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Indices of icosahedral numbers (A006564) which are semiprimes.
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%I #27 Feb 16 2025 08:32:56

%S 37,61,157,193,229,313,373,397,409,433,457,601,613,673,877,997,1009,

%T 1321,1429,1453,1489,1549,1657,1741,1777,1861,2017,2293,2377,2557,

%U 2677,2689,2713,2749,2797,2857,2917,2953,3109,3169,3181,3361,3433,3517,4021

%N Indices of icosahedral numbers (A006564) which are semiprimes.

%D J. H. Conway and R. K. Guy, The Book of Numbers, New York, Springer-Verlag, p. 50, 1996.

%H Amiram Eldar, <a href="/A103946/b103946.txt">Table of n, a(n) for n = 1..10000</a>

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2003), 65-75.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>.

%F Numbers k such that A006564(k) is a term of A001358.

%F Numbers k such that A102294(k) = 2.

%F Numbers k such that A001222(A006564(k)) = 2.

%F Numbers k such that Bigomega(k*(5*k^2 - 5*k + 2)/2) = 2.

%e a(69) = 7333 because the 69th icosahedral number to be a semiprime is A006564(7333) = 7333 * (5*73332 - 5*7333 + 2)/2 = 985657062703 = 7333 * 134413891, which is a term of A001358, a semiprime because both 7333 and 134413891 are primes.

%t Select[ Prime[ Range[ 557]], PrimeQ[(5#^2 - 5# + 2)/2] &] (* _Robert G. Wilson v_, Feb 21 2005 *)

%o (PARI) isok(n) = bigomega(n*(5*n^2 -5*n + 2)/2) == 2; \\ _Michel Marcus_, Dec 14 2015

%Y Cf. A001222, A001358, A006564, A099186, A102294.

%K nonn,easy,changed

%O 1,1

%A _Jonathan Vos Post_, Feb 20 2005

%E Edited and extended by _Robert G. Wilson v_, Feb 21 2005