A349563
Dirichlet convolution of right-shifted Catalan numbers with A349452 (Dirichlet inverse of A011782, 2^(n-1)).
Original entry on oeis.org
1, -1, -2, -1, -2, 18, 68, 311, 1182, 4370, 15772, 56754, 203916, 734636, 2658096, 9661591, 35292134, 129511602, 477376556, 1766730706, 6563071700, 24464139348, 91478369336, 343051112482, 1289887370140, 4861912443284, 18367285959072, 69533415236716, 263747683314904, 1002241674463968, 3814985428350480, 14544633872450487
Offset: 1
-
s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, CatalanNumber[# - 1] * s[n/#] &]; Array[a, 32] (* Amiram Eldar, Nov 22 2021 *)
-
A000108(n) = (binomial(2*n, n)/(n+1));
A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349563(n) = sumdiv(n,d,A000108(d-1)*A349452(n/d));
A349569
Dirichlet convolution of A000027 (identity function) with A349452 (Dirichlet inverse of A011782, 2^(n-1)).
Original entry on oeis.org
1, 0, -1, -4, -11, -24, -57, -112, -243, -480, -1013, -1964, -4083, -8064, -16309, -32496, -65519, -130440, -262125, -523156, -1048263, -2095104, -4194281, -8383760, -16777015, -33546240, -67107609, -134200860, -268435427, -536835096, -1073741793, -2147417216, -4294962187, -8589803520, -17179867533, -34359463812
Offset: 1
-
s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#]*2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, # * s[n/#] &]; Array[a, 36] (* Amiram Eldar, Nov 22 2021 *)
-
A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349569(n) = sumdiv(n,d,d * A349452(n/d));
A371283
Heinz numbers of sets of divisors of positive integers. Numbers whose prime indices form the set of divisors of some positive integer.
Original entry on oeis.org
2, 6, 10, 22, 34, 42, 62, 82, 118, 134, 166, 218, 230, 254, 314, 358, 382, 390, 422, 482, 554, 566, 662, 706, 734, 798, 802, 862, 922, 1018, 1094, 1126, 1174, 1198, 1234, 1418, 1478, 1546, 1594, 1718, 1754, 1838, 1914, 1934, 1982, 2062, 2126, 2134, 2174, 2306
Offset: 1
The terms together with their prime indices begin:
2: {1}
6: {1,2}
10: {1,3}
22: {1,5}
34: {1,7}
42: {1,2,4}
62: {1,11}
82: {1,13}
118: {1,17}
134: {1,19}
166: {1,23}
218: {1,29}
230: {1,3,9}
254: {1,31}
314: {1,37}
358: {1,41}
382: {1,43}
390: {1,2,3,6}
Partitions of this type are counted by
A054973.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A355741 counts choices of a prime factor of each prime index.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],SameQ[prix[#],Divisors[Last[prix[#]]]]&]
A371285
Heinz number of the multiset union of the divisor sets of each prime index of n.
Original entry on oeis.org
1, 2, 6, 4, 10, 12, 42, 8, 36, 20, 22, 24, 390, 84, 60, 16, 34, 72, 798, 40, 252, 44, 230, 48, 100, 780, 216, 168, 1914, 120, 62, 32, 132, 68, 420, 144, 101010, 1596, 2340, 80, 82, 504, 4386, 88, 360, 460, 5170, 96, 1764, 200, 204, 1560, 42294, 432, 220, 336
Offset: 1
The prime indices of 105 are {2,3,4}, with divisor sets {{1,2},{1,3},{1,2,4}}, with multiset union {1,1,1,2,2,3,4}, with Heinz number 2520, so a(105) = 2520.
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
4: {1,1}
10: {1,3}
12: {1,1,2}
42: {1,2,4}
8: {1,1,1}
36: {1,1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
390: {1,2,3,6}
84: {1,1,2,4}
60: {1,1,2,3}
16: {1,1,1,1}
34: {1,7}
72: {1,1,1,2,2}
Product of
A275700 applied to each prime index.
