A109701 -id:A109701 - cp's OEIS frontend

A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 26, 30, 34, 38, 42, 47, 53, 60, 67, 74, 82, 91, 102, 114, 126, 139, 153, 169, 187, 207, 228, 250, 274, 301, 331, 364, 399, 436, 476, 520, 569, 622, 679, 739, 804, 875, 953, 1038, 1128, 1224, 1327

Offset: 0

N. J. A. Sloane, Herman P. Robinson

There are many (at least 8) equivalent definitions of this sequence (besides the comments below, see also Schur, Alladi, Andrews). - N. J. A. Sloane, Jun 17 2011
Coefficients of replicable function number 72e. - N. J. A. Sloane, Jun 10 2015
Also number of partitions of n into odd parts in which no part appears more than twice, cf. A070048 and A096938. - Vladeta Jovovic, Jan 18 2005
Also number of partitions of n into distinct parts congruent to 1 or 2 modulo 3. (Follows from second g.f.) - N. Sato, Jul 20 2005
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A262928 and A261612. - Vaclav Kotesovec, Jan 13 2017
Convolution of A109702 and A109701. - Vaclav Kotesovec, Jan 21 2017

			G.f: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
T72e = 1/q + q^11 + q^23 + q^35 + q^47 + 2*q^59 + 2*q^71 + 3*q^83 + ...
The logarithm of the g.f. begins:
log(A(x)) = x + x^2/2 + x^3/3 + x^4/4 + 6*x^5/5 + x^6/6 + 8*x^7/7 + x^8/8 + x^9/9 + 6*x^10/10 + 12*x^11/11 + x^12/12 + ... + A186099(n)*x^n/n + ... . - _Paul D. Hanna_, Feb 17 2013

K. Alladi, Refinements of Rogers-Ramanujan type identities. In Special Functions, q-Series and Related Topics (Toronto, ON, 1995), 1-35, Fields Inst. Commun., 14, Amer. Math. Soc., Providence, RI, 1997.
G. E. Andrews, Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. In q-Series From a Contemporary Perspective (South Hadley, MA, 1998), 45-56, Contemp. Math., 254, Amer. Math. Soc., Providence, RI, 2000.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
I. Schur, Zur Additiven Zahlentheorie, Ges. Abh., Vol. 2, Springer, pp. 43-50.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from R. Zumkeller)
K. Alladi and B. Gordon, Generalizations of Schur's partition theorem, Manuscr. Math. 79 (1993) 113-126.
K. Alladi and B. Gordon, Schur's partition theorem, companions, refinements and generalizations, Trans. Amer. Math. Soc. 347 (1995) 1591-1608.
G. E. Andrews, K. Alladi, B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, J. Reine Angew. Math. 460 (1995) 165-188.
D. M. Bressoud, A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer, Math. Soc. 79 (1980) 338-340.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. Vol. 225 (1967), 154-190.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
Padmavathamma, R. Raghavendra and B. M. Chandrashekara, A new bijective proof of a partition theorem of K. Alladi, Discrete Math., 237 (2004), 125-128.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Schur's Partition Theorem
James J. Y. Zhao, A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem, Elect. J. Combin, Volume 22, Issue 1 (2015) Paper #P1.68.
Index entries for McKay-Thompson series for Monster simple group

Cf. A000041, A001651, A003114, A000726, A109389, A109697, A132462, A132463, A285219, A304047.

Cf. A186099 (log).

a003105 n = p 1 n where
   p k m | m == 0 = 1 | m < k = 0 | otherwise = q k (m-k) + p (k+2) m
   q k m | m == 0 = 1 | m < k = 0 | otherwise = p (k+2) (m-k) + p (k+2) m
-- Reinhard Zumkeller, Nov 12 2011

with(combinat);
A:=proc(n) local i, j, t3, t2, t1;
    t2:=0;
    t1:=firstpart(n);
    for j from 1 to numbpart(n)+2 do
        t3:=1;
        for i from 1 to nops(t1) do
            if (t1[i] mod 6) <> 1 and (t1[i] mod 6) <> 5 then t3:=0; fi;
        od;
        if t3=1 then t2:=t2+1; fi;
        if nops(t1) = 1 then RETURN(t2); fi;
        t1:=nextpart(t1);
    od;
end;
# brute-force Maple program from N. J. A. Sloane, Jun 17 2011

max = 63; f[x_] := 1/Product[1 - x^k + x^(2k), {k, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 01 2011, after Vladeta Jovovic *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] / QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 3] != 0, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 63; kmax = nmax/6;
s = Flatten[{Range[0, kmax]*6 + 1}~Join~{Range[kmax]*6 - 1}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Jan 09 2005 */

