A001274 -id:A001274 - cp's OEIS frontend

A256510 Primes p such that phi(p-2) = phi(p-1).

3, 5, 17, 257, 977, 3257, 5189, 11717, 13367, 22937, 65537, 307397, 491537, 589409, 983777, 1659587, 2822717, 3137357, 5577827, 6475457, 7378373, 8698097, 10798727, 32235737, 37797437, 39220127, 39285437, 51555137, 52077197, 56992553, 63767927, 70075997

Offset: 1

Jaroslav Krizek, Mar 31 2015

nonn

First 5 Fermat primes from A019434 are terms of this sequence.
a(2) = 5 is only term of a(n) such that a(n) - 2 is a prime q, i.e., prime 3 is only prime q such that phi(q) = phi(q+1).
If there are any other Fermat primes, they will not be in the sequence. - Robert Israel, Mar 31 2015

			Prime 17 is in the sequence because phi(15) = phi(16) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..819 (terms below 10^13, calculated from the b-file at A001274)

Cf. A000010, A000215, A001274, A019434.

[n: n in [3..10^7] |  IsPrime(n) and EulerPhi(n-2) eq EulerPhi(n-1)];

with(numtheory): A256510:=n->`if`(isprime(n) and phi(n-2) = phi(n-1), n, NULL): seq(A256510(n), n=1..10^5); # Wesley Ivan Hurt, Mar 31 2015

Select[Prime@ Range@ 100000, EulerPhi[# - 2] == EulerPhi[# - 1] &] (* Michael De Vlieger, Mar 31 2015 *)

A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

Original entry on oeis.org

1, 5, 119, 629, 17907119

Offset: 1

Felix Fröhlich, May 11 2015

From Manfred Scheucher, May 27 2015: (Start)
a(6)>=3*10^8 (calculation)
a(7)>=3.5*10^13, a(8)>=4.5*10^25, a(9)>=3.0*10^47, and so on... (doubly exponential lower bound, see uploaded pdf)
239719159679 and 239742643139 admit a ratio of 5.998... and 6.008..., resp.
There might be a relation to the sequence A098026. (End)

			a(3) = 119, because phi(119) == 3*phi(120) = 96 and 119 is the smallest k where this equality holds for n = 3.

Manfred Scheucher, A lower bound on A257865(n)

Cf. A001274, A171262, A256907, A256937, A257550.

Table[k = 1; While[EulerPhi[k] != n EulerPhi[k + 1], k++]; k, {n, 4}] (* Michael De Vlieger, May 12 2015 *)

a(n) = my(k=1); while(eulerphi(k)!=n*eulerphi(k+1), k++); k

a(n) >= exp(exp(c(n-3))) with c=exp(gamma) and gamma being the Euler-Mascheroni_constant (see pdf). - Manfred Scheucher, May 27 2015

A259397 Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have phi(a + b) = phi(n), where phi(n) is the Euler totient function of n.

Original entry on oeis.org

6, 12, 14, 28, 30, 48, 62, 124, 126, 222, 224, 254, 448, 476, 496, 510, 768, 876, 1022, 1792, 1806, 2032, 2034, 2046, 2625, 2850, 2898, 3204, 3246, 3560, 3705, 3850, 4064, 4094, 7722, 7744, 7920, 7980, 7992, 8060, 8094, 8136, 8148, 8150, 8164, 8190, 11880, 13365

Offset: 1

Paolo P. Lava, Jun 26 2015

It appears that a or b is equal to 1. In particular, if b=1 we have 2625, 3705, 13365, 25545, 57645, ... that are a subset of A001274.

			6 in base 2 is 110. If we take 110 = concat(1,10) then 1 and 10 converted to base 10 are 1 and 2. Finally phi(1 + 2) = 2 = phi(6).
12 in base 2 is 1100. If we take 1100 = concat(1,100) then 1 and 100 converted to base 10 are 1 and 4. Finally phi(1 + 4) = 4 = phi(12);
2625 in base 2 is 101001000001. If we take 101001000001 = concat(10100100000,1) then 10100100000 and 1 converted to base 10 are 1312 and 1. Finally phi(1312 + 1) = 1200 = phi(2625); etc.

Paolo P. Lava, Table of n, a(n) for n = 1..150

Cf. A000010, A001274, A258813, A258843, A258844.

with(numtheory): P:=proc(q) local a,b,c,k,n;
for n from 1 to q do c:=convert(n,binary,decimal);
for k from 1 to ilog10(c) do
a:=convert(trunc(c/10^k),decimal,binary);
b:=convert((c mod 10^k),decimal,binary);
if a*b>0 then if phi(a+b)=phi(n) then print(n); break;
fi; fi; od; od; end: P(10^8);

A280450 Smallest k > 2n+1 such that phi(k) = phi(2n+1).

