A001838 -id:A001838 - cp's OEIS frontend

A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

3, 7, 12, 13, 18, 19, 24, 28, 36, 37, 40, 43, 66, 67, 79, 88, 97, 103, 109, 124, 127, 163, 184, 193, 223, 229, 232, 277, 307, 313, 328, 349, 379, 397, 424, 439, 457, 463, 487, 499, 508, 613, 643, 664, 673, 712, 739, 757, 769, 823, 853, 859, 877, 883, 904, 907

Offset: 1

Labos Elemer, Aug 17 2000

nonn

In contrast with A015913, composite solutions are not rare. Prime solutions are common.
From Kevin J. Gomez, Mar 02 2016: (Start)
Composite solutions have two known forms:
  n such that n = 4 * (2^p - 1) where 2^p - 1 is a Mersenne prime. (A001348)
  n such that n = 8q where q is a Sophie Germain prime. (A005394)
There are composite solutions (such as 36) that do not fit either of these forms. (End)

			For k = 1048: phi(1048) = 520, phi(1048+4) = 524.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Cf. A000010, A015913 (sigma(k+4) = sigma(k) + 4).

Cf. A001838 (k=2), this sequence (k=4), A262084 (k=6), A262085 (k=8), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+4) eq EulerPhi(n)+4]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 4)== EulerPhi[#] + 4 &] (* Vincenzo Librandi, Sep 11 2015 *)
Position[Partition[EulerPhi[Range[1000]],5,1],?(#[[1]]+4==#[[5]]&),1, Heads-> False]//Flatten (* _Harvey P. Dale, Dec 18 2019 *)

isok(n) = eulerphi(n+4) == eulerphi(n) + 4; \\ Michel Marcus, Sep 11 2015

A050475 Numbers k such that x = 2^k-2 satisfies phi(x)+2 = phi(x+2).

Original entry on oeis.org

3, 4, 6, 8, 14, 18, 20, 32, 62, 90, 108, 128, 522, 608, 1280, 2204, 2282, 3218, 4254, 4424, 9690, 9942, 11214, 19938, 21702, 23210, 44498, 86244, 110504, 132050, 216092, 756840, 859434, 1257788, 1398270, 2976222, 3021378, 6972594, 13466918, 20996012, 24036584, 25964952, 30402458, 32582658

Offset: 1

Jud McCranie, Dec 24 1999

nonn

Other solutions of this equation are in A001838.

Also, numbers k such that 2^(k-1)-1 is prime. Proof: If x = 2^k-2, phi(x)+2 = phi(x+2) <==> phi(2^k-2)+2 = phi(2^k) <==> phi(2(2^(k-1)-1)) + 2 = 2^k(1-1/2) <==> phi(2)*phi(2^(k-1)-1)+2 = 2^(k-1) <==> phi(2^(k-1)-1) = 2^(k-1)-2 if y = 2^(k-1)-1. We have phi(y) = y-1 <==> y=2^(k-1)-1 is prime. Therefore a(n) = A000043(n)+1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 19 2004

			phi(2^18-2)+2 = 131072 = phi(2^18), so 18 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)

Cf. A000043, A001838.

Flatten[Position[EulerPhi[2^# - 2] + 2 == EulerPhi[2^# ] & /@ Range[1, 250], True]] (* Vit Planocka *)

a(n) = A000043(n) + 1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 19 2004

a(39)-a(44) from Ivan Panchenko, Apr 11 2018

A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

Original entry on oeis.org

5, 7, 11, 13, 17, 21, 23, 31, 37, 40, 41, 47, 53, 56, 61, 67, 73, 83, 88, 97, 98, 101, 103, 107, 131, 136, 151, 152, 156, 157, 167, 173, 191, 193, 223, 227, 233, 237, 248, 251, 257, 263, 271, 277, 296, 307, 311, 328, 331, 347, 353, 367, 373, 376, 383, 433, 443

Offset: 1

Kevin J. Gomez, Sep 10 2015

The majority of solutions can be predicted by known properties of the equality. There are several solutions that do not fit these parameters.
An odd natural number m is a solution if m and m + 6 are both prime (sexy primes) (A023201).
Among the solutions for even natural numbers are all m = 8*p with odd primes p such that 4*p+3 is a prime number. Proof: From A000010 we can learn that the formula phi(p*2) = floor(((2 + p - 1) mod p)/(p - 1)) + p - 1 is known. If we define p = 4*q+3 and m = 8*q and insert, we will obtain phi(8*q+6) = 4*q+2. Also it is known that phi(8*q) = 4*q-4 if q is any odd prime. - Thomas Scheuerle, Dec 20 2024

			5 is a term since phi(5+6) = 10 = 6 + 4 = phi(5) + 6.

