A280954 -id:A280954 - cp's OEIS frontend

A305614 Expansion of Sum_{p prime} x^p/(1 + x^p).

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

Offset: 0

Gus Wiseman, Jun 06 2018

sign

a(n) is the number of prime divisors p|n such that n/p is odd, minus the number of prime divisors p|n such that n/p is even.

			The prime divisors of 12 are 2, 3. We see that 12/2 = 6, 12/3 = 4. None of those are odd, but both of them are even, so a(12) = -2.
The prime divisors of 30 are {2,3,5} with quotients {15,10,6}. One of these is odd and two are even, so a(30) = 1 - 2 = -1.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000005, A000607, A001221, A008683, A048165, A083399, A088705, A106404, A205745, A280954.

a:= n-> -add((-1)^(n/i[1]), i=ifactors(n)[2]):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 07 2018
# Alternative
N:= 1000: # to get a(0)..a(N)
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  R:= [seq(i,i=p..N,p)];
  W:= ;
  V[R]:= V[R]+W;
od:
[0,seq(V[i],i=1..N)]; # Robert Israel, Jun 07 2018

Table[Sum[If[PrimeQ[d], (-1)^(n/d - 1), 0], {d, Divisors[n]}], {n, 30}]

a(n) = -Sum_{p|n prime} (-1)^(n/p).
From Robert Israel, Jun 07 2018: (Start)
If n is odd, a(n) = A001221(n).
If n == 2 (mod 4), a(n) = 2 - A001221(n).
If n == 0 (mod 4) and n > 0, a(n) = -A001221(n). (End)
L.g.f.: log(Product_{k>=1} (1 + x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018

A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, 0, -1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -3, 3, -3, 4, -4, 5, -6, 6, -6, 8, -9, 9, -11, 12, -13, 14, -16, 19, -19, 21, -25, 26, -28, 32, -36, 38, -41, 46, -50, 55, -60, 65, -70, 77, -85, 91, -99, 108, -116, 126, -138, 149, -160, 174, -188, 202, -219, 237, -255, 274, -296

Offset: 0

N. J. A. Sloane

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005, A000040, A000586, A000607, A001221, A008683, A083399, A088705, A280954, A305614.

nn=20;
ser=Product[1/(1+x^p),{p,Select[Range[nn],PrimeQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}] (* Gus Wiseman, Jun 06 2018 *)

a(n) = A184198(n) - A184199(n). - Vaclav Kotesovec, Jan 11 2021

A298423 Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 7, 8, 10, 2, 15, 2, 16, 11, 9, 2, 28, 7, 10, 14, 22, 2, 37, 2, 25, 18, 12, 17, 55, 2, 13, 23, 52, 2, 55, 2, 40, 51, 15, 2, 95, 13, 44, 34, 53, 2, 79, 37, 85, 41, 18, 2, 185, 2, 19, 80, 91, 54, 112, 2, 87, 56, 122, 2

Offset: 0

Gus Wiseman, Jan 19 2018

nonn

Note that n is automatically divisible by the number of parts.

			The a(9) = 4 partitions: (9), (441), (711), (111111111).

Cf. A000005, A000041, A067538, A143773, A280954, A298422, A298424, A298426.

Table[Length[Select[IntegerPartitions[n],Function[ptn,And@@(Divisible[#-1,Length[ptn]]&/@ptn)]]],{n,60}]

G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(k*i)).

A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 48, 61, 78, 99, 124, 156, 195, 241, 299, 367, 450, 549, 670, 811, 982, 1183, 1422, 1704, 2040, 2431, 2894, 3435, 4070, 4811, 5679, 6684, 7858, 9217, 10797, 12623, 14738, 17174, 19988, 23225, 26951, 31227, 36141, 41759

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n such that each part is either 1 or not a perfect power (A001597, A007916).

			The a(5) = 6 integer partitions whose parts are 1's or not perfect powers are (5), (32), (311), (221), (2111), (11111).

