A038348 -id:A038348 - cp's OEIS frontend

A379317 Positive integers with a unique even prime index.

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142

Offset: 1

Gus Wiseman, Dec 29 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
   3: {2}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}

Partitions of this type are counted by A038348 (strict A096911).
For all even parts we have A066207, counted by A035363 (strict A000700).
For no even parts we have A066208, counted by A000009 (strict A035457).
Positions of 1 in A257992.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A349158.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A038550, A087436.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],EvenQ]]==1&]

A116482 Triangle read by rows: T(n,k) is the number of partitions of n having k even parts (n>=0, 0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 4, 4, 2, 1, 5, 6, 3, 1, 6, 8, 5, 2, 1, 8, 11, 7, 3, 1, 10, 14, 10, 5, 2, 1, 12, 19, 14, 7, 3, 1, 15, 24, 19, 11, 5, 2, 1, 18, 31, 26, 15, 7, 3, 1, 22, 39, 34, 21, 11, 5, 2, 1, 27, 49, 45, 29, 15, 7, 3, 1, 32, 61, 58, 39, 22, 11, 5, 2, 1, 38, 76, 75, 52, 30

Offset: 0

Emeric Deutsch, Feb 17 2006

Row n has 1 + floor(n/2) terms. Row sums are the partition numbers (A000041).
Column 0 yields A000009. Column 1 yields A038348. Column 2 yields A096778.
Sum_{k=0..floor(n/2)}k*T(n,k) = A066898(n).
From Gregory L. Simay, Nov 02 2015: (Start)
If n<=2k+1, T(n+2k,k) = A000041(n), the number of partitions of n.
T(n+2k,k) = the convolution of A000009(n-2j),which are the strict partitions of (n-2j), and p(j+k,k), which are the number of partitions of j+k having exactly k parts.
T(n+2k,k) = e(n,k) where e(n,0)= A000009(n) and e(n,k) = e(n,k-1) + e(n-2k,k-1) + e(n-4k,k-1) + ... .(End)

			T(7,2) = 3 because we have [4,2,1], [3,2,2] and [2,2,1,1,1].
Triangle starts:
   1;
   1;
   1,  1;
   2,  1;
   2,  2,  1;
   3,  3,  1;
   4,  4,  2,  1;
   5,  6,  3,  1;
   6,  8,  5,  2,  1;
   8, 11,  7,  3,  1;
  10, 14, 10,  5,  2, 1;
  12, 19, 14,  7,  3, 1;
  15, 24, 19, 11,  5, 2, 1;
  18, 31, 26, 15,  7, 3, 1;
  22, 39, 34, 21, 11, 5, 2, 1;
  27, 49, 45, 29, 15, 7, 3, 1;
Added entries for n=8 through n=15. - _Gregory L. Simay_, Nov 03 2015
From _Gregory L. Simay_, Nov 03 2015: (Start)
T(15,4) = T(7+2*4,4) = p(7) = 15, since 7 < 2*4 + 1.
T(15,3) = T(13,2) + T(9,3) = 26 + 3 = 29.
T(10,1) = T(8+2*1,1) = T(8,0) + T(6,0) + T(4,0) + T(2,0) + T(0,0) = 6 + 4 + 2 + 1 + 1 = 14.
T(15,3) = T(9+2*3) = e(9,3) = e(9,2) + e(3,2) = (e(9,1) + e(5,1) + e(1,1)) + e(3,1) = q(9) + q(7) + q(5) + q(3) + q(1) + q(5) + q(3) + q(1) + q(1) + q(3) + q(1) = q(9) + q(7) + 2*q(5) + 3*q(3) + 4*q(1) = 8 + 5 + 2*3 + 3*2 + 4*1 = 29 = the convolution sum of q(9-2j) with p(3+j,3).
(End)

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000041, A038348, A066898, A103919.

g:=1/product((1-x^(2*j-1))*(1-t*x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser,x^n) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1
      then 0 else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [`if`(irem(i, 2)=0, 0$j, [][]), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq (T(n), n=0..30);  # Alois P. Heinz, Jan 07 2013

nn=8;CoefficientList[Series[Product[1/(1-x^(2i-1))/(1-y x^(2i)),{i,1,nn}],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Jan 07 2013 *)

G.f.: G(t,x) = 1/Product_{j>=1}((1-x^(2j-1))(1-tx^(2j))).
From Gregory L. Simay, Nov 03 2015: (Start)
G.f.: T(n+2k,k) = g.f.: e(n,k) = Product_{j>=1}(1-x^2*(k+j))*p(x), where p(x) is the g.f. of the partitions of x. If n<=2k+1, then the g.f. reduces to p(x).
T(n+2k,k) = T(n+2k-2,k-1) + T(n,k).
(End)

A096778 Number of partitions of n with at most two even parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 45, 58, 75, 95, 121, 151, 189, 234, 289, 354, 433, 526, 637, 768, 923, 1105, 1319, 1569, 1861, 2202, 2597, 3056, 3587, 4201, 4908, 5723, 6658, 7732, 8961, 10367, 11971, 13802, 15884, 18253, 20942, 23992, 27445, 31353

Offset: 0

Vladeta Jovovic, Aug 16 2004

Also number of partitions of n+4 with exactly two even parts. Example: a(3)=3 because the partitions of 7 with exactly two even parts are [4,2,1], [3,2,2] and [2,2,1,1,1]. a(n)=A116482(n+4,2). - Emeric Deutsch, Feb 21 2006

			a(3)=3 because we have [3],[2,1] and [1,1,1].

