Reinhard Zumkeller, Jun 17 2009 reinhard.zumkeller@gmail.com Enumerations of Divisors ======================== EDP(n, x) := interpolating polynomial for the divisors of n. {EDP(n, k): 0<=k1 (*largest proper divisor of n *) EDP(n, A000005(n)-1) = n EDP(n, A000005(n)) = A161700(n) EDP(A000040(n), x) = A006093(n)*x + 1 ---+--------------------------------------------------------------------+------------- n | EDP(n, x) | ---+--------------------------------------------------------------------+------------- 1 | 1 | A000012(x) 2 | x + 1 | A000027(x+1) 3 | 2*x + 1 | A005408(x) 4 | (x^2 + x + 2)/ 2 | A000124(x) 5 | 4*x + 1 | A016813(x) 6 | (x^3 - 3*x^2 + 5*x + 3)/3 | A086514(x+1) 7 | 6*x + 1 | A016921(x) 8 | (x^3 + 5*x + 6)/6 | A000125(x) 9 | 2*x^2 + 1 | A058331(x) 10 | x^2 + 1 | A002522(x) 11 | 10*x + 1 | A017281(x) 12 | (x^5 - 5*x^4 + 5*x^3 + 5*x^2 + 114*x + 120)/120 | A161701(x) 13 | 12*x + 1 | A017533(x) 14 | (-x^3 + 9*x^2 - 5*x + 3)/3 | A161702(x) 15 | (4*x^3 - 12*x^2 + 14*x + 3)/3 | A161703(x) 16 | (x^4 - 2*x^3 + 11*x^2 + 14*x + 24)/24 | A000127(x+1) 17 | 16*x + 1 | A158057(x) 18 | (3*x^5 - 35*x^4 + 145*x^3 - 235*x^2 + 152*x + 30)/30 | A161704(x) 19 | 18*x + 1 | A161705(x) 20 | (-11*x^5 + 145*x^4 - 635*x^3 + 1115*x^2 - 494*x + 120)/120 | A161706(x) 21 | (4*x^3 - 9*x^2 + 11*x + 3)/3 | A161707(x) 22 | -x^3 + 7*x^2 - 5*x + 1 | A161708(x) 23 | 22*x + 1 | A161709(x) 24 | (-6*x^7 + 154*x^6 - 1533*x^5 + 7525*x^4 | A161710(x) | - 18879*x^ 3 + 22561*x^2 - 7302*x + 2520)/2520 | 25 | 8*x^2 - 4*x + 1 | A080856(x) 26 | (-4*x^3 + 27*x^2 - 20*x + 3)/3 | A161711(x) 27 | (4*x^3 - 6*x^2 + 8*x + 3)/3 | A161712(x) 28 | (-x^5 + 15*x^4 - 65*x^3 + 125*x^2 - 34*x + 40)/40 | A161713(x) 29 | 28*x + 1 | A161714(x) 30 | (50*x^7 - 1197*x^6 + 11333*x^5 - 53655*x^4 | A161715(x) | + 132125*x^3 - 156828*x^2 + 73212*x + 5040)/5040 | 31 | 30*x + 1 | A128470(x+1) 32 | (x^5 - 5*x^4 + 25*x^3 + 5*x^2 + 94*x + 120)/120 | A006261(x) ---+--------------------------------------------------------------------+------------- pFad - Phonifier reborn

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