A245202 Numbers k such that tau(k) + phi(k) is a perfect square.
3, 9, 21, 24, 26, 30, 51, 72, 77, 84, 90, 93, 100, 119, 122, 162, 168, 174, 194, 210, 213, 221, 276, 282, 291, 301, 381, 384, 386, 408, 414, 437, 469, 510, 527, 533, 564, 594, 597, 616, 723, 731, 744, 770, 791, 794, 858, 869, 896, 917, 930, 948, 952, 954
Offset: 1
Keywords
Examples
3 is in the sequence because phi(3) + tau(3) = 2 + 2 = 4^2. 9 is in the sequence because phi(9) + tau(9) = 6 + 3 = 3^2. 15 is not in the sequence because phi(15) + tau(15) = 8 + 4 = 12 = 2^2 * 3, which is not a perfect square.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Chai Wah Wu)
Programs
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Mathematica
Select[Range[1000], IntegerQ[Sqrt[DivisorSigma[0, #] + EulerPhi[#]]] &] (* Amiram Eldar, Apr 27 2024 *)
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PARI
isok(n) = issquare(numdiv(n) + eulerphi(n)); \\ Michel Marcus, Jul 23 2014
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Python
from sympy import totient, divisor_count from gmpy2 import is_square [n for n in range(1,10**4) if is_square(int(divisor_count(n)+totient(n)))] # Chai Wah Wu, Aug 04 2014
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