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=2. - N. J. A. Sloane, Aug 31 2014

Cf. A038348.

Cf. A116482.

CoefficientList[ Series[(1/((1 - x^2)*(1 - x^4)))/Product[1 - x^(2i + 1), {i, 0, 50}], {x, 0, 48}], x] (* Robert G. Wilson v, Aug 16 2004 *)

G.f.: (1/((1-x^2)*(1-x^4)))/Product(1-x^(2*i+1), i=0..infinity). More generally, g.f. for number of partitions of n with at most k even parts is (1/Product(1-x^(2*i), i=1..k))/Product(1-x^(2*i+1), i=0..infinity).

a(n) ~ 3^(3/4) * n^(1/4) * exp(Pi*sqrt(n/3)) / (8*Pi^2). - Vaclav Kotesovec, May 29 2018

More terms from Robert G. Wilson v, Aug 17 2004

More terms from Emeric Deutsch, Feb 21 2006

A173305 Triangle by columns, A000009 in every column shifted down twice for k > 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 4, 2, 1, 1, 5, 3, 2, 1, 6, 4, 2, 1, 1, 8, 5, 3, 2, 1, 10, 6, 4, 2, 1, 1, 12, 8, 5, 3, 2, 1, 15, 10, 6, 4, 2, 1, 1, 18, 12, 8, 5, 3, 2, 1, 22, 15, 10, 6, 4, 2, 1, 1, 27, 18, 12, 8, 5, 3, 2, 1, 32, 22, 15, 10, 6, 4, 2, 1, 1

Offset: 0

Gary W. Adamson, Feb 15 2010

Row sums = A038348.
Let the triangle = M. Limit_{n->oo} M^n = the partition numbers, A000041;
equivalent to the statement A000009(x) = A000041(x)/A000041(x^2), or
(1 + x + x^2 + 2x^3 + 2x^4 +3x^5 + 4x^6 + ...) = (1 + x + 2x^2 + 3x^3 + ...)/(1 + x^2 + 2x^4 + 3x^6 + 5x^8 + 7x^10 + ...).

			Triangle begins:
   1;
   1;
   1,  1;
   2,  1;
   2,  1,  1;
   3,  2,  1;
   4,  2,  1,  1;
   5,  3,  2,  1;
   6,  4,  2,  1,  1;
   8,  5,  3,  2,  1;
  10,  6,  4,  2,  1,  1;
  12,  8,  5,  3,  2,  1;
  15, 10,  6,  4,  2,  1,  1;
  18, 12,  8,  5,  3,  2,  1;
  22, 15, 10,  6,  4,  2,  1,  1;
  27, 18, 12,  8,  5,  3,  2,  1;
  32, 22, 15, 10,  6,  4,  2,  1,  1;
  38, 27, 18, 12,  8,  5,  3,  2,  1;
  46, 32, 22, 15, 10,  6,  4,  2,  1,  1;
  54, 38, 27, 18, 12,  8,  5,  3,  2,  1;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..10301 (rows 0..200 of the triangle, flattened).

Cf. A173306, A038348, A000009, A000041.

Table[PartitionsQ[n-2*k], {n, 0, 15}, {k, 0, n/2}] (* Paolo Xausa, Feb 21 2024 *)

Triangle by columns, A000009 in every column shifted down twice for k > 0.

T(n,k) = A000009(n-2*k). - Paolo Xausa, Feb 21 2024

A331852 a(n) is the number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then applying the bitwise XOR operator to the numbers represented by the blocks.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 8, 8, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 9, 9, 10, 10, 9, 9, 10, 10, 11, 11, 11, 11, 9, 9, 8, 8, 7, 7, 10, 10, 10, 10, 11, 11, 10, 10, 11, 11, 11, 11, 12

Offset: 0

Rémy Sigrist, Jan 29 2020

			For n = 6:
- the binary representation of 6 is "110",
- we can split it in 4 ways:
      "110" -> 6
      "1" and "10" -> 1 XOR 2 = 3
      "11" and "0" -> 3 XOR 0 = 3
      "1" and "1" and "0" -> 1 XOR 1 XOR 0 = 0
- we have 3 distinct values,
- hence a(6) = 3.

Rémy Sigrist, PARI program for A331852
Index entries for sequences related to binary expansion of n

See A331851 for similar sequences.

Cf. A038348.

PARI
```
See Links section.
```

a(2^k) = k+1 for any k >= 0.

Apparently a(2^k-1) = A038348(k) for any k >= 0.