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

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305631-A305635.

q:= n-> is(n=1 or 1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1-x^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, 12, 13, 17, 21, 25, 32, 39, 46, 58, 68, 83, 99, 121, 141, 171, 201, 239, 282, 336, 391, 463, 541, 635, 741, 868, 1005, 1174, 1359, 1580, 1826, 2115, 2436, 2814, 3237, 3726, 4276, 4914, 5618, 6445, 7359, 8414, 9594, 10947, 12453

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

a(n) is the number of integer partitions of n whose parts are not perfect powers (A001597, A007916).

			The a(9) = 5 integer partitions whose parts are not perfect powers are (72), (63), (522), (333), (3222).

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

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

q:= n-> is(1=igcd(map(i-> i[2], ifactors(n)[2])[])):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(q(d), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=100;
wadQ[n_]:=n>1&&GCD@@FactorInteger[n][[All,2]]==1;
ser=Product[1/(1-x^p),{p,Select[Range[nn],wadQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A280962 Number of integer partitions of the n-th even number or the n-th odd number using predecessors of prime numbers.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 26, 37, 53, 74, 101, 137, 183, 240, 314, 406, 520, 662, 837, 1049, 1311, 1627, 2008, 2469, 3021, 3678, 4466, 5397, 6499, 7804, 9338, 11137, 13251, 15715, 18589, 21938, 25823, 30322, 35535, 41544, 48471, 56448, 65602, 76097, 88128, 101867

Offset: 0

Gus Wiseman, Jan 11 2017

nonn

a(n) is both the number of integer partitions of even numbers {0, 2, 4, 6, ...} = A005843 using primes minus one {1, 2, 4, 6, ...} = A006093 and the number of integer partitions of odd numbers {1, 3, 5, 7, ...} = A005408 using primes minus one.

			The a(4)=11 partitions of 9 are:
(621),   (6111),
(441),   (4221),   (42111),   (411111),
(22221), (222111), (2211111), (21111111),
(111111111).

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

Cf. A005408, A005843, A006093, A280954.

b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,
      b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))
    end:
a:= n-> b(2*n, nextprime(2*n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 12 2017

nn=60;invser=Product[1-x^(Prime[n]-1),{n,PrimePi[2nn-1]}];
Table[SeriesCoefficient[1/invser,{x,0,n}],{n,1,2nn-1,2}]

G.f. G(x) satisfies: (1+x)*G(x^2) = Product_{p prime} 1/(1-x^(p-1)).

a(n) = A280954(A005408(n)) = A280954(A005843(n)).

A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 1, 2, 4, 3, 2, 4, 6, 5, 4, 7, 10, 8, 7, 11, 15, 13, 12, 17, 22, 19, 18, 25, 30, 28, 26, 35, 42, 39, 38, 49, 59, 56, 54, 69, 81, 77, 76, 94, 110, 105, 105, 127, 147, 141, 142, 171, 195, 189, 190, 227, 257, 250, 254, 299, 335, 328, 334, 390, 432

Offset: 0

Gus Wiseman, Jun 07 2018

nonn

			O.g.f.: 1/((1 - x)(1 + x^2)(1 - x^3)(1 - x^5)(1 + x^6)(1 - x^7)(1 + x^10)...).

Cf. A000607, A001597, A005117, A007916, A048165, A081362, A091050, A280954, A303707, A304779, A304817, A305614, A305630-A305635.

nn=20;
radQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]==1];
ser=Product[1/(1+(-x)^p),{p,Select[Range[nn],radQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

A341154 Number of partitions of 2*n into exactly n prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 71, 94, 116, 151, 182, 234, 282, 354, 424, 528, 627, 773, 914, 1113, 1311, 1585, 1854, 2227, 2599, 3095, 3597, 4262, 4931, 5811, 6704, 7855, 9035, 10542, 12080, 14036, 16047, 18561, 21161, 24397, 27736, 31866

Offset: 0

Ilya Gutkovskiy, Feb 06 2021

nonn

Cf. A000961, A023893, A181062, A280954, A341153.

nmax = 48; CoefficientList[Series[Product[1/(1 - Boole[PrimePowerQ[k + 1]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Boole[PrimePowerQ[d + 1]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

G.f.: Product_{p prime, k>=1} 1 / (1 - x^(p^k-1)).

