This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(3)=4 partitions of 3 into powers of 2 or 3: 3.2^0, 1.2^0+1.2^1, 3.3^0, 1.3^1
a(12) = #{3+3+3+3, 6+3+3, 6+6, 7+5, 9+3, 12} = 6.
From _Gus Wiseman_, Jan 07 2019: (Start)
The a(3) = 1 through a(14) = 5 integer partitions (A = 10, ..., E = 14):
(3) (5) (6) (7) (53) (9) (A) (B) (C) (D) (E)
(33) (63) (55) (65) (66) (76) (77)
(333) (73) (533) (75) (A3) (95)
(93) (553) (B3)
(633) (733) (653)
(3333) (5333)
(End)
g:= product(1-x^(2^j),j=0..15)/product(1-x^i,i=1..75): gser:= series(g, x=0,62): seq(coeff(gser,x,n),n=0..59); # Emeric Deutsch, Mar 29 2006
Table[Length[Select[IntegerPartitions[n],And@@Not/@IntegerQ/@Log[2,#]&]],{n,20}] (* Gus Wiseman, Jan 07 2019 *)
a(5)=1 since the only partition without 1 as a part is 5 (a power of 5). a(6)=4 since 6 can be written as 6 (powers of 6), 3+3 (powers of 3) and 4+2 and 2+2+2 (both powers of 2).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):
(2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
(22) (33) (44) (333) (55) (66)
(42) (422) (82) (84)
(222) (2222) (442) (93)
(4222) (444)
(22222) (822)
(3333)
(4422)
(42222)
(222222)
(End)
Compare above to the example section of A379957. - _Antti Karttunen_, Jan 23 2025
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],SameQ@@radbase/@#]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)
a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&!ispower(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025
seq(n)={Vec(1 + sum(d=2, n, if(!ispower(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025
a(6)=10 since 6 can be written as 6 (powers of 6), 5+1 (5), 4+1+1 (4 or 2), 3+3 (3), 3+1+1+1 (3), 4+2 (2), 2+2+2 (2), 2+2+1+1 (2), 2+1+1+1+1 (2) and 1+1+1+1+1+1 (powers of anything).
From _Gus Wiseman_, Jan 01 2019: (Start)
The a(1) = 1 through a(8) = 15 integer partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (41) (33) (61) (44)
(111) (31) (221) (42) (331) (71)
(211) (311) (51) (421) (422)
(1111) (2111) (222) (511) (611)
(11111) (411) (2221) (2222)
(2211) (4111) (3311)
(3111) (22111) (4211)
(21111) (31111) (5111)
(111111) (211111) (22211)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
(End)
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@DeleteCases[#,1]&]],{n,30}] (* Gus Wiseman, Jan 01 2019 *)
The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (111) (22) (11111) (33) (1111111) (44) (333)
(1111) (42) (422) (111111111)
(222) (2222)
(111111) (11111111)
.
(A) (B) (C) (D) (E)
(55) (11111111111) (66) (1111111111111) (77)
(82) (84) (842)
(442) (93) (4442)
(4222) (444) (8222)
(22222) (822) (44222)
(1111111111) (3333) (422222)
(4422) (2222222)
(42222) (11111111111111)
(222222)
(111111111111)
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@#&]],{n,30}]
a(4)=6 distinct polynomials: 4 (for all n), n+2, 2n, n^2 (for n=2), n+1 (for n=3), n (for n=4).
a(2) = 5; 3 partitions for k=1: 2.1^0, 1.1^1+1.1^0, 2.1^1; and 2 for k=2: 2.2^0, 1.2^1
a(2)=4 distinct polynomials: 2 (for all n), n+1, 2n (for n=1), n (for n=2).
The a(2) = 1 through a(11) = 12 integer partitions (A = 10, B = 11):
(2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(22) (32) (33) (43) (44) (54) (55) (65)
(222) (52) (53) (63) (64) (74)
(322) (62) (72) (73) (83)
(332) (333) (433) (92)
(2222) (522) (532) (443)
(3222) (622) (533)
(3322) (632)
(22222) (722)
(3332)
(5222)
(32222)
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]
The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
(2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (66)
(22) (33) (44) (333) (55) (84)
(42) (422) (82) (93)
(222) (2222) (442) (444)
(4222) (822)
(22222) (3333)
(4422)
(42222)
(222222)
The a(20) = 16 integer partitions:
(10,10), (16,4),
(8,8,4), (16,2,2),
(5,5,5,5), (8,4,4,4), (8,8,2,2),
(4,4,4,4,4), (8,4,4,2,2),
(4,4,4,4,2,2), (8,4,2,2,2,2),
(4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
(4,4,2,2,2,2,2,2),
(4,2,2,2,2,2,2,2,2),
(2,2,2,2,2,2,2,2,2,2).
radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],And@@powsqfQ/@#,SameQ@@radbase/@#]&]],{n,30}]
a(n)={if(n==0, 1, sumdiv(n, d, if(d>1&&issquarefree(d), polcoef(1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)), n))))} \\ Andrew Howroyd, Jan 23 2025
seq(n)={Vec(1 + sum(d=2, n, if(issquarefree(d), -1 + 1/prod(j=1, logint(n, d), 1 - x^(d^j), Ser(1, x, 1+n)))))} \\ Andrew Howroyd, Jan 23 2025
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