The squarefree case is also
A275700.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A355741 counts choices of a prime factor of each prime index.
Cf.
A000720,
A003963,
A005179,
A007416,
A034729,
A054973,
A056239,
A321898,
A370820,
A371165,
A371181,
A371284,
A371288.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@Join@@Divisors/@prix[n],{n,100}]
A371286
Products of elements of A275700 (Heinz numbers of divisor sets). Numbers with a (necessarily unique) factorization into elements of A275700.
Original entry on oeis.org
1, 2, 4, 6, 8, 10, 12, 16, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 60, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 118, 120, 124, 128, 132, 134, 136, 144, 160, 164, 166, 168, 176, 192, 200, 204, 216, 218, 220, 230, 236, 240, 248, 252, 254, 256, 264, 268, 272, 288
Offset: 1
The terms together with their prime factorizations and unique factorizations into terms of A275700 begin:
1 = = ()
2 = 2 = (2)
4 = 2*2 = (2*2)
6 = 2*3 = (6)
8 = 2*2*2 = (2*2*2)
10 = 2*5 = (10)
12 = 2*2*3 = (2*6)
16 = 2*2*2*2 = (2*2*2*2)
20 = 2*2*5 = (2*10)
22 = 2*11 = (22)
24 = 2*2*2*3 = (2*2*6)
32 = 2*2*2*2*2 = (2*2*2*2*2)
34 = 2*17 = (34)
36 = 2*2*3*3 = (6*6)
40 = 2*2*2*5 = (2*2*10)
42 = 2*3*7 = (42)
44 = 2*2*11 = (2*22)
48 = 2*2*2*2*3 = (2*2*2*6)
60 = 2*2*3*5 = (6*10)
62 = 2*31 = (62)
64 = 2*2*2*2*2*2 = (2*2*2*2*2*2)
68 = 2*2*17 = (2*34)
72 = 2*2*2*3*3 = (2*6*6)
80 = 2*2*2*2*5 = (2*2*2*10)
82 = 2*41 = (82)
84 = 2*2*3*7 = (2*42)
88 = 2*2*2*11 = (2*2*22)
96 = 2*2*2*2*2*3 = (2*2*2*2*6)
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A355741 counts choices of a prime factor of each prime index.
-
nn=100;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1, {{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
s=Table[Times@@Prime/@Divisors[n],{n,nn}];
m=Max@@Table[Select[Range[2,k],prix[#] == Divisors[Last[prix[#]]]&],{k,nn}];
Join@@Position[Table[Length[Select[facs[n], SubsetQ[s,Union[#]]&]],{n,m}],1]
A245282
G.f.: Sum_{n>=1} Fibonacci(n+1) * x^n / (1 - x^n).
Original entry on oeis.org
1, 3, 4, 8, 9, 19, 22, 42, 59, 100, 145, 257, 378, 634, 999, 1639, 2585, 4255, 6766, 11051, 17736, 28804, 46369, 75316, 121402, 196798, 317870, 514868, 832041, 1347372, 2178310, 3526217, 5703035, 9230052, 14930382, 24162310, 39088170, 63252754, 102334536, 165591226, 267914297
Offset: 1
G.f.: A(x) = x + 3*x^2 + 4*x^3 + 8*x^4 + 9*x^5 + 19*x^6 + 22*x^7 +...
where by definition
A(x) = 1*x/(1-x) + 2*x^2/(1-x^2) + 3*x^3/(1-x^3) + 5*x^4/(1-x^4) + 8*x^5/(1-x^5) + 13*x^6/(1-x^6) + 21*x^7/(1-x^7) + 34*x^8/(1-x^8) + 55*x^9/(1-x^9) + 89*x^10/(1-x^10) + 144*x^11/(1-x^11) +...+ Fibonacci(n+1)*x^n/(1-x^n) +...
