A167932 -id:A167932 - cp's OEIS frontend

A167934 a(n) = A000041(n) - A032741(n).

1, 1, 1, 2, 3, 6, 8, 14, 19, 28, 39, 55, 72, 100, 132, 173, 227, 296, 380, 489, 622, 789, 999, 1254, 1568, 1956, 2433, 3007, 3713, 4564, 5597, 6841, 8344, 10140, 12307, 14880, 17969, 21636, 26012, 31182, 37331, 44582, 53167, 63260, 75170

Offset: 0

Omar E. Pol, Nov 16 2009

nonn

a(n) is also the number of partitions of n whose parts are not all equal, (including however the partition with a single part of size n). Note that the number of partitions of n whose parts are all equal gives the number of divisors of n, for n>0. (See also A144300.)

			The partitions of n = 6 are:
6 ....................... All parts are equal, but included .. (1).
5 + 1 ................... All parts are not equal ............ (2).
4 + 2 ................... All parts are not equal ............ (3).
4 + 1 + 1 ............... All parts are not equal ............ (4).
3 + 3 ................... All parts are equal, not included.
3 + 2 + 1 ............... All parts are not equal ............ (5).
3 + 1 + 1 + 1 ........... All parts are not equal ............ (6).
2 + 2 + 2 ............... All parts are equal, not included.
2 + 2 + 1 + 1 ........... All parts are not equal ............ (7).
2 + 1 + 1 + 1 + 1 ....... All parts are not equal ............ (8).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal, not included.
Then a(6) = 8.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A144300, A135010, A138121, A167930, A167932, A167935.

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then `if`(k=0, 1, 0)
    elif i=0 then 0
    else b(n, i-1, k)+
         b(n-i, i, `if`(k<0, i, `if`(k<>i, 0, k)))
      fi
    end:
a:= n-> 1 +b(n, n-1, -1):
seq(a(n), n=0..50);  #  Alois P. Heinz, Dec 01 2010

a[0] = 1; a[n_] := PartitionsP[n] - DivisorSigma[0, n] + 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 08 2016 *)

a(n) = A000041(n) - A032741(n).

More terms from Alois P. Heinz, Dec 01 2010

A167930 Number of partitions of n in which some but not all parts are equal.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 4, 9, 13, 20, 29, 43, 57, 82, 110, 146, 195, 258, 334, 435, 558, 713, 910, 1150, 1446, 1814, 2268, 2815, 3491, 4308, 5301, 6501, 7954, 9692, 11795, 14295, 17301, 20876, 25148, 30200, 36218, 43322, 51741, 61650, 73354

Offset: 0

Omar E. Pol, Nov 15 2009

nonn

The parts may not all be equal, and at least one part must occur at least twice. - N. J. A. Sloane, May 30 2024

			The partitions of 6 are:
6 ....................... All parts are distinct.
5 + 1 ................... All parts are distinct.
4 + 2 ................... All parts are distinct.
4 + 1 + 1 ............... Only some parts are equal ...... (1).
3 + 3 ................... All parts are equal.
3 + 2 + 1 ............... All parts are distinct.
3 + 1 + 1 + 1 ........... Only some parts are equal ...... (2).
2 + 2 + 2 ............... All parts are equal.
2 + 2 + 1 + 1 ........... Only some parts are equal ...... (3).
2 + 1 + 1 + 1 + 1 ....... Only some parts are equal ...... (4).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
Then a(6) = 4.
a(7) = 9 from 511  4111  331  322  3211  31111  2221  22111  211111. - _N. J. A. Sloane_, May 30 2024

Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167931, A167932, A167933.

f[lst_]:=With[{c=Split[lst]},Length[lst]>2&&Max[Length/@c]>1&&Length[c]>1]; Table[Length[ Select[ IntegerPartitions[n],f]],{n,0,50}] (* Harvey P. Dale, May 30 2024 *)

a(n) = A047967(n) - A032741(n).
a(n) = A000041(n) - A000009(n) - A032741(n).
a(0) = 0: For n>0, a(n) = A000041(n) - A000009(n) - A000005(n) + 1.

Edited by Omar E. Pol, Nov 16 2009

More terms from Max Alekseyev, May 02 2011

A167928 Number of partitions of n that do not contain 1 as a part and whose parts are not the same divisor of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 3, 4, 6, 9, 13, 16, 23, 31, 38, 51, 65, 83, 104, 132, 162, 207, 252, 313, 381, 475, 571, 703, 846, 1032, 1237, 1502, 1791, 2164, 2570, 3086, 3659, 4375, 5167, 6146, 7244, 8584, 10086, 11909, 13954, 16421, 19195, 22510, 26250, 30696, 35714

Offset: 0

Omar E. Pol, Nov 17 2009

nonn

Note that these partitions are located in the head of the last section of the set of partitions of n (see the shell model of partitions, here).

			The partitions of 6 are:
6 ....................... All parts are the same divisor of n.
5 + 1 ................... Contains 1 as a part.
4 + 2 ................... All parts are not the same divisor of n. <------(1)
4 + 1 + 1 ............... Contains 1 as a part.
3 + 3 ................... All parts are the same divisor of n.
3 + 2 + 1 ............... Contains 1 as a part.
3 + 1 + 1 + 1 ........... Contains 1 as a part.
2 + 2 + 2 ............... All parts are the same divisor of n.
2 + 2 + 1 + 1 ........... Contains 1 as a part.
2 + 1 + 1 + 1 + 1 ....... Contains 1 as a part.
1 + 1 + 1 + 1 + 1 + 1 ... Contains 1 as a part.
Then a(6) = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000041, A002865, A032741, A144300, A135010, A138121, A167929, A167930, A167932, A167934.

b:= proc(n, i, t) option remember;
      `if`(n=0, `if`(t<>1, 1, 0), `if`(i<2, 0,
      add(b(n-i*j, i-1, `if`(j=0, t, max(0, t-1))), j=0..n/i)))
    end:
a:= n-> b(n, n, 2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 24 2013

Prepend[Array[ n \[Function] Length@Select[IntegerPartitions[n, All, Range[2, n - 1]], Length[Union[ # ]] > 1 &], 40], 1] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t != 1, 1, 0], If[i < 2, 0, Sum[b[n - i*j, i - 1, If[j == 0, t, Max[0, t - 1]]], {j, 0, n/i}]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = A002865(n) - A032741(n).

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms from Alois P. Heinz, May 24 2013

cp's OEIS Frontend

A167934 a(n) = A000041(n) - A032741(n).

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A167930 Number of partitions of n in which some but not all parts are equal.

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A167928 Number of partitions of n that do not contain 1 as a part and whose parts are not the same divisor of n.

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