A053211 Cototients of consecutive pure powers of primes.

2, 4, 3, 8, 5, 9, 16, 7, 32, 27, 11, 25, 64, 13, 81, 128, 17, 49, 19, 256, 23, 125, 243, 29, 31, 512, 121, 37, 41, 43, 1024, 729, 169, 47, 343, 53, 625, 59, 61, 2048, 67, 289, 71, 73, 79, 2187, 361, 83, 89, 4096, 97, 101, 103, 107, 109, 529, 113, 1331, 3125, 127

Offset: 1

Author

Labos Elemer, Mar 03 2000

Keywords

• nonn

Comments

Cototients of prime powers do not remain always prime powers, but are primes if their exponent is 2.

Examples

			The 10th pure power of prime (but not a prime) is 81, so a(10) = 81 - EulerPhi(81) = 81 - 54 = 27. For n=p^2, a(n)=p.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^20, showing even a(n) in blue, 3 | a(n) in green, and prime a(n) in red, else black.

Crossrefs

Cf. A000010, A051953, A001248, A002618, A036689, A053650, A053191, A053192, A246547.

Programs

Mathematica

Map[# - EulerPhi@ # &, Select[Range[16200], And[! PrimeQ@ #, PrimePowerQ@ #] &]] (* Michael De Vlieger, Jun 11 2018 *)
With[{nn = 2^14}, Map[Times @@ Map[#1^(#2 - 1) & @@ FactorInteger[#][[1]]] &, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] ] (* Michael De Vlieger, Mar 11 2023 *)

Formula

a(n) = A051953(A025475(n+1)) = cototient(p^k) = p^(k-1).