A352064 Irregular triangle T(n,k) where row n lists the positions of n in A275314.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 7, 15, 20, 27, 36, 48, 64, 14, 30, 40, 54, 72, 96, 128, 21, 25, 28, 45, 60, 80, 81, 108, 144, 192, 256, 42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512, 11, 35, 63, 75, 84, 100, 112, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024

Offset: 1

Michael De Vlieger, Mar 02 2022

A table by Leonhard Euler.
Let L(n-1) be a partition of (n-1) whose parts m are restricted to predecessors of primes. Row n lists the products (m+1) for all such partitions L(n-1).
Greatest term in row n is 2^(n-1).
Least term in row p prime is p.

			Triangle begins:
   1;
   2;
   3,  4;
   6,  8;
   5,  9, 12, 16;
  10, 18, 24, 32;
   7, 15, 20, 27,  36,  48,  64;
  14, 30, 40, 54,  72,  96, 128;
  21, 25, 28, 45,  60,  80,  81, 108, 144, 192, 256;
  42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512;
  ...
Illustration of relationship of terms of row n and partitions of (n-1) such that all parts m are restricted to prime predecessors. We show the partitions in parentheses, adding 1 to each part m below in brackets to take the product. The products are terms in row n in this sequence.
      1 = (1);
          [2]
row 2:     2;
.
      2 = (2),    (1+1);
          [3]     [2*2]
row 3:     3,       4;
.
      3 = (2+1),  (1+1+1);
          [3*2]   [2*2*2]
row 4:     6,       8;
.
      4 = (4),    (2+2),    (2+1+1),     (1+1+1+1);
          [5]     [3*3]     [3*2*2]      [2*2*2*2]
row 5:     5,       9,        12,           16;
.
      5 = (4+1),  (2+2+1),  (2+1+1+1),   (1+1+1+1+1);
          [5*2]   [3*3*2]   [3*2*2*2]    [2*2*2*2*2]
row 6:    10,      18,        24,           32;
etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10673 (rows n = 1..40, flattened)
Leonhard Euler, Tentamen novae theoriae mvsicae ex certissimis harmoniae principiis dilvcide expositae, Petropoli, ex typographia Academiae scientiarvm (1739), 41.

Cf. A001414, A006093, A275314, A280954.

With[{n = 12}, Take[#, n] &@ Values@ KeySort@ PositionIndex@ Array[Total[Flatten[ConstantArray[#1 - 1, #2] & @@@ FactorInteger[#]]] &, 2^n]] // TableForm (* syntactically simple, or, more efficiently *)
f[m_] := Block[{s = {Prime@ PrimePi[m + 1] - 1}}, KeySort@ Merge[#, Identity] &@ Join[{1 -> {}}, Reap[Do[If[# <= m, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 1, s = DeleteCases[s, 1]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]]] &@ Total[s], {i, Infinity}]][[-1, -1]] ]]; Map[Union[Times @@ # & /@ #] &, Values@ f[40] + 1] // Flatten

A280954(n) = length of row n.

cp's OEIS Frontend

A305614 Expansion of Sum_{p prime} x^p/(1 + x^p).

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A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

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A298423 Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.

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A305630 Expansion of Product_{r = 1 or not a perfect power} 1/(1 - x^r).

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A305631 Expansion of Product_{r not a perfect power} 1/(1 - x^r).

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A280962 Number of integer partitions of the n-th even number or the n-th odd number using predecessors of prime numbers.

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A305632 Expansion of Product_{r = 1 or not a perfect power} 1/(1 + (-x)^r).

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A341154 Number of partitions of 2*n into exactly n prime powers (including 1).

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