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A053211
Cototients of consecutive pure powers of primes.
2
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, 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 *)
KEYWORD
nonn
AUTHOR
Labos Elemer, Mar 03 2000
STATUS
approved