The g.f. is also given by the series identity:
A(x) = x*(1+x)/(1-x-x^2) + x^2*(1+x^2)/(1-x^2-x^4) + x^3*(1+x^3)/(1-x^3-x^6) + x^4*(1+x^4)/(1-x^4-x^8) + x^5*(1+x^5)/(1-x^5-x^10) + x^6*(1+x^6)/(1-x^6-x^12) + x^7*(1+x^7)/(1-x^7-x^14) +...+ x^n*(1+x^n)/(1-x^n-x^(2*n)) +...
And also we have the series:
A(x) = x*(1 + x) + x^2*((1+x)^2 + (1+x^2)) + x^3*((1+x)^3 + (1+x^3))
+ x^4*((1+x)^4 + (1+x^2)^2 + (1+x^4)) + x^5*((1+x)^5 + (1+x^5))
+ x^6*((1+x)^6 + (1+x^2)^3 + (1+x^3)^2 + (1+x^6))
+ x^7*((1+x)^7 + (1+x^7))
+ x^8*((1+x)^8 + (1+x^2)^4 + (1+x^4)^2 + (1+x^8))
+ x^9*((1+x)^9 + (1+x^3)^3 + (1+x^9))
+ x^10*((1+x)^10 + (1+x^2)^5 + (1+x^5)^2 + (1+x^10))
+ x^11*((1+x)^11 + (1+x^11))
+ x^12*((1+x)^12 + (1+x^2)^6 + (1+x^3)^4 + (1+x^4)^3 + (1+x^6)^2 + (1+x^12))
+...+ x^n * Sum_{d|n} (1 + x^d)^(n/d) +...
or, more explicitly,
A(x) = x*(1 + x) + x^2*(2 + 2*x + 2*x^2) + x^3*(2 + 3*x + 3*x^2 + 2*x^3)
+ x^4*(3 + 4*x + 8*x^2 + 4*x^3 + 3*x^4)
+ x^5*(2 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + 2*x^5)
+ x^6*(4 + 6*x + 18*x^2 + 22*x^3 + 18*x^4 + 6*x^5 + 4*x^6)
+ x^7*(2 + 7*x + 21*x^2 + 35*x^3 + 35*x^4 + 21*x^5 + 7*x^6 + 2*x^7)
+ x^8*(4 + 8*x + 32*x^2 + 56*x^3 + 78*x^4 + 56*x^5 + 32*x^6 + 8*x^7 + 4*x^8)
+ x^9*(3 + 9*x + 36*x^2 + 87*x^3 + 126*x^4 + 126*x^5 + 87*x^6 + 36*x^7 + 9*x^8 + 3*x^9)
+ x^10*(4 + 10*x + 50*x^2 + 120*x^3 + 220*x^4 + 254*x^5 + 220*x^6 + 120*x^7 + 50*x^8 + 10*x^9 + 4*x^10)
+ x^11*(2 + 11*x + 55*x^2 + 165*x^3 + 330*x^4 + 462*x^5 + 462*x^6 + 330*x^7 + 165*x^8 + 55*x^9 + 11*x^10 + 2*x^11)
+ x^12*(6 + 12*x + 72*x^2 + 224*x^3 + 513*x^4 + 792*x^5 + 952*x^6 + 792*x^7 + 513*x^8 + 224*x^9 + 72*x^10 + 12*x^11 + 6*x^12) + ...
-
with(numtheory): seq(add(combinat[fibonacci](d+1), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 10 2025
-
{a(n)=polcoeff(sum(m=1, n, fibonacci(m+1)*x^m/(1-x^m +x*O(x^n))), n)}
for(n=1,50, print1(a(n), ", "))
-
{a(n)=polcoeff(sum(m=1, n, x^m*(1+x^m)/(1-x^m-x^(2*m) +x*O(x^n)) ), n)}
for(n=1, 50, print1(a(n), ", "))
-
{a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,(1 + x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,50,print1(a(n),", "))
A323766
Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.
