A144121 -id:A144121 - cp's OEIS frontend

A144118 Number of non-Fibonacci parts in the last section of the set of partitions of n.

0, 0, 0, 1, 0, 2, 2, 4, 5, 9, 11, 20, 22, 37, 45, 68, 83, 122, 149, 210, 259, 353, 436, 585, 717, 941, 1161, 1497, 1835, 2344, 2862, 3612, 4403, 5496, 6678, 8279, 10010, 12314, 14857, 18148, 21811, 26503, 31739, 38356, 45803, 55066, 65553, 78488, 93129

Offset: 1

Omar E. Pol, Sep 11 2008

nonn

First differences of A144116.

Cf. A001690, A135010, A138121, A138137, A144115, A144116, A144117, A144121.

b:= proc(n) option remember; true end: l:= [0, 1]: for k to 100 do b(l[1]):= false; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2] -aa(n-1, n-1)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009

Clear[b]; b[] = True; l = {0, 1}; For[k = 1, k <= 100, k++, b[l[[1]]] = False; l = {l[[2]], l[[1]] + l[[2]]}]; a[n, i_] := aa[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i == 0 || n < 0, {0, 0}, g = aa[n, i-1]; h = aa[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]]; a[n_] := aa[n, n][[2]] - aa[n-1, n-1][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 05 2016 after Alois P. Heinz *)

a(n) = A138137(n)-A144117(n) = A144116(n)-A144116(n-1).

More terms from Alois P. Heinz, Jul 28 2009

A144119 Total number of nonprime parts in all partitions of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Sep 11 2008

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

			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)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A006128, A018252, A037032, A144116, A144121.

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

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 *)

vector(100, n, sum(k=1, n, (numdiv(k)-omega(k))*numbpart(n-k))) \\ Altug Alkan, Oct 29 2015

a(n) = A006128(n)-A037032(n).

A144120 Number of prime parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 32, 48, 61, 88, 113, 154, 198, 267, 337, 446, 563, 730, 915, 1174, 1460, 1853, 2294, 2878, 3545, 4416, 5404, 6679, 8144, 9991, 12125, 14791, 17866, 21677, 26084, 31478, 37733, 45340

Offset: 1

Omar E. Pol, Sep 11 2008

First differences of A037032.

Cf. A037032, A135010, A138121, A138137, A144117, A144118, A144121.

a(n) = A037032(n)-A037032(n-1).

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A144118 Number of non-Fibonacci parts in the last section of the set of partitions of n.

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A144119 Total number of nonprime parts in all partitions of n.

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A144120 Number of prime parts in the last section of the set of partitions of n.

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