Original entry on oeis.org

4, 8, 9, 14, 22, 21, 16, 32, 27, 26, 46, 33, 38, 58, 62, 44, 39, 57, 45, 55, 49, 52, 94, 86, 64, 106, 75, 63, 118, 77, 74, 104, 134, 92, 142, 91, 82, 93, 158, 162, 166, 128, 116, 115, 95, 99, 111, 119, 122, 125, 206, 112, 214, 133, 117, 145, 178, 135, 153, 242

Offset: 1

Thomas Ordowski, Jan 03 2017

nonn

All terms are composite.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A001274, A005384, A140141.

f:= n -> min(select(`>`,numtheory:-invphi(numtheory:-phi(2*n+1)),2*n+1)):
map(f, [$1..100]); # Robert Israel, Jan 03 2017

Table[k = 2 n + 2; While[EulerPhi@ k != #, k++] &@ EulerPhi[2 n + 1]; k, {n, 120}] (* Michael De Vlieger, Jan 03 2017 *)

a(n) = my(k=2*n+2); while(eulerphi(k)!=eulerphi(2*n+1), k++); k \\ Felix Fröhlich, Jan 05 2017

2n+1 < a(n) < 4n+3.
From Robert Israel, Jan 03 2017: (Start)
a(n)=2n+2 if and only if 2n+1 is in A001274.
If n > 3 is in A005384, then a(n)=4n+2. (End)

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

Amiram Eldar, Jul 26 2017

nonn

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

The unitary version of A003275.

Amiram Eldar, Table of n, a(n) for n = 1..2198

Cf. A001274, A003275, A047994, A287055.

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

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

A291177 Numbers k such that s(k) = s(k+1) but phi(k) != phi(k+1), where s(k) = phi(k) + phi(phi(k)) + ... + 1 is the sum of iterated phi (A092693).

Original entry on oeis.org

45, 297, 356, 375, 1335, 1935, 3915, 4743, 5271, 6015, 6375, 6903, 20894, 22311, 25347, 28118, 31664, 32384, 39632, 49155, 50954, 55935, 59984, 64514, 70275, 119324, 125054, 162944, 209715, 334304, 342975, 472718, 767584, 798567, 862802, 908775, 1280096

Offset: 1

Amiram Eldar, Aug 19 2017

nonn

The restriction phi(k) != phi(k+1) is intended to exclude all the (trivial) terms of A001274.

Amiram Eldar, Table of n, a(n) for n = 1..200

Cf. A001274, A092693.

s[n_]:=Plus @@ FixedPointList[EulerPhi, n] - (n + 1);seqQ[n_]:=(s[n+1]==s[n])&&(EulerPhi[n+1]!=EulerPhi[n]);Select[Range[10^5],seqQ]

A305235 Smallest positive number k such that there are exactly n successive equal values of A001221 starting at k, i.e., such that A305234(k) = n.

Original entry on oeis.org

1, 4, 3, 2, 54, 91, 142, 141, 44360, 48919, 218972, 526097, 526096, 526095, 17233173, 127890362, 29138958036, 118968284929, 118968284928, 585927201065, 585927201064, 585927201063, 585927201062, 313978488186061, 453918847597185, 453918847597184, 455626105596320

Offset: 0

Felix Fröhlich, May 28 2018

nonn

a(27) > 2 * 10^15. - Toshitaka Suzuki, Jun 22 2025

			For n = 5: A001221(91+k) = 2 for k = 0..5 and 91 is the smallest number x with exactly 5 successors that have the same value of A001221 as x, so a(5) = 91.

Cf. A001221, A001274, A045983, A048932, A048971, A077657, A305234.

a305234(n) = my(k=n+1, i=0); while(omega(k)==omega(n), i++; k++); i
a(n) = my(k=1); while(1, if(a305234(k)==n, return(k)); k++)

a(16)-a(22) from Toshitaka Suzuki, Apr 01 2025

a(23)-a(26) from Toshitaka Suzuki, Jun 22 2025

A308378 Numbers k such that phi(2k+1) = phi(2k+2).

