A237280 -id:A237280 - cp's OEIS frontend

A237278 Numbers k such that A000041(k) == 0 (mod 4).

11, 15, 21, 26, 30, 55, 58, 59, 62, 66, 70, 74, 75, 78, 80, 84, 94, 96, 98, 100, 106, 108, 109, 112, 113, 116, 117, 120, 122, 124, 125, 126, 128, 130, 131, 133, 135, 136, 137, 141, 142, 149, 153, 154, 171, 179, 180, 187, 191, 200, 205, 206, 230, 231, 236

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(11) = 56 == 0 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237279, A237280, A237281, A243935 (see crossrefs).

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

is(n)=numbpart(n)%4==0 \\ Charles R Greathouse IV, Apr 08 2015

A278478 a(n) is the 2-adic valuation of A000041(n).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 0, 1, 1, 6, 0, 0, 0, 5, 0, 0, 0, 2, 3, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 11, 1, 3, 0, 2, 1, 0, 1, 0, 0, 4, 0, 2, 7, 1, 0, 2, 2, 0, 0, 3, 2, 0

Offset: 0

Joerg Arndt, Nov 23 2016

nonn

Write A000041(n) = 2^k * s where s is odd, then a(n) = k.

Joerg Arndt, Table of n, a(n) for n = 0..10000

Cf. A000041, A007814, A069935, A278479.
Cf. A052002, A237280, A278779, A278780, A278781, A278782, A278783, A278784 (positions of terms 0, 1, 2, ..., 7 in this sequence).
Cf. also A278241.

a:= n-> padic[ordp](combinat[numbpart](n), 2):
seq(a(n), n=0..120);  # Alois P. Heinz, Nov 23 2016

a[n_] := IntegerExponent[PartitionsP[n], 2]; Array[a, 100, 0] (* Amiram Eldar, May 25 2024 *)

{ my( x='x+O('x^100), v=Vec(1/eta(x)) ); vector(#v,n,valuation(v[n],2)) }

From Amiram Eldar, May 25 2024: (Start)
a(n) = A007814(A000041(n)).
a(n) = log_2(A069935(n)). (End)

A237281 Numbers k such that A000041(k) == 3 (mod 4).

Original entry on oeis.org

3, 5, 6, 7, 14, 16, 20, 23, 24, 33, 35, 38, 41, 44, 51, 53, 54, 56, 60, 63, 68, 72, 76, 77, 81, 82, 91, 92, 95, 99, 102, 111, 115, 118, 121, 127, 134, 138, 139, 140, 146, 156, 159, 161, 164, 165, 166, 168, 169, 173, 177, 178, 182, 183, 188, 192, 196, 201

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(6) = 11 == 3 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237278, A237279, A237280.

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

A275029 Partition numbers (A000041) congruent to 2 (mod 4).

Original entry on oeis.org

2, 22, 30, 42, 490, 1002, 1958, 3010, 3718, 6842, 12310, 37338, 53174, 89134, 105558, 124754, 204226, 614154, 1741630, 2012558, 13848650, 34262962, 133230930, 214481126, 271248950, 607163746, 4835271870, 30388671978, 45060624582, 88751778802, 107438159466

Offset: 1

Colin Barker, Nov 13 2016

nonn

Partition numbers having the same number of even divisors as odd divisors.
The corresponding indices are in A237280.
The intersection of A000041 and A016825.

			30 is in the sequence because it is a partition number, and its divisors are [1,2,3,5,6,10,15,30].

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000041, A016825, A213179, A237280.

select(t -> t mod 4 = 2, map(combinat:-numbpart, [$1..500])); # Robert Israel, Nov 14 2016

Select[PartitionsP@ Range@ 160, Mod[#, 4] == 2 &] (* Michael De Vlieger, Nov 15 2016 *)

a000041(n) = numbpart(n)
terms(n) = my(i=0, k=2); while(1, if(Mod(a000041(k), 4)==2, print1(a000041(k), ", "); i++); if(i==n, break); k++)
/* Print initial 50 terms as follows */
terms(50) \\ Felix Fröhlich, Nov 15 2016

A121062 Partition numbers mod 4.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 3, 3, 2, 2, 2, 0, 1, 1, 3, 0, 3, 1, 1, 2, 3, 0, 2, 3, 3, 2, 0, 2, 2, 1, 0, 2, 1, 3, 2, 3, 1, 1, 3, 1, 2, 3, 2, 1, 3, 2, 2, 2, 1, 1, 2, 3, 1, 3, 3, 0, 3, 2, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 3, 1, 0, 1, 3, 1, 0, 0, 3, 3, 0, 2, 0, 3, 3, 1, 0, 1, 2, 1, 1, 1, 1, 3, 3, 1, 0, 3, 0, 2, 0, 3, 0, 2, 3, 2, 1

Offset: 0

Jonathan Vos Post, Aug 10 2006

P_n==0: 11, 15, 21, 26, 30, 55, 58, 59, 62, 66, 70, 74, 75, 78, 80, 84, 94, 96, 98, 100, ... A237278.
P_n==1: 0, 1, 4, 12, 13, 17, 18, 29, 32, 36, 37, 39, 43, 48, 49, 52, 61, 67, 69, 71, 73, ... A237279.
P_n==2: 2, 8, 9, 10, 19, 22, 25, 27, 28, 31, 34, 40, 42, 45, 46, 47, 50, 57, 64, 65, 79, ... A237280.
P_n==3: 3, 5, 6, 7, 14, 16, 20, 23, 24, 33, 35, 38, 41, 44, 51, 53, 54, 56, 60, 63, 68, ... A237281.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A010873, A049575.

Cf. A237278, A237279, A237280, A237281.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     (b(n, i-1)+b(n-i, min(n-i, i))) mod 4))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..104);  # Alois P. Heinz, Dec 20 2024

f[n_] := Mod[PartitionsP@n, 4]; Table[f@n, {n, 0, 104}] (* Robert G. Wilson v *)

a(n) = numbpart(n) % 4; \\ Michel Marcus, Jun 29 2016

a(n) = A000041(n) mod 4 = A010873(A000041(n)).

a(n) = A000025(n) mod 4. - John M. Campbell, Jun 29 2016

More terms from Robert G. Wilson v, Aug 17 2006

A237279 Numbers k such that A000041(k) == 1 (mod 4).

Original entry on oeis.org

1, 4, 12, 13, 17, 18, 29, 32, 36, 37, 39, 43, 48, 49, 52, 61, 67, 69, 71, 73, 83, 85, 87, 88, 89, 90, 93, 104, 105, 107, 114, 119, 123, 132, 143, 144, 145, 148, 150, 152, 155, 157, 162, 172, 181, 185, 186, 189, 190, 193, 194, 195, 199, 203, 208, 216, 219

Offset: 1

Clark Kimberling, Feb 05 2014

The set of positive integers is partitioned by the sequences A237278-A237281.

			A000041(4) = 5 == 1 (mod 4).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000041, A237276, A237278, A237280, A237281.

Cf. A121062.

f[n_, k_] := Select[Range[250], Mod[PartitionsP[#], n] == k &]
Table[f[4, k], {k, 0, 3}] (* A237278-A237281 *)

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A237278 Numbers k such that A000041(k) == 0 (mod 4).

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A278478 a(n) is the 2-adic valuation of A000041(n).

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A237281 Numbers k such that A000041(k) == 3 (mod 4).

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A275029 Partition numbers (A000041) congruent to 2 (mod 4).

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A121062 Partition numbers mod 4.

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A237279 Numbers k such that A000041(k) == 1 (mod 4).

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