A281573
Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) / Product_{j>=1} (1 - x^j), where mu() is the Moebius function (A008683).
Original entry on oeis.org
1, 3, 6, 11, 19, 33, 51, 79, 118, 176, 252, 362, 505, 705, 965, 1314, 1765, 2365, 3127, 4124, 5387, 7012, 9052, 11653, 14893, 18982, 24048, 30378, 38176, 47857, 59704, 74302, 92099, 113879, 140300, 172463, 211297, 258325, 314887, 383037, 464684, 562653, 679566, 819269, 985449, 1183242, 1417738, 1695886
Offset: 1
a(5) = 19 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 3 + 3 + 4 + 5 = 19.
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nmax = 48; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 - x^i), {i, 1, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]
A284943
Expansion of Sum_{p prime, k>=1} x^(p^k)*(1 - x)^2/(1 - 2*x)^2.
Original entry on oeis.org
0, 1, 3, 8, 20, 47, 110, 251, 564, 1251, 2750, 5994, 12978, 27934, 59825, 127565, 270959, 573575, 1210466, 2547562, 5348385, 11203292, 23419629, 48865346, 101782870, 211670094, 439548898, 911515214, 1887865266, 3905400206, 8070139762, 16658958223, 34355273843
Offset: 1
a(5) = 20 because we have [5], [4, 1], [3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 4], [1, 3, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 3], [1, 1, 2, 1], [1, 1, 1, 2], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 0 = 20.
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b:= proc(n) option remember; nops(ifactors(n)[2])=1 end:
a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
`if`(b(j), ceil(2^(n-j-1)), 0), j=1..n))
end:
seq(a(n), n=1..33); # Alois P. Heinz, Aug 07 2019
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nmax = 33; Rest[CoefficientList[Series[Sum[Floor[1/PrimeNu[k]] x^k (1 - x)^2/(1 - 2 x)^2, {k, 2, nmax}], {x, 0, nmax}], x]]
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x='x+O('x^34); concat([0], Vec(sum(k=2, 34, (1\omega(k))*x^k*(1 - x)^2/(1 - 2*x)^2))) \\ Indranil Ghosh, Apr 06 2017
A281611
Expansion of Sum_{p prime, i>=2} x^(p^i)/(1 - x^(p^i)) / Product_{j>=1} (1 - x^j).
Original entry on oeis.org
0, 0, 0, 1, 1, 2, 3, 7, 10, 16, 23, 36, 50, 73, 100, 144, 193, 267, 355, 481, 631, 838, 1088, 1426, 1833, 2368, 3019, 3861, 4879, 6178, 7751, 9737, 12131, 15120, 18721, 23181, 28535, 35110, 42991, 52606, 64090, 78015, 94609, 114621, 138398, 166927, 200737, 241131, 288864, 345649, 412592, 491931
Offset: 1
a(6) = 2 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 0 + 0 + 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 2.
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nmax = 52; Rest[CoefficientList[Series[Sum[Sign[PrimeOmega[i] - 1] Floor[1/PrimeNu[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]
A281612
Expansion of Sum_{i = p*q, p prime, q prime} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).
Original entry on oeis.org
0, 0, 0, 1, 1, 3, 4, 8, 12, 20, 28, 45, 62, 92, 127, 181, 244, 340, 452, 614, 809, 1077, 1401, 1841, 2371, 3071, 3923, 5026, 6363, 8078, 10149, 12769, 15939, 19899, 24676, 30604, 37726, 46489, 57007, 69849, 85211, 103871, 126119, 152987, 184955, 223349, 268898, 323384, 387830, 464587, 555168, 662619, 789084
Offset: 1
a(6) = 3 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 1 + 0 + 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 3.
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nmax = 53; Rest[CoefficientList[Series[Sum[Floor[PrimeOmega[i]/2] Floor[2/PrimeOmega[i]] x^i/(1 - x^i), {i, 2, nmax}]/Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]]
Showing 1-4 of 4 results.
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