Original entry on oeis.org

0, 1, 7, 127, 247, 487, 1312, 1627, 1852, 2593, 5857, 6682, 9157, 11467, 12772, 23107, 24607, 24667, 28822, 32767, 82087, 92317, 99157, 107887, 143497, 153697, 159637, 194122, 198742, 207637, 245767, 284407, 294703, 343492, 420127

Offset: 1

Torlach Rush, May 24 2019

nonn

For n > 0, 2*a(n) + 1 is a term of A020884. This is because 2*a(n) + 1 is odd and every odd number is the difference of the squares of two consecutive numbers and hence are coprime.
For n > 0, (2*a(n) + 1) * (2*a(n) + 2) is a term of A024364. This is because (2*a(n) + 1) * (2*a(n) + 2) = 2*((a(n) + 1)^2 + (a(n) + 1) * a(n)) and gcd((a(n) + 1), a(n)) = 1.
For n > 0, a(n) is congruent to 1 or 4 mod 6.
2*a(n) + 1 is congruent to 1 or 3 mod 6 and is a term of A047241.
2*a(n) + 2 is congruent to 2 or 4 mod 6 and is a term of A047235.

			0 is a term because phi(1) = phi(2) = 1.
1 is a term because phi(3) = phi(4) = 2.
7 is a term because phi(15) = phi(16) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..5416 (calculated using the b-file at A001274)

Cf. A000010, A004767, A020884, A024364, A047235, A047241, A299535.
Subset of A047234.
Subset of A001274.

Select[Range[0, 9999], EulerPhi[2# + 1] == EulerPhi[2# + 2] &] (* Alonso del Arte, Jul 05 2019 *)
Select[(#-1)/2&/@SequencePosition[EulerPhi[Range[900000]],{x_,x_}][[All,1]],IntegerQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2019 *)

lista(nn) = for(n=0, nn, if(eulerphi(2*n+1) == eulerphi(2*n+2), print1(n, ", ")));
lista(430000)

a(n) = (A299535(n) - 2) / 2.

A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

3, 5, 8720288051472, 9134280520365, 41544070492925, 42466684755492, 51363581614342, 68616494581632, 113312918293575, 210911076210835, 215517565688425, 294988451482725, 383617980270525, 432759876053505, 442863123838135, 532068058516992, 892813363927485, 923102743748185, 929531173876305

Offset: 1

Michel Marcus and Giovanni Resta, Feb 29 2020

nonn

10^15 < a(20) <= 1089641067389872.
Also terms: 1248817919303952, 1332436545865422, 1394926716616125, 1868522795664525, 1950445682260072.
a(4) and a(9) appear in Kevin Ford's paper.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.
Mathematics StackExchange, Conjecture on the gap between integers having the same number of co-primes, Sep 25 2019.

Cf. A000010, A007015.
Cf. A001274, A001494, A179186, A179187, A179188, A179189, A179202, A330429.
Cf. A276503, A276504, A217139.

Select[Range[100000], EulerPhi[#] == EulerPhi[# + 3] &] (* Alonso del Arte, Mar 01 2020 *)

isok(k) = eulerphi(k) == eulerphi(k+3); \\ Michel Marcus, Feb 29 2020

A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

9, 15, 1005079920836, 13695542245376, 26160864154416, 27402841561095, 27599063056565, 110263115897935, 124632211478775, 127400054266476, 154090744843026, 205849483744896, 231019991767556, 339938754880725, 459718637643265, 632733228632505, 646552697065275, 683008674773416, 884965354448175

Offset: 1

Giovanni Resta, Mar 01 2020

nonn

a(20) > 10^15.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.

Cf. A000010, A007015.
Cf. A001274, A001494, A330251, A179186, A179187, A179188, A179189, A179202.
Cf. A276503, A276504, A217139.

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A256510 Primes p such that phi(p-2) = phi(p-1).

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A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

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A259397 Numbers n with the property that it is possible to write the base 2 expansion of n as concat(a_2,b_2), with a_2>0 and b_2>0 such that, converting a_2 and b_2 to base 10 as a and b, we have phi(a + b) = phi(n), where phi(n) is the Euler totient function of n.

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A280450 Smallest k > 2n+1 such that phi(k) = phi(2n+1).

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

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A291177 Numbers k such that s(k) = s(k+1) but phi(k) != phi(k+1), where s(k) = phi(k) + phi(phi(k)) + ... + 1 is the sum of iterated phi (A092693).

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A305235 Smallest positive number k such that there are exactly n successive equal values of A001221 starting at k, i.e., such that A305234(k) = n.

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A308378 Numbers k such that phi(2k+1) = phi(2k+2).

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