Peter Kagey, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010, A023201.

Cf. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).

[n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // Vincenzo Librandi, Sep 11 2015

Select[Range@500, EulerPhi@(# + 6)== EulerPhi[#] + 6 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ Anders Hellström, Sep 11 2015

[n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # Tom Edgar, Sep 10 2015

A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

3, 5, 11, 23, 24, 29, 36, 42, 48, 50, 53, 56, 59, 71, 72, 80, 89, 101, 102, 125, 131, 132, 149, 173, 176, 191, 230, 233, 248, 263, 269, 359, 368, 389, 401, 431, 449, 464, 479, 491, 563, 569, 593, 599, 638, 653, 656, 683, 701, 719, 743, 761, 821, 848, 911, 929, 983

Offset: 1

Kevin J. Gomez, Sep 10 2015

Sequence includes numbers n such that n and n + 8 are both prime (A023202).
Sequence also includes numbers n equal to 8*(a Mersenne prime) (cf A000668).
Sequence also includes n such that n/16 and n/8 + 1 are both odd primes.
Contains more composites than sequences A262084 and A262086. This is most likely due to the fact that 8 is a power of 2, as in A001838.

			3 since phi(11) = phi(3) + 8 (3 and 11 are both prime).
24 is a solution since phi(32) = phi(24) + 8 (24 is 8 * 3; 3 is a Mersenne prime).

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+8) eq EulerPhi(n)+8]; // Vincenzo Librandi, Sep 11 2015

select(t -> numtheory:-phi(t+8) = numtheory:-phi(t)+8, [$1..1000]); # Robert Israel, Mar 04 2016

Select[Range@1000, EulerPhi@(# + 8)== EulerPhi[#] + 8 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 8) == eulerphi(n) + 8 \\ Anders Hellström, Sep 11 2015

[n for n in (1..1000) if euler_phi(n+8) == euler_phi(n)+8] # Bruno Berselli, Mar 04 2016

A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

Original entry on oeis.org

3, 7, 13, 19, 31, 36, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967

Offset: 1

Kevin J. Gomez, Sep 10 2015

The only composite term less than 10^11 is 36. - Giovanni Resta, Sep 14 2015

			3 is in the sequence since phi(13) = phi(3) + 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)
Wikipedia, Euler's totient function.

Cf. A000010, A023203.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262085 (k=8), this sequence (k=10).

[n: n in [1..1000] | EulerPhi(n+10) eq EulerPhi(n)+10]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 10) == EulerPhi[#] + 10 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 10) == eulerphi(n) + 10 \\ Anders Hellström, Sep 11 2015

A056774 Composite n such that phi(n+2) = phi(n)+2.

Original entry on oeis.org

6, 12, 14, 18, 20, 44, 62, 92, 116, 164, 212, 254, 332, 356, 452, 524, 692, 716, 764, 932, 956, 1004, 1124, 1172, 1436, 1676, 1724, 1772, 1964, 2036, 2372, 2564, 2612, 2636, 2732, 2876, 2972, 3044, 3236, 3644, 3812, 4052, 4076, 4124, 4196, 4412, 4892

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Below 100000 no common composite solutions with sigma(n+2)=sigma(n)+2, while prime solutions are common.

phi(x)+2=phi(x+2) is equivalent to cototient(x+2)=cototient(x), so also defines closest numbers with identical value of cototients (A051953), either primes or composites.

			n=254, phi(254+2) = phi(256) = 128 = phi(254)+2 = 126+2.

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

Cf. A000010, A001838, A055458, A015913-A015917, A050507, A054799.

Select[Range[5000],CompositeQ[#]&&EulerPhi[#]+2==EulerPhi[#+2]&] (* Harvey P. Dale, Jul 10 2017 *)

isok(n) = !isprime(n) && (eulerphi(n+2) == eulerphi(n)+2); \\ Michel Marcus, Aug 30 2019

A056853 Numbers n satisfying phi(n+1) - phi(n-1) = 2.

