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 A144119 Total number of nonprime parts in all partitions of n. 5
 1, 2, 4, 8, 13, 22, 34, 54, 80, 119, 170, 246, 342, 478, 653, 894, 1198, 1610, 2127, 2813, 3672, 4789, 6181, 7975, 10192, 13010, 16488, 20861, 26224, 32918, 41086, 51199, 63494, 78599, 96888, 119235, 146167, 178879, 218181, 265662, 322487, 390834, 472343 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is also the sum of the differences between the sum of m-th largest and the sum of (m+1)st largest elements in all partitions of n for all nonprimes m. - Omar E. Pol, Oct 27 2012 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 FORMULA a(n) = A006128(n)-A037032(n). EXAMPLE From Omar E. Pol, Nov 20 2011 (Start): For n = 6 we have: -------------------------------------- .                        Number of Partitions            nonprime parts -------------------------------------- 6 .......................... 1 3 + 3 ...................... 0 4 + 2 ...................... 1 2 + 2 + 2 .................. 0 5 + 1 ...................... 1 3 + 2 + 1 .................. 1 4 + 1 + 1 .................. 3 2 + 2 + 1 + 1 .............. 2 3 + 1 + 1 + 1 .............. 3 2 + 1 + 1 + 1 + 1 .......... 4 1 + 1 + 1 + 1 + 1 + 1 ...... 6 ------------------------------------ Total ..................... 22 So a(6) = 22. (End) MAPLE b:= proc(n, i) option remember; local g;       if n=0 then [1, 0]     elif i<1 then [0, 0]     else g:= `if`(i>n, [0\$2], b(n-i, i));          b(n, i-1) +g +[0, `if`(isprime(i), 0, g[1])]       fi     end: a:= n-> b(n, n)[2]: seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012 MATHEMATICA b[n_, i_] := b[n, i] = Module[{g}, If[n == 0, {1, 0}, If[i<1, {0, 0}, g = If[i>n, {0, 0}, b[n-i, i]]; b[n, i-1] + g + {0, If[PrimeQ[i], 0, g[[1]]]} ]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *) PROG (PARI) vector(100, n, sum(k=1, n, (numdiv(k)-omega(k))*numbpart(n-k))) \\ Altug Alkan, Oct 29 2015 CROSSREFS Cf. A006128, A018252, A037032, A144116, A144121. Sequence in context: A164480 A134035 A078157 * A207033 A291553 A330153 Adjacent sequences:  A144116 A144117 A144118 * A144120 A144121 A144122 KEYWORD easy,nonn AUTHOR Omar E. Pol, Sep 11 2008 STATUS approved

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Last modified April 18 07:59 EDT 2021. Contains 343084 sequences. (Running on oeis4.)