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A123650
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a(n) = 1 + n^2 + n^3 + n^5.
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3
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4, 45, 280, 1105, 3276, 8029, 17200, 33345, 59860, 101101, 162504, 250705, 373660, 540765, 762976, 1052929, 1425060, 1895725, 2483320, 3208401, 4093804, 5164765, 6449040, 7977025, 9781876, 11899629, 14369320, 17233105, 20536380, 24327901
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = 1 + n^2 + n^3 + n^5 = 101101 (base n) = +/- (n+1)*(n^2-n+1)*(n^2+1).
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
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MATHEMATICA
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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 *)
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PROG
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(PARI) for(n=1, 25, print1(1+n^2+n^3+n^5, ", ")) \\ G. C. Greubel, Oct 17 2017
(Magma) [1+n^2+n^3+n^5: n in [1..25]]; // G. C. Greubel, Oct 17 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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