A087177 -id:A087177 - cp's OEIS frontend

A087180 Number partition numbers <= P(n) of the form 3*k (P = A000041).

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 7, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 15, 15, 16, 16, 16, 16, 17, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 28, 28, 28, 29, 29, 30, 31, 31, 31, 31, 32

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A068907, A071754, A087177, A087181, A087182, A087183.

Accumulate[Table[Boole[Mod[PartitionsP[n], 3] == 0], {n, 0, 100}]] (* Amiram Eldar, May 22 2025 *)

a(n) + A087181(n) + A087182(n) = n + 1.

A087181 Number partition numbers <= P(n) of the form 3*k+1 (P = A000041).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 23, 23, 23

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A068907, A071754, A087177, A087180, A087182, A087184.

Accumulate[Table[Boole[Mod[PartitionsP[n], 3] == 1], {n, 0, 100}]] (* Amiram Eldar, May 22 2025 *)

A087180(n) + a(n) + A087182(n) = n + 1.

A087182 Number partition numbers <= P(n) of the form 3*k+2 (P = A000041).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 17, 17, 18, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 24

Offset: 0

Reinhard Zumkeller, Aug 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Partition Function.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.

Cf. A000041, A068907, A071754, A087177, A087180, A087181, A087185.

Accumulate[Table[Boole[Mod[PartitionsP[n], 3] == 2], {n, 0, 100}]] (* Amiram Eldar, May 22 2025 *)

A087180(n) + A087181(n) + a(n) = n + 1.

A071640 a(n) = Sum_{i=1..n} A040051(i).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 13, 13, 14, 15, 15, 15, 15, 15, 16, 16, 16, 17, 18, 18, 19, 20, 21, 22, 23, 23, 24, 24, 25, 26, 26, 26, 26, 27, 28, 28, 29, 30, 31, 32, 32, 33, 33, 33, 33, 34, 35, 35, 36, 36, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42

Offset: 1

Benoit Cloitre, Jun 22 2002

Does 2*a(n) > n for n>2? Cf. A086144. - Benoit Cloitre

No. First failure is at n = 6662. - Peter Luschny, Oct 05 2011

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000041, A086144, A040051, A087177.

A071640 := proc(n) option remember; if n=1 then 1 else A071640(n-1)+
(combinat[numbpart](n) mod 2) fi end: # Peter Luschny, Oct 05 2011

a[n_] := Sum[Mod[PartitionsP[i], 2], {i, 1, n}];
Array[a, 80] (* Jean-François Alcover, Jun 03 2019 *)

a(n) = my(x='x+O('x^(n+1)), p = 1/eta(x)); sum(i=1, n, (1-(-1)^(polcoeff(p, i))))/2; \\ corrected by Michel Marcus, Jun 11 2019

A086144 a(n) = 2*A071640(n) - n.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 5, 4, 5, 6, 5, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 7, 6, 7, 8, 7, 6, 5, 6, 7, 6, 7, 8, 9, 10, 9, 10, 9, 8, 7, 8, 9, 8, 9, 8, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 9, 10, 11, 10, 9, 8, 9, 10, 11, 10, 11, 10, 11, 12, 13, 14

Offset: 1

Benoit Cloitre, Sep 06 2003

sign

It is conjectured that A071640(n)/n -> 1/2. - Benoit Cloitre

a(6671) < 0. - Peter Luschny, Oct 05 2011

Cf. A071640, A040051, A000041, A087177.

A086144 := proc(n) option remember; if n=1 then 1 else
if combinat[numbpart](n) mod 2 = 1 then 1 else -1 fi;
% + A086144(n-1) fi end: seq(A086144(i),i=1..90); # Peter Luschny, Oct 05 2011

a071640[n_] := Sum[Mod[PartitionsP[i], 2], {i, 1, n}];
a[n_] := 2 a071640[n] - n;
Array[a, 100] (* Jean-François Alcover, Jun 11 2019 *)

a(n) = my(x='x+O('x^(n+1)), p = 1/eta(x)); sum(i=1, n, (1-(-1)^(polcoeff(p, i)))) - n; \\ Michel Marcus, Jun 11 2019

Erroneous data for n>55 replaced, keyword sign added by Peter Luschny, Oct 05 2011

cp's OEIS Frontend

A087180 Number partition numbers <= P(n) of the form 3*k (P = A000041).

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A087181 Number partition numbers <= P(n) of the form 3*k+1 (P = A000041).

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A087182 Number partition numbers <= P(n) of the form 3*k+2 (P = A000041).

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A071640 a(n) = Sum_{i=1..n} A040051(i).

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A086144 a(n) = 2*A071640(n) - n.

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