{S(n,x)=sumdiv(n,d,d*(1-x^d)^(n/d))}
{a(n)=polcoeff(exp(sum(k=1,n,S(k,x)*x^k/k)+x*O(x^n)),n)}
for(n=0,60,print1(a(n),", "))
/* Paul D. Hanna, Feb 17 2013 */

G.f.: 1/Product_{k>=0} (1-x^(6*k+1))*(1-x^(6*k+5)) = Product_{k>=0} (1+x^(3*k+1))*(1+x^(3*k+2)) = 1/Product_{k>=0} (1-x^k+x^(2*k)). - Vladeta Jovovic, Jun 08 2003
Expansion of chi(-x^3) / chi(-x) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Mar 04 2012
Expansion of f(x, x^2) / f(-x^3) = f(-x^6) / f(-x, -x^5) in powers of x where f() is Ramanujan theta function. - Michael Somos, Jul 05 2014
Expansion of q^(1/12) * eta(q^2) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jan 09 2005
Euler transform of period 6 sequence [1, 0, 0, 0, 1, 0, ...]. - Michael Somos, Jan 09 2005
Given g.f. A(x), then B(q) = (A(q^12) / q)^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v^4 + (1 - u^3) * v^3 + 6*u^2*v^2 + (u^4 - u)*v + u^3. - Michael Somos, Jan 09 2005
The logarithmic derivative equals A186099. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} A186099(n) * x^n/n ) where A186099(n) = sum of divisors of n congruent to 1 or 5 mod 6. - Paul D. Hanna, Feb 17 2013
G.f.: exp( Sum_{n>=1} S(n,x) * x^n/n ) where S(n,x) = Sum_{d|n} d*(1-x^d)^(n/d). - Paul D. Hanna, Feb 17 2013
a(n) ~ Pi*sqrt(2) / sqrt(3*(12*n-1)) * BesselI(1, Pi*sqrt(12*n-1) / (3*sqrt(6))) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)) * (1 - (9/(8*Pi) + Pi/36)/sqrt(2*n) + (5 - 135/(4*Pi^2) + Pi^2/81)/(64*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
a(n) = (1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

More terms from Vladeta Jovovic, Jun 08 2003

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

A109697 Number of partitions of n into parts each equal to 1 mod 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 7, 7, 8, 10, 11, 12, 12, 13, 15, 17, 18, 19, 20, 23, 26, 28, 29, 31, 34, 38, 41, 43, 45, 50, 55, 60, 63, 66, 71, 79, 85, 90, 94, 101, 110, 120, 127, 133, 141, 153, 165, 176, 184, 195, 210, 227, 241, 254, 267, 286, 307, 327

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(11)=3 since 11 = 11 = 6+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1

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

Cf. A000041, A003105, A284097.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), this sequence (m=5), A109701 (m=6), A109703 (m=7), A277090 (m=8).

g:=1/product(1-x^(1+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); # Emeric Deutsch, Mar 30 2006

Table[Count[IntegerPartitions[n],?(Union[Mod[#,5]]=={1}&)],{n,0,75}] (* _Harvey P. Dale, Oct 08 2011 *)

G.f.: 1/product(1-x^(1+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(1/5) * exp(Pi*sqrt(2*n/15)) / (2^(8/5) * 3^(1/10) * 5^(2/5) * Pi^(4/5) * n^(3/5)) * (1 - (3*sqrt(3/10)/(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} A284097(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(5*j)). - Ilya Gutkovskiy, Jul 17 2019

More terms from Emeric Deutsch, Mar 30 2006

A284098 a(n) = Sum_{d|n, d == 1 (mod 6)} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 14, 8, 1, 1, 1, 1, 20, 1, 8, 1, 1, 1, 26, 14, 1, 8, 1, 1, 32, 1, 1, 1, 8, 1, 38, 20, 14, 1, 1, 8, 44, 1, 1, 1, 1, 1, 57, 26, 1, 14, 1, 1, 56, 8, 20, 1, 1, 1, 62, 32, 8, 1, 14, 1, 68, 1, 1, 8, 1, 1, 74, 38, 26, 20, 8, 14, 80, 1, 1

Offset: 1

Seiichi Manyama, Mar 20 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A086729, A109701.

Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), this sequence (k=6), A284099 (k=7), A284100 (k=8).

Table[Sum[If[Mod[d, 6] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 21 2017 *)

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 6)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%6==1]) # Indranil Ghosh, Mar 21 2017

G.f.: Sum_{k>=0} (6*k + 1)*x^(6*k+1)/(1 - x^(6*k+1)). - Ilya Gutkovskiy, Mar 21 2017
G.f.: Sum_{n >= 1} x^n*(1 + 5*x^(6*n))/(1 - x^(6*n))^2. - Peter Bala, Dec 19 2021
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - Amiram Eldar, Nov 26 2023

A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 6.

Convolution of A281244 and A280456 is A098884. - Vaclav Kotesovec, Jan 18 2017

			a(32) = 3 because we have [31, 1], [25, 7] and [19, 13].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Index entries for related partition-counting sequences

Cf. A000700, A016921, A109701, A169975, A261612, A280454, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 105; CoefficientList[Series[Product[(1 + x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 6] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(6*k+1)).

a(n) ~ exp(Pi*sqrt(n)/(3*sqrt(2)))/(2*2^(5/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/(144*sqrt(2)) - 9/(4*sqrt(2)*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 18 2017

A109703 Number of partitions of n into parts each equal to 1 mod 7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 31, 34, 38, 41, 43, 44, 45, 46, 50, 55, 60, 63, 65, 66, 68, 72, 79, 85, 90, 93, 95, 97, 103, 111, 120, 127, 132, 135

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(15)=3 because we have 15=8+1+1+1+1+1+1+1=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A284099.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), A109697 (m=5), A109701 (m=6), this sequence (m=7), A277090 (m=8).

g:=1/product(1-x^(1+7*j),j=0..20): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=0..77); # Emeric Deutsch, Apr 14 2006

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

G.f.: 1/product(1-x^(1+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(1/7) * exp(Pi*sqrt(2*n/21)) / (2^(11/7) * 3^(1/14) * 7^(3/7) * Pi^(6/7) * n^(4/7)) * (1 - (2*sqrt(6/7)/(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} A284099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Jul 17 2019

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

A277090 Expansion of Product_{k>=0} 1/(1 - x^(8*k+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 30, 31, 34, 38, 41, 43, 44, 45, 45, 46, 50, 55, 60, 63, 65, 66, 67, 68, 72, 79, 85, 90, 93, 95, 96, 98, 103, 111, 120, 127, 132, 135, 137, 139, 145

Offset: 0

Ilya Gutkovskiy, Sep 29 2016

nonn

Number of partitions of n into parts congruent to 1 mod 8.

More generally, the ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0) is Product_{k>=0} 1/(1 - x^(m*k+1)).

			a(10) = 2, because we have [9, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Index entries for related partition-counting sequences

Cf. A017077, A284100.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), A109697 (m=5), A109701 (m=6), A109703 (m=7).

CoefficientList[Series[QPochhammer[x, x^8]^(-1), {x, 0, 90}], x]

G.f.: Product_{k>=0} 1/(1 - x^(8*k+1)).
a(n) ~ exp((Pi*sqrt(n))/(2*sqrt(3)))*Gamma(1/8)/(4*3^(1/16)*(2*Pi)^(7/8)*n^(9/16)).
a(n) = (1/n)*Sum_{k=1..n} A284100(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

cp's OEIS Frontend

A003105 Schur's 1926 partition theorem: number of partitions of n into parts 6n+1 or 6n-1.

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

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A109697 Number of partitions of n into parts each equal to 1 mod 5.

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A284098 a(n) = Sum_{d|n, d == 1 (mod 6)} d.

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A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

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A109703 Number of partitions of n into parts each equal to 1 mod 7.

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A277090 Expansion of Product_{k>=0} 1/(1 - x^(8*k+1)).

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A304883 Expansion of Product_{k>=1} 1/(1-x^(3k-1)) Product_{k>=1} 1/(1-x^(6*k-5)).