Original entry on oeis.org

4, 6, 7, 12, 13, 15, 18, 19, 21, 30, 42, 45, 60, 63, 72, 93, 102, 108, 117, 138, 150, 165, 180, 192, 198, 213, 228, 240, 255, 270, 282, 312, 333, 348, 357, 420, 432, 453, 462, 522, 525, 570, 600, 618, 642, 660, 693, 717, 765, 810, 822, 828, 858, 882, 933, 957

Offset: 1

Joseph L. Pe, Feb 12 2002

nonn

			phi(13+1)-phi(13-1) = 2, so 13 is a term of the sequence.

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

Cf. A000010. Essentially the same as A001838.

Select[Range[10^3], EulerPhi[ # + 1] - EulerPhi[ # - 1] == 2 &]
Flatten[Position[Partition[EulerPhi[Range[1000]],3,1],?(Last[#]-First[#] == 2&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jul 14 2014 *)

A056773 Composite n such that phi(n+4) = phi(n)+4.

Original entry on oeis.org

12, 18, 24, 28, 36, 40, 66, 88, 124, 184, 232, 328, 424, 508, 664, 712, 904, 1048, 1384, 1432, 1528, 1864, 1912, 2008, 2248, 2344, 2586, 2872, 3352, 3448, 3544, 3928, 4072, 4744, 5128, 5224, 5272, 5464, 5752, 5944, 6088, 6472, 7288, 7624, 8104, 8152, 8248

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Are all terms even? - Robert Israel, Apr 28 2020

			24 is in the sequence because 24 is composite and phi(24)+4 = 12 = phi(24+4).

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

A001838, A015913, A055458. Composites in A056772. Primes in A056772 are A023200.

filter:= n -> not isprime(n) and numtheory:-phi(n+4)=numtheory:-phi(n)+4:
select(filter, [$1..10000]); # Robert Israel, Apr 28 2020

Select[Range[9000],CompositeQ[#]&&EulerPhi[#]+4==EulerPhi[#+4]&] (* Harvey P. Dale, Feb 12 2015 *)

is(n)=!isprime(n) && eulerphi(n+4)==eulerphi(n)+4 \\ Charles R Greathouse IV, Apr 28 2020

Edited by Robert Israel, Apr 28 2020

A056775 Numbers k such that phi(k+12) = phi(k) + 12.

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 41, 45, 47, 59, 61, 65, 67, 71, 80, 89, 97, 99, 101, 112, 117, 127, 135, 137, 139, 151, 167, 171, 176, 179, 181, 196, 199, 207, 209, 211, 227, 229, 239, 251, 257, 269, 271, 272, 279, 281, 294, 304, 310, 312, 337, 347, 367, 369, 389

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Prime solutions are in A046133, common with primes in A015917.

			65 is a term since phi(65) = 48, phi(65+12) = phi(77) = 60 = 48 + 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000010, A001838, A015917, A054902, A046133, A046133.

Select[Range[400],EulerPhi[#]+12==EulerPhi[#+12]&] (* Harvey P. Dale, Jan 21 2013 *)

A056776 Composite numbers k such that phi(k+12) = phi(k) + 12.

Original entry on oeis.org

45, 65, 80, 99, 112, 117, 135, 171, 176, 196, 207, 209, 272, 279, 294, 304, 310, 312, 369, 406, 429, 477, 496, 531, 592, 656, 657, 711, 752, 801, 909, 927, 944, 981, 1014, 1072, 1078, 1179, 1251, 1359, 1424, 1557, 1611, 1629, 1712, 1719, 1744, 1786, 1791

Offset: 1

Labos Elemer, Aug 17 2000

nonn

There are common cases with A054902.

			656 is a term since it is composite and phi(656) = 320, phi(656+12) = phi(668) = 332 = 320 + 12.
657 is a term since it is composite and phi(657) = 432, phi(657+12) = phi(669) = 444 = 432 + 12.

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

Cf. A000010, A001838, A015917, A054902, A046133, A046133.

Select[Range[1800], CompositeQ[#] && EulerPhi[# + 12] == EulerPhi[#] + 12 &] (* Amiram Eldar, Mar 01 2020 *)

cp's OEIS Frontend

A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

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A050475 Numbers k such that x = 2^k-2 satisfies phi(x)+2 = phi(x+2).

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A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

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A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

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A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

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A056774 Composite n such that phi(n+2) = phi(n)+2.

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A056853 Numbers n satisfying phi(n+1) - phi(n-1) = 2.

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