A056772 -id:A056772 - cp's OEIS frontend

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

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

A056675 Number of non-unitary but squarefree divisors of n!. Also number of unitary but not-squarefree divisors of n!.

Original entry on oeis.org

0, 0, 0, 2, 4, 6, 12, 12, 12, 14, 28, 28, 56, 60, 60, 60, 120, 120, 240, 240, 240, 248, 496, 496, 496, 504, 504, 504, 1008, 1008, 2016, 2016, 2016, 2032, 2032, 2032, 4064, 4080, 4080, 4080, 8160, 8160, 16320, 16320, 16320, 16352, 32704, 32704, 32704, 32704

Offset: 1

Labos Elemer, Aug 10 2000

nonn

			n=10: 10! = 2*2*2*2*2*2*2*2*3*3*3*3*5*5*7 = 256*81*25*7, which has 270 divisors, of which 16 are unitary and 16 are squarefree; overlap={1,7}. The set {2, 3, 5, 6, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210} represents the squarefree non-unitary divisors of 10!, so a(10)=14.

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

Cf. A000005, A000142, A007913, A048656, A055229, A055231, A056674, A056772.

a(n) = my(f=n!); sumdiv(f, d, issquarefree(d) && (gcd(d, f/d) != 1)); \\ Michel Marcus, Sep 05 2017

a(n) = A048656(n) - A000005(A055231(n!)) = A048656(n) - A000005(A007913(n!)/A055229(n!)).
a(n) = A056674(A000142(n)). - Amiram Eldar, Aug 06 2019
a(n) = A048656(n) - A056672(n). - Sean A. Irvine, May 02 2022

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

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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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A056675 Number of non-unitary but squarefree divisors of n!. Also number of unitary but not-squarefree divisors of n!.

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

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