A001838 -id:A001838 - cp's OEIS frontend

A320391 Numbers k such that phi(k - 2) = phi(k) - 2.

5, 7, 8, 13, 14, 16, 19, 20, 22, 31, 43, 46, 61, 64, 73, 94, 103, 109, 118, 139, 151, 166, 181, 193, 199, 214, 229, 241, 256, 271, 283, 313, 334, 349, 358, 421, 433, 454, 463, 523, 526, 571, 601, 619, 643, 661, 694, 718, 766, 811, 823, 829, 859, 883, 934, 958

Offset: 1

Vincenzo Librandi, Oct 13 2018

nonn

			7 is in the sequence because phi(5) = 4 = phi(7) - 2.
8 is in the sequence because phi(6) = 2 = phi(8) - 2.
9 is not in the sequence because phi(7) = 6 but phi(9) - 2 = 4 instead.

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

Cf. A001838. Contains A006512 and terms > 10 in A194593.

Filtered([1..960],k->Phi(k-2)=Phi(k)-2); # Muniru A Asiru, Oct 28 2018

[n: n in [3..1000] | EulerPhi(n-2) eq EulerPhi(n)-2];

with(numtheory): select(k->phi(k-2)=phi(k)-2,[$1..960]); # Muniru A Asiru, Oct 28 2018

Select[Range@1000, EulerPhi@(# - 2) == EulerPhi[#] - 2 &]
Flatten[Position[Partition[EulerPhi[Range[1000]],3,1],?(#[[1]]==#[[3]]-2&),1,Heads->False]]+2 (* _Harvey P. Dale, Oct 24 2020 *)

isok(n) = eulerphi(n-2) == eulerphi(n)-2; \\ Michel Marcus, Oct 14 2018

a(n) = A001838(n)+2. - Robert Israel, Oct 30 2018

A056777 Composite numbers k such that both phi(k+12) = phi(k) + 12 and sigma(k+12) = sigma(k) + 12.

Original entry on oeis.org

65, 209, 11009, 38009, 680609, 2205209, 3515609, 4347209, 10595009, 12006209, 31979009, 89019209, 169130009, 244766009, 247590209, 258084209, 325622009, 357777209, 377330609, 441630209, 496175609, 640343009, 1006475609

Offset: 1

Labos Elemer, Aug 17 2000

nonn

It is easy to show that if p, p+2, p+6 and p+8 are all prime (a prime quadruple as defined in A007530, which lists the values of p) with x=p(p+8), x+12=(p+2)(p+6), then x is in the sequence. I conjecture that all members of the sequence are of this form. - Jud McCranie, Oct 11 2000

Numbers so far are all congruent to 65 (mod 72). - Ralf Stephan, Jul 07 2003

			k = 209 = 11*19, k + 12 = 221 = 13*17, phi(k + 12) = 192 = 180 + 12 = phi(k) + 12, also sigma(221) = 252 = sigma(209) + 12 = 240 + 12.
phi(65) + 12 = 60 = phi(65 + 12), sigma(65) + 12 = 96 = sigma(65 + 12), 65 is composite.

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

isok(n) = !isprime(n) && (sigma(n+12) == sigma(n)+12) && (eulerphi(n+12)==eulerphi(n)+12); \\ Michel Marcus, Jul 14 2017

More terms from Jud McCranie, Oct 11 2000

cp's OEIS Frontend

A320391 Numbers k such that phi(k - 2) = phi(k) - 2.

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