A035462 -id:A035462 - cp's OEIS frontend

A050452 a(n) = Sum_{d|n, d == 3 (mod 4)} d.

0, 0, 3, 0, 0, 3, 7, 0, 3, 0, 11, 3, 0, 7, 18, 0, 0, 3, 19, 0, 10, 11, 23, 3, 0, 0, 30, 7, 0, 18, 31, 0, 14, 0, 42, 3, 0, 19, 42, 0, 0, 10, 43, 11, 18, 23, 47, 3, 7, 0, 54, 0, 0, 30, 66, 7, 22, 0, 59, 18, 0, 31, 73, 0, 0, 14, 67, 0, 26, 42, 71, 3, 0, 0, 93, 19, 18

Offset: 1

N. J. A. Sloane, Dec 23 1999

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Mariusz Skałba, A Note on Sums of Two Squares and Sum-of-divisors Functions, INTEGERS 20A (2020) A92.

Cf. A000593, A050449, A001842, A035462, A245058.

Cf. Sum_{d|n, d=k-1 mod k} d: A000593 (k=2), A078182 (k=3), this sequence (k=4).

A050452 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if d mod 4 = 3 then
                        a := a+d ;
                end if;
        end do:
        a;
end proc:
seq(A050452(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

Table[Total[Select[Divisors[n],Mod[#,4]==3&]],{n,80}] (* Harvey P. Dale, Jul 07 2013 *)

a(n) = sumdiv(n, d, d*((d % 4) == 3)); \\ Amiram Eldar, Nov 26 2023

a(n) = A000593(n) - A050449(n). - Reinhard Zumkeller, Apr 18 2006
G.f.: Sum_{k>=1} (4*k - 1)*x^(4*k-1)/(1 - x^(4*k-1)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/48 = 0.205616... (A245058). - Amiram Eldar, Nov 26 2023

A035451 Number of partitions of n into parts congruent to 1 mod 4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 15, 17, 18, 20, 23, 26, 28, 30, 34, 38, 41, 44, 49, 55, 60, 64, 70, 78, 85, 91, 99, 109, 119, 128, 138, 151, 164, 176, 190, 207, 225, 241, 259, 281, 304, 326, 349, 377, 408, 437, 467, 503, 542, 581

Offset: 0

Olivier Gérard

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
James Mc Laughlin, Andrew V. Sills, Peter Zimmer, Rogers-Ramanujan-Slater Type Identities , arXiv:1901.00946 [math.NT]

Cf. A035462, A035382, A050449.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), this sequence (m=4), A109697 (m=5), A109701 (m=6), A109703 (m=7), A277090 (m=8).

 g := add(x^(n*(4*n-3))/mul((1-x^(4*k))*(1-x^(4*k-3)), k = 1..n), n = 0..5): gser := series(g,x,101): seq(coeff(gser,x,n), n = 0..100); # Peter Bala, Feb 02 2021

nmax=100; CoefficientList[Series[Product[1/(1-x^(4*k+1)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 + 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 03 2020 *)

G.f.: 1/Product_{k>=0} (1 - x^(4*k+1)). - Vladeta Jovovic, Nov 22 2002
G.f.: Sum_{n>=0} (x^n / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Apr 07 2011
G.f.: 1 + Sum_{n>=0} (x^(4*n+1) / Product_{k>=n} (1 - x^(4*k+1))) = 1 + Sum_{n>=0} (x^(4*n+1) / Product_{k=0..n} (1 - x^(4*k+1))). - Joerg Arndt, Apr 08 2011
a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n/6)) / (2^(19/8) * 3^(1/8) * n^(5/8) * Pi^(3/4)) * (1 + (Pi/(96*sqrt(6)) - 5*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
G.f.: Sum_{n>=0} x^(n*(4*n-3))/Product_{k = 1..n} ( (1-x^(4*k))*(1-x^(4*k-3)) ). (Set z = x and q = x^4 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021

Offset changed by N. J. A. Sloane, Apr 11 2010

A109700 Number of partitions of n into parts each equal to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 3, 4, 2, 2, 3, 5, 4, 3, 3, 6, 6, 6, 4, 6, 7, 9, 7, 7, 8, 11, 11, 11, 9, 12, 14, 16, 13, 14, 16, 21, 20, 19, 18, 24, 26, 27, 24, 27, 31, 36, 34, 34, 35, 43, 45, 47, 43, 49, 55, 62, 58, 59, 63, 75, 77, 77, 75, 87

