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A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

%I A360944 #46 Mar 30 2023 05:15:17
%S A360944 1,2,7,9,11,14,18,22,29,37,57,58,63,67,74,76,79,108,114,126,134,137,
%T A360944 143,155,158,175,183,191,211,225,231,244,248,274,277,286,308,310,329,
%U A360944 341,350,366,372,379,382,396,417,422,423,450,453,462,554,556,604,623,631,658,682
%N A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).
%C A360944 Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.
%C A360944 Subsequence of triangular numbers is A287472.
%H A360944 Robert Israel, <a href="/A360944/b360944.txt">Table of n, a(n) for n = 1..10000</a>
%e A360944 phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.
%p A360944 filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);
%t A360944 Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* _Amiram Eldar_, Feb 27 2023 *)
%o A360944 (PARI) isok(m) = ispolygonal(eulerphi(m), 3); \\ _Michel Marcus_, Feb 27 2023
%o A360944 (Python)
%o A360944 from itertools import islice, count
%o A360944 from sympy.ntheory.primetest import is_square
%o A360944 from sympy import totient
%o A360944 def A360944_gen(startvalue=1): # generator of terms >= startvalue
%o A360944     return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
%o A360944 A360944_list = list(islice(A360944_gen(),20)) # _Chai Wah Wu_, Feb 28 2023
%Y A360944 Cf. A000010, A000217, A055469, A287472.
%Y A360944 Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).
%K A360944 nonn
%O A360944 1,2
%A A360944 _Bernard Schott_, Feb 26 2023