A276305 - OEIS

%I #48 Sep 08 2022 08:46:17

%S 31,37,73,103,137,139,181,193,211,269,373,433,463,541,563,571,587,733,

%T 751,859,887,929,1021,1129,1151,1381,1399,1489,1637,1723,1993,2053,

%U 2083,2087,2237,2521,2621,2731,2837,2843,2909,3109,3137,3209,3271,3313,3323,3343,3541,4091

%N Primes p such that d(p*(2p+1)) = 12 where d(n) is the number of divisors of n (A000005).

%C Conjecture: this sequence is infinite.

%C Each number p * (2p + 1) is of the form p * q * r^2 but not of the form p * q^5. - _David A. Corneth_, Aug 30 2016

%H Amiram Eldar, <a href="/A276305/b276305.txt">Table of n, a(n) for n = 1..10000</a>

%e Consider 31. Then 31*((2*31)+1) = 2*(31^2) + 31 = 1953 = 3*3*7*31 and d(1953) = 12.

%t Select[Prime@ Range@ 576, DivisorSigma[0, # (2 # + 1)] == 12 &] (* _Michael De Vlieger_, Aug 30 2016 *)

%o (Scheme, with Antti Karttunen's IntSeq-library)

%o (define A276305 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (= 12 (A000005 (* n (+ n n 1))))))))

%o ;; _Antti Karttunen_, Aug 29 2016

%o (PARI) is(n) = ispseudoprime(n) && numdiv(n*(2*n+1))==12 \\ _Felix Fröhlich_, Aug 29 2016

%o (PARI) is(n)=numdiv(2*n+1)==6 && isprime(n) \\ _Charles R Greathouse IV_, Aug 29 2016

%o (Magma) [n: n in [0..5000] | NumberOfDivisors(2*n+1) eq 6 and IsPrime(n)]; // _Vincenzo Librandi_, Aug 30 2016

%Y Cf. A000005, A005384, A030630, A054753, A276045, A276307.

%K nonn

%O 1,1

%A _Anthony Hernandez_, Aug 29 2016

%E More terms from _Antti Karttunen_, Aug 29 2016