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

1,1

Comments

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

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.

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A297075 A306458 A287637 * A131390 A131395 A352809

Adjacent sequences: A053208 A053209 A053210 * A053212 A053213 A053214

Keyword

nonn

Author

Labos Elemer, Mar 03 2000

Status

approved