 2,1], ?(#[[1]]==#[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Nov 20 2019 *)

A241003 Numbers k such that anti-phi(k) = anti-phi(k+1).

Original entry on oeis.org

2, 8, 14, 20, 27, 32, 284, 297, 362, 717, 842, 1322, 1377, 1725, 1802, 1917, 1982, 2222, 2637, 3410, 4094, 4149, 4850, 5288, 5642, 5654, 5660, 5690, 5750, 5937, 5949, 6237, 7017, 7245, 7377, 7490, 8097, 8217, 8277, 8462, 8774, 9117, 9542, 9897, 10034, 11409, 11810

Offset: 1

Paolo P. Lava, Aug 07 2014

nonn

Like A001274 but using anti-phi, as defined in A066452, instead of phi, per A000010.

			anti-phi(2) = anti-phi(3) = 1.
anti-phi(8) = anti-phi(9) = 4.
anti-phi(14) = anti-phi(15) = 7. Etc.

Amiram Eldar, Table of n, a(n) for n = 1..512

Cf. A000010, A001274, A066418, A066452.

for n from 1 do
    if A066452(n) = A066452(n+1) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Aug 07 2014

A247174 Numbers k such that phi(k) = phi(k+1) and simultaneously Product_{d|k} phi(d) = Product_{d|(k+1)} phi(d) where phi(x) = Euler totient function (A000010).

Original entry on oeis.org

1, 3, 15, 255, 65535, 2200694, 2619705, 6372794, 40588485, 76466985, 81591194, 118018094, 206569605, 470542485, 525644385, 726638834, 791937614, 971122514, 991172805

Offset: 1

Jaroslav Krizek, Nov 22 2014

nonn

Numbers n such that A000010(n) = A000010(n+1) and simultaneously A029940(n) = A029940(n+1).
4294967295 is also a term of this sequence.
Intersection of A001274 and A248795.

			15 is in the sequence because phi(15) = phi(16) = 8 and simultaneously Product_{d|15} phi(d) = Product_{d|(15+1)} phi(d) = 64.

Amiram Eldar, Table of n, a(n) for n = 1..213 (terms below 10^13, calculated using the b-file at A001274)

Cf. A000010, A001274, A029940, A248795.

[n: n in [1..100000] |  (&*[EulerPhi(d): d in Divisors(n)]) eq (&*[EulerPhi(d): d in Divisors(n+1)]) and EulerPhi(n) eq EulerPhi(n+1)]

[n: n in [A248795(n)] | EulerPhi(n) eq EulerPhi(n+1)]

a247174[n_Integer] := Module[{a001274, a248795},
  a001274[m_] := Select[Range[m], EulerPhi[#] == EulerPhi[# + 1] &];
  a248795[m_] :=
   Select[Range[m],
    Product[EulerPhi[i], {i, Divisors[#]}] ==
      Product[EulerPhi[j], {j, Divisors[# + 1]}] &];
Intersection[a001274[n], a248795[n]]] (* Michael De Vlieger, Dec 01 2014 *)

a(6)-a(19) using A248795 by Jaroslav Krizek, Nov 25 2014

cp's OEIS Frontend

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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A333741 Odd numbers k such that phi(k) = phi(k+2), where phi is the Euler totient function (A000010).

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A333874 Numbers k such that A173557(k) = A173557(k+1).

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A333875 Numbers k such that both k and k+1 are squarefree and phi(k) = phi(k+1), where phi is the Euler totient function (A000010).

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

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A003276 Numbers k such that the multiplicative group of residues prime to k, M_k, is isomorphic to M_{k+1}.

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A063739 Squarefree numbers k such that phi(k) = phi(k+1).

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A217773 Numbers n such that tau(n) = tau(n+1) and phi(n) = phi(n+1).

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