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(30)=2 since 30 = 14+4+4+4+4 = 9+9+4+4+4

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284103.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), this sequence (m=5), A109702 (m=6), A109708 (m=7).

g:=1/product(1-x^(4+5*j),j=0..25): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..90); # Emeric Deutsch, Mar 30 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+4)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(4+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(4/5) * exp(Pi*sqrt(2*n/15)) / (2^(19/10) * 3^(2/5) * 5^(1/10) * Pi^(1/5) * n^(9/10)) * (1 - (9*sqrt(6/5)/(5*Pi) + Pi/(120*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284103(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

More terms from Michael Somos, Aug 10 2005

A109702 Number of partitions of n into parts each equal to 5 mod 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 4, 2, 2, 3, 5, 6, 5, 3, 3, 5, 7, 8, 6, 4, 5, 8, 10, 10, 8, 6, 8, 11, 13, 13, 10, 9, 12, 15, 18, 17, 14, 13, 16, 21, 23, 22, 18, 18, 23, 28, 31, 28, 24, 25, 31, 38, 39, 36, 32, 34

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(40)=4 since 40 = 35+5 = 29+11 = 23+17 = 5+5+5+5+5+5+5+5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284104.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), this sequence (m=6), A109708 (m=7).

g:=1/product(1-x^(5+6*j),j=0..20): gser:=series(g,x=0,92): seq(coeff(gser,x,n),n=0..89); # Emeric Deutsch, Apr 14 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(6*k+5)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
Table[Count[IntegerPartitions[n],?(Union[Mod[#,6]]=={5}&)],{n,0,90}] (* _Harvey P. Dale, Mar 08 2022 *)

G.f.: 1/product(1-x^(5+6j),j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(5/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(1/6) * n^(11/12)) * (1 - (55/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284104(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
Euler transform of period 6 sequence [ 0, 0, 0, 0, 1, 0, ...]. - Kevin T. Acres, Apr 28 2018

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

A109708 Number of partitions of n into parts each equal to 6 mod 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 5, 5, 5, 2, 2, 3, 5, 6, 8, 5, 3, 3, 5, 7, 10, 9, 7, 4, 5, 7, 11, 12, 12, 8, 6, 7, 12, 14, 17, 15, 11, 8, 12, 15, 20, 21, 19, 13, 13, 16, 22, 26, 28, 23

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(45)=3 because we have 45=27+6+6+6=20+13+6+6=13+13+13+6.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284105.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), A109702 (m=6), this sequence (m=7).

g:=1/product(1-x^(6+7*j),j=0..20): gser:=series(g,x=0,98): seq(coeff(gser,x,n),n=0..95); # Emeric Deutsch, Apr 14 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+6)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(6+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(6/7) * exp(Pi*sqrt(2*n/21)) / (2^(27/14) * 3^(3/7) * 7^(1/14) * Pi^(1/7) * n^(13/14)) * (1 - (39*sqrt(3/14)/(7*Pi) + 13*Pi/(168*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284105(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

A339060 Number of compositions (ordered partitions) of n into distinct parts congruent to 3 mod 4.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 4, 1, 0, 6, 4, 1, 0, 6, 6, 1, 0, 12, 6, 1, 0, 18, 8, 1, 24, 24, 8, 1, 24, 30, 10, 1, 48, 42, 10, 1, 72, 48, 12, 1, 120, 60, 12, 121, 144, 72, 14, 121, 216, 84, 14, 241, 264, 96, 16, 361, 360, 114, 16, 601, 432, 126, 18, 841

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(21) = 6 because we have [11, 7, 3], [11, 3, 7], [7, 11, 3], [7, 3, 11], [3, 11, 7] and [3, 7, 11].

Index entries for sequences related to compositions

Cf. A004767, A017817, A032020, A032021, A035462, A147599, A337547, A337548, A339059.

nmax = 75; CoefficientList[Series[Sum[k! x^(k (2 k + 1))/Product[1 - x^(4 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(2*k + 1)) / Product_{j=1..k} (1 - x^(4*j)).