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

Or, positive integers whose prime indices include at least one 1 or prime number.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]

A246579 G.f.: x^(k^2)/(mul(1-x^(2i),i=1..k)mul(1+x^(2*r-1),r=1..oo)) with k=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 2, -3, 5, -7, 11, -15, 21, -29, 39, -52, 69, -90, 116, -150, 190, -241, 303, -379, 470, -583, 716, -878, 1071, -1302, 1575, -1902, 2285, -2739, 3273, -3899, 4631, -5489, 6486, -7647, 8996, -10557, 12363, -14450, 16853, -19618, 22798, -26441

Offset: 0

N. J. A. Sloane, Aug 31 2014

sign

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=3.

k=0,1,2 give (apart perhaps from signs) A081360, A038348, A096778. Cf. A246590.

fU:=proc(k) local a,i,r;
a:=x^(k^2)/mul(1-x^(2*i),i=1..k);
a:=a/mul(1+x^(2*r-1),r=1..101);
series(a,x,101);
seriestolist(%);
end;
fU(3);

A246580 G.f.: x^(k^2)/(mul(1-x^(2i),i=1..k)mul(1+x^(2*r-1),r=1..oo)) with k=4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 2, -3, 5, -7, 11, -15, 22, -30, 41, -55, 74, -97, 127, -165, 212, -271, 344, -434, 544, -680, 843, -1043, 1283, -1573, 1919, -2336, 2829, -3419, 4116, -4942, 5914, -7062, 8405, -9983, 11825, -13976, 16479, -19392, 22767

Offset: 0

N. J. A. Sloane, Aug 31 2014

sign

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=4.

k=0,1,2 give (apart perhaps from signs) A081360, A038348, A096778. Cf. A246589.

fU:=proc(k) local a,i,r;
a:=x^(k^2)/mul(1-x^(2*i),i=1..k);
a:=a/mul(1+x^(2*r-1),r=1..101);
series(a,x,101);
seriestolist(%);
end;
fU(4);

A303904 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k^3)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Partial sums of A279329.

Cf. A000578, A003997, A036469, A038348, A248801, A279329, A302834.

b:= proc(n, i) option remember; `if`(n<0, 0,
     `if`(n=0, 1, `if`(n>i^2*(i+1)^2/4, 0, (t->
       b(t, min(t, i-1)))(n-i^3)+b(n, i-1))))
    end:
a:= proc(n) option remember; `if`(n<0, 0,
       b(n, iroot(n, 3))+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 02 2018

nmax = 91; CoefficientList[Series[1/(1 - x) Product[1 + x^k^3, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]

a(n) ~ exp(2^(7/4) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * 3^(5/4) / (2^(15/8) * sqrt(Pi) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/8) * n^(1/8)). - Vaclav Kotesovec, May 04 2018

A303905 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(k(k+1)/2)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 24, 24, 26, 29, 30, 31, 34, 36, 37, 41, 44, 44, 47, 50, 52, 56, 59, 62, 65, 67, 70, 73, 75, 79, 85, 89, 91, 96, 100, 102, 108, 113, 116, 123, 129, 132, 137, 142, 147, 153, 158, 162, 169, 176, 182, 190, 196, 201

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Partial sums of A024940.

Index entries for sequences related to partitions

Cf. A000217, A024940, A036469, A038348, A061208, A248801, A302835, A303668.

nmax = 69; CoefficientList[Series[1/(1 - x) Product[1 + x^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) / (2^(1/3) * (sqrt(2) - 1)^(1/3) * sqrt(3) * Pi^(2/3) * Zeta(3/2)^(1/3) * n^(1/6)). - Vaclav Kotesovec, May 04 2018

cp's OEIS Frontend

A379317 Positive integers with a unique even prime index.

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A116482 Triangle read by rows: T(n,k) is the number of partitions of n having k even parts (n>=0, 0<=k<=floor(n/2)).

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A096778 Number of partitions of n with at most two even parts.

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A173305 Triangle by columns, A000009 in every column shifted down twice for k > 0.

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A331852 a(n) is the number of distinct values obtained by partitioning the binary representation of n into consecutive blocks, and then applying the bitwise XOR operator to the numbers represented by the blocks.

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A379313 Positive integers whose prime indices are not all composite.

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A246579 G.f.: x^(k^2)/(mul(1-x^(2*i),i=1..k)*mul(1+x^(2*r-1),r=1..oo)) with k=3.

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A246580 G.f.: x^(k^2)/(mul(1-x^(2*i),i=1..k)*mul(1+x^(2*r-1),r=1..oo)) with k=4.

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A246579 G.f.: x^(k^2)/(mul(1-x^(2i),i=1..k)mul(1+x^(2*r-1),r=1..oo)) with k=3.

A246580 G.f.: x^(k^2)/(mul(1-x^(2i),i=1..k)mul(1+x^(2*r-1),r=1..oo)) with k=4.

A303905 Expansion of (1/(1 - x))Product_{k>=1} (1 + x^(k(k+1)/2)).