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A123650 a(n) = 1 + n^2 + n^3 + n^5.

%I A123650 #18 Sep 08 2022 08:45:28
%S A123650 4,45,280,1105,3276,8029,17200,33345,59860,101101,162504,250705,
%T A123650 373660,540765,762976,1052929,1425060,1895725,2483320,3208401,4093804,
%U A123650 5164765,6449040,7977025,9781876,11899629,14369320,17233105,20536380,24327901
%N A123650 a(n) = 1 + n^2 + n^3 + n^5.
%C A123650 3rd row, A(3,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 4th row, A(4,n), is A123111 1 + n^2 + n^3 + n^5 + n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The current sequence, A(3,n), can never be prime because of the polynomial factorization a(n) = 1 + n^2 + n^3 + n^5 = +/- (n+1)*(n^2-n+1)*(n^2+1). Its fewest prime factors are 2 for the semiprime a(1) = 4. We similarly have polynomial factorizations for A123651 = A(7,n) = 1+n^2+n^3+n^5+n^7+n^11+n^13+n^17 and A123652 = A(13,n) = 1+n^2+n^3+n^5+...+n^41.
%H A123650 G. C. Greubel, <a href="/A123650/b123650.txt">Table of n, a(n) for n = 1..1000</a>
%H A123650 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A123650 a(n) = 1 + n^2 + n^3 + n^5 = 101101 (base n) = +/- (n+1)*(n^2-n+1)*(n^2+1).
%F A123650 G.f.: x*(4 +21*x +70*x^2 +20*x^3 +6*x^4 -x^5)/(1-x)^6. - _Colin Barker_, May 25 2012
%t A123650 Table[1+n^2+n^3+n^5,{n,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{4,45,280,1105,3276,8029},30] (* _Harvey P. Dale_, Jan 18 2014 *)
%o A123650 (PARI) for(n=1,25, print1(1+n^2+n^3+n^5, ", ")) \\ _G. C. Greubel_, Oct 17 2017
%o A123650 (Magma) [1+n^2+n^3+n^5: n in [1..25]]; // _G. C. Greubel_, Oct 17 2017
%Y A123650 Cf. A002522, A098547, A123111, A123113, A123651, A123652.
%K A123650 easy,nonn
%O A123650 1,1
%A A123650 _Jonathan Vos Post_, Oct 04 2006