A307978 Expansion of e.g.f. exp((sinh(x) - sin(x))/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 10, 1, 0, 280, 120, 1, 15400, 17160, 2080, 1401401, 3203200, 1290640, 190623040, 775975201, 712150400, 36321556720, 239000886400, 413465452401, 9339501072000, 91625659447400, 266045692290560, 3216459513124001, 42923384190336000, 193108117771690680

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

Number of partitions of n-set into blocks congruent to 3 mod 4.

Seiichi Manyama, Table of n, a(n) for n = 0..608

Cf. A002017, A003724, A004767, A005046, A013369, A035462, A291975, A306347, A307979.

nmax = 29; CoefficientList[Series[Exp[(Sinh[x] - Sin[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{3}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 29}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sinh(x)-sin(x))/2))) \\ Seiichi Manyama, Mar 17 2022

a(n) = if(n==0, 1, sum(k=0, (n-3)\4, binomial(n-1, 4*k+2)*a(n-4*k-3))); \\ Seiichi Manyama, Mar 17 2022

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/4)} binomial(n-1,4*k+2) * a(n-4*k-3). - Seiichi Manyama, Mar 17 2022

A117957 Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 4, 4, 3, 3, 5, 6, 5, 4, 6, 8, 7, 6, 8, 10, 10, 9, 10, 13, 13, 12, 14, 17, 18, 16, 18, 22, 23, 22, 23, 28, 31, 29, 30, 36, 39, 39, 39, 45, 51, 50, 51, 57, 64, 65, 65, 73, 81, 83, 84, 91, 102, 106, 106

Offset: 0

Emeric Deutsch, Apr 05 2006

nonn

Also number of partitions of n such that 2k and 2k+1 occur with the same multiplicities. Example: a(26)=3 because we have [11,10,3,2], [9,8,5,4] and [7,7,6,6]. It is easy to find a bijection between these partitions and those described in the definition.

			a(26)=3 because we have [21,5],[17,9] and [13,13].

Cf. A035451, A035462.

g:=1/product(1-x^(4*i+1),i=1..50): gser:=series(g,x=0,93): seq(coeff(gser,x,n),n=0..88);

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: 1/product(1-x^(4i+1), i=1..infinity).

a(n) ~ exp(sqrt(n/6)*Pi) * Pi^(1/4) * Gamma(1/4) / (2^(31/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Mar 07 2016

A374019 Expansion of Product_{k>=1} 1 / (1 - x^(4*k-1))^2.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 3, 2, 0, 4, 4, 2, 5, 6, 7, 8, 8, 12, 15, 12, 17, 26, 23, 24, 37, 40, 39, 50, 62, 66, 74, 86, 101, 116, 122, 144, 175, 184, 202, 246, 274, 294, 340, 388, 432, 480, 533, 610, 684, 742, 835, 956, 1045, 1144, 1299, 1450, 1586, 1758, 1965, 2182, 2400, 2638, 2941, 3268, 3560, 3922

Offset: 0

Ilya Gutkovskiy, Jun 25 2024

nonn

Cf. A000712, A022567, A035462, A050452, A261629, A374018.

nmax = 65; CoefficientList[Series[Product[1/(1 - x^(4 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A050452(k) * a(n-k).
a(n) = Sum_{k=0..n} A035462(k) * A035462(n-k).
a(n) ~ Pi^(3/2) * exp(Pi*sqrt(n/3)) / (2*sqrt(3) * Gamma(1/4)^2 * n). - Vaclav Kotesovec, Jun 25 2024

cp's OEIS Frontend

A050452 a(n) = Sum_{d|n, d == 3 (mod 4)} d.

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A035451 Number of partitions of n into parts congruent to 1 mod 4.

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A109700 Number of partitions of n into parts each equal to 4 mod 5.

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Maple

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A109702 Number of partitions of n into parts each equal to 5 mod 6.

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Maple

Mathematica

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A109708 Number of partitions of n into parts each equal to 6 mod 7.

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Maple

Mathematica

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Extensions

A339060 Number of compositions (ordered partitions) of n into distinct parts congruent to 3 mod 4.

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Mathematica

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A307978 Expansion of e.g.f. exp((sinh(x) - sin(x))/2).

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