Original entry on oeis.org
1, 1, 4, 5, 12, 9, 25, 17, 42, 39, 64, 58, 132, 103, 173, 200, 303, 299, 491, 492, 756, 832, 1122, 1257, 1858, 1975, 2646, 3083, 4057, 4567, 6118, 6844, 8913, 10265, 12912, 14931, 19089, 21639, 27003, 31397, 38830, 44585, 55138, 63263, 77371, 89585, 108076
Offset: 0
The a(6) = 25 constant multiset partitions of constant multiset partitions of integer partitions of 6:
((6))
((52))
((42))
((33))
((3)(3))
((3))((3))
((411))
((321))
((222))
((2)(2)(2))
((2))((2))((2))
((3111))
((2211))
((21)(21))
((21))((21))
((21111))
((111111))
((111)(111))
((11)(11)(11))
((111))((111))
((11))((11))((11))
((1)(1)(1)(1)(1)(1))
((1)(1)(1))((1)(1)(1))
((1)(1))((1)(1))((1)(1))
((1))((1))((1))((1))((1))((1))
Cf.
A000005,
A000041,
A000837,
A001970,
A034729,
A047968,
A306017,
A319066,
A323764,
A323765,
A323774.
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Table[If[n==0,1,Sum[PartitionsP[d]*DivisorSigma[0,n/d],{d,Divisors[n]}]],{n,0,30}]
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a(n) = if (n==0, 1, sumdiv(n, d, numbpart(d)*numdiv(n/d))); \\ Michel Marcus, Jan 28 2019
A331638
Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.
Original entry on oeis.org
1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093
Offset: 1
For constant instead of strict blocks we have
A034729.
Without equal sizes we have
A116540 (normal set multipartitions).
Without strict blocks we have
A317583.
For distinct instead of equal sizes we have
A382428, non-strict blocks
A326517.
For equal sums instead of sizes we have
A382429, non-strict blocks
A326518.
A349564
Dirichlet convolution of A011782 [2^(n-1)] with A349450 [Dirichlet inverse of right-shifted Catalan numbers].
Original entry on oeis.org
1, 1, 2, 2, 2, -14, -68, -308, -1178, -4366, -15772, -56780, -203916, -734772, -2658088, -9662208, -35292134, -129514026, -477376556, -1766739436, -6563071972, -24464170892, -91478369336, -343051227304, -1289887370136, -4861912851116, -18367285963792, -69533416706328, -263747683314904, -1002241679797688
Offset: 1
-
s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * CatalanNumber[n/# - 1] &, # < n &]; a[n_] := DivisorSum[n, 2^(# - 1) * s[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
-
A000108(n) = (binomial(2*n, n)/(n+1));
memoA349450 = Map();
A349450(n) = if(1==n,1,my(v); if(mapisdefined(memoA349450,n,&v), v, v = -sumdiv(n,d,if(dA000108((n/d)-1)*A349450(d),0)); mapput(memoA349450,n,v); (v)));
A349564(n) = sumdiv(n,d,2^(d-1)*A349450(n/d));
A339916
The sum of 2^((d-1)/2) over all divisors of 2n+1.
Original entry on oeis.org
1, 3, 5, 9, 19, 33, 65, 135, 257, 513, 1035, 2049, 4101, 8211, 16385, 32769, 65571, 131085, 262145, 524355, 1048577, 2097153, 4194455, 8388609, 16777225, 33554691, 67108865, 134217765, 268435971, 536870913, 1073741825, 2147484699, 4294967365, 8589934593, 17179871235, 34359738369, 68719476737
Offset: 0
For n=7, a(7)=2^7+2^2+2^1+2^0=135 because the divisors of 15 are 15,5,3,1.
-
seq(add(2^((d-1)/2),d=numtheory:-divisors(2*n+1)),n=0..100); # Robert Israel, Dec 24 2020
-
A339916[n_]:=Block[{d=Divisors[2n+1]},Sum[2^((d[[k]]-1)/2),{k,Length[d]}]];Array[A339916,50,0]
-
a(n) = sumdiv(2*n+1, d, 2^((d-1)/2)); \\ Michel Marcus, Dec 23 2020
-
from sympy import divisors
def a(n): return sum(2**((d-1)//2) for d in divisors(2*n+1))
print([a(n) for n in range(37)]) # Michael S. Branicky, Dec 24 2020
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