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Squarefree semiprimes k such that k+1 is the product of three distinct primes and k+2 is the product of four distinct primes.
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%I #27 Oct 04 2024 16:35:08

%S 2413,6193,6697,9469,11065,11233,11893,12153,13333,13393,14005,14089,

%T 14233,15293,17113,17533,17833,17869,18613,18653,19693,20053,20557,

%U 20613,20733,20893,20993,21145,22033,22285,22405,22693,22753,22969,23329,23413,24033,24493,26101,26453,27113,27553,28117,28453,28741,29053,29353,29713

%N Squarefree semiprimes k such that k+1 is the product of three distinct primes and k+2 is the product of four distinct primes.

%H Alois P. Heinz, <a href="/A376352/b376352.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) == 1 (mod 4).

%e 2413 is a term because 2413 = 19*127 is the product of two distinct primes, 2414 = 2*17*71 is the product of three distinct primes and 2415 = 3*5*7*23 is the product of four distinct primes.

%e 6193 is a term because 6193 = 11*563 is the product of two distinct primes, 6194 = 2*19*163 is the product of three distinct primes and 6195 = 3*5*7*59 is the product of four distinct primes.

%p q:= n-> andmap(j-> map(i-> i[2], ifactors(n+j-2)[2])=[1$j], [$2..4]):

%p select(q, [$1..30000])[]; # _Alois P. Heinz_, Sep 21 2024

%t Position[Partition[FactorInteger[#][[;; , 2]] & /@ Range[30000], 3, 1], {{1, 1}, {1, 1, 1}, {1, 1, 1, 1}}] // Flatten (* _Amiram Eldar_, Sep 21 2024 *)

%Y Cf. A006881, A007304, A046386, A328137.

%K nonn

%O 1,1

%A _Massimo Kofler_, Sep 21 2024