This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085410 #10 Feb 22 2025 18:48:56
%S A085410 1,2,5,9,19,27,53,74,122,170,274,355,555,724,1043,1377,1964,2487,3497,
%T A085410 4429,5993,7622,10205,12701,16831,20964,27166,33756,43452,53296,68134,
%U A085410 83464,105086,128495,160803,195006,242811,293701,362026,436842,536103
%N A085410 Total number of parts in all partitions of n into relatively prime parts.
%F A085410 Moebius transform of A006128: Sum_{d|n} mu(n/d)*A006128(d).
%e A085410 Partitions of 6 into relatively prime parts are: 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 1+1+1+3, 1+2+3, 1+1+4, 1+5; total number of parts is a(6)=6+5+4+4+3+3+2=27.
%t A085410 f[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; MT[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu /@ (n/d)*f /@ d)]; Table[ MT[n], {n, 1, 41}]
%o A085410 (PARI) a006128(n) = sum(m=1, n, numdiv(m)*numbpart(n - m));
%o A085410 a(n) = sumdiv(n, d, moebius(n/d)*a006128(d)); \\ _Indranil Ghosh_, Apr 25 2017
%o A085410 (Python)
%o A085410 from sympy import divisors, divisor_count, npartitions, mobius
%o A085410 def a006128(n): return sum([divisor_count(m)*npartitions(n - m) for m in range(1, n + 1)])
%o A085410 def a(n): return sum([mobius(n/d)*a006128(d) for d in divisors(n)]) # _Indranil Ghosh_, Apr 25 2017
%Y A085410 Cf. A000837, A006128.
%K A085410 easy,nonn
%O A085410 1,2
%A A085410 _Vladeta Jovovic_, Aug 13 2003
%E A085410 More terms from _Robert G. Wilson v_, Aug 17 2003