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Numbers n such that n^2 + 3, n^3 + 3, n^4 + 3, n^5 + 3, n^6 + 3 and n^7 + 3 are semiprime.
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%I #30 Jul 30 2022 18:45:15

%S 1,976,5380,16582,17864,22316,27922,34930,44954,50744,61264,72670,

%T 107534,147776,193774,195266,240170,260920,265292,281582,314462,

%U 337832,342014,367060,379784,383930,384704,392050,421226,455734,463790,498134,499306,510194,538384

%N Numbers n such that n^2 + 3, n^3 + 3, n^4 + 3, n^5 + 3, n^6 + 3 and n^7 + 3 are semiprime.

%C All terms in this sequence, except a(1), are even.

%C Subsequence of A253906.

%H K. D. Bajpai, <a href="/A253907/b253907.txt">Table of n, a(n) for n = 1..555</a>

%e a(2) = 976;

%e 976^2 + 3 = 952579 = 43 * 22153;

%e 976^3 + 3 = 929714179 = 1013 * 917783;

%e 976^4 + 3 = 907401035779 = 7 * 129628719397;

%e 976^5 + 3 = 885623410917379 = 2224441 * 398133019;

%e 976^6 + 3 = 864368449055358979 = 97327 * 8881075642477;

%e 976^7 + 3 = 843623606278030360579 = 16403765263 * 51428656333;

%e All six are semiprime.

%t Select[Range[10^5], k = 3; PrimeOmega[(#^2 + k)] == 2 && PrimeOmega[(#^3 + k)] == 2 && PrimeOmega[(#^4 + k)] == 2 && PrimeOmega[(#^5 + k)] == 2 && PrimeOmega[(#^6 + k)] == 2 && PrimeOmega[(#^7 + k)] == 2 &]

%t Select[Range[54*10^4],Union[PrimeOmega[#^Range[2,7]+3]]=={2}&] (* _Harvey P. Dale_, Jul 30 2022 *)

%Y Cf. A001358, A108868, A242331, A253906.

%K nonn

%O 1,2

%A _K. D. Bajpai_, Jan 24 2015