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A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

%I A101696 #13 Feb 16 2025 08:32:55
%S A101696 2,8,26,66,174,398,878,2174,4862,10494,22014,45054,98302,222718,
%T A101696 480766,1021438,2127358,4355582,8943102,18773502,38696446,79590910,
%U A101696 175142398,368080382,764442110,1586525694,3247470078,6644856318,13489960446
%N A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.
%C A101696 It seems that this sum can never be a prime after a(1) = 2, since the n-th n-almost prime is always even. The number of prime factors (with multiplicity) of a(n) is 1, 3, 2, 3, 3, 2, 2, 2, 4, 5, 4, 4, 3, 3, 5, 4, 3, 4, 7, 4, 2, 5, 5, 2, 3, 7, 4, 3, 4.
%C A101696 This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. a(1)=2 is prime. a(2)=8 is a 3-almost prime. a(3)=26 is a semiprime. a(4)=66 is a 3-almost prime. a(5)= 174 is a 3-almost prime. a(6)=398 is a semiprime. a(7)=878 is a semiprime. a(8)=2174 is a semiprime. a(9)=4862 is a 4-almost prime. a(10)=10494 is a 5-almost prime. a(11)=22014 is a 4-almost prime. a(12)=45054 is a 3-almost prime. a(13)=98302 is a 3-almost prime. a(14)=222718 is a 3-almost prime. a(15)=480766 is a 5-almost prime. a(16)=1021438 is a 4-almost prime. a(17)=2127358 is a 3-almost prime. a(18)=4355582 is a 4-almost prime. a(19)=8943102 is a 7-almost prime. a(20)=18773502 is a 4-almost prime. 21-almost numbers are not yet listed in the OEIS.
%H A101696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%F A101696 a(1) = first 1-almost prime = first prime = A000040(1). a(2) = a(1) + 2nd 2-almost prime = a(1) + 2nd semiprime = A000040(1)+A001358(2). a(3) = a(2) + 3rd 3-almost prime = a(2) + A014612(3). a(4) = a(3) + 4th 4-almost prime = a(3) + A014613(4)... a(n) = a(n-1) + n-th n-almost prime.
%e A101696 a(1) = first 1-almost prime = first prime = A000040(1) = 2.
%e A101696 a(2) = 2 + 2nd 2-almost prime = 2 + A001358(2) = 2+ 6 = 8.
%e A101696 a(3) = a(2) + 3rd 3-almost prime = 8+A014612(3) = 8+18 = 26.
%e A101696 a(4) = a(3) + 4th 4-almost prime = 26+A014613(4) = 26+40 = 66.
%e A101696 a(5) = a(4) + 5th 5-almost prime = 66+A014614(5) = 66+108=174.
%e A101696 ...
%e A101696 a(12) = a(11) + 12th 12-almost prime = 22014 + 23040 = 45054 (the first nontrivial palindrome in the sequence).
%Y A101696 Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606, A101695.
%K A101696 nonn
%O A101696 1,1
%A A101696 _Jonathan Vos Post_, Dec 12 2004, Sep 28 2006
%E A101696 Edited by _N. J. A. Sloane_, Jul 03 2008 at the suggestion of _R. J. Mathar_