cp's OEIS Frontend

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.

Showing 1-10 of 31 results. Next

A002095 Number of partitions of n into nonprime parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 8, 12, 13, 17, 19, 26, 28, 37, 40, 52, 58, 73, 79, 102, 113, 139, 154, 191, 210, 258, 284, 345, 384, 462, 509, 614, 679, 805, 893, 1060, 1171, 1382, 1528, 1792, 1988, 2319, 2560, 2986, 3304, 3823, 4231, 4888, 5399, 6219, 6870
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Partial sums of A023895. - Emeric Deutsch, Apr 19 2006
Column k=0 of A222656. - Alois P. Heinz, May 29 2013

Examples

			a(6) = 3 from the partitions 6 = 1+1+1+1+1+1 = 4+1+1.

References

  • L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.
  • A. Murthy, Some new Smarandache sequences, functions and partitions, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000 (but beware errors).
  • Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 2.6.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000607, A018252, A096258.

Programs

  • Haskell
    a002095 = p a018252_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 15 2012
  • Maple
    g:=product((1-x^ithprime(j))/(1-x^j),j=1..60): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Apr 19 2006
  • Mathematica
    NonPrime[n_Integer] := FixedPoint[n + PrimePi[ # ] &, n + PrimePi[n]]; CoefficientList[ Series[1/Product[1 - x^NonPrime[i], {i, 1, 50}], {x, 0, 50}], x]
  • PARI
    first(n)=my(x='x+O('x^(n+1)),pr=1); forprime(p=2,n+1, pr *= (1-x^p)); pr/prod(i=1,n+1, 1-x^i) \\ Charles R Greathouse IV, Jun 23 2017

Formula

G.f.: Product_{i>0} (1-x^prime(i))/(1-x^i). - Vladeta Jovovic, Jul 31 2004

Extensions

More terms from James Sellers, Dec 23 1999
Corrected by Robert G. Wilson v, Feb 11 2002

A204389 Number of partitions of n into distinct composite parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 3, 2, 3, 1, 5, 3, 5, 4, 7, 4, 9, 7, 10, 9, 13, 10, 17, 14, 18, 18, 25, 22, 30, 27, 34, 36, 44, 40, 53, 52, 62, 65, 76, 74, 93, 95, 107, 113, 131, 133, 158, 164, 182, 195, 221, 229, 264, 276, 304, 329, 367, 383, 431
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 15 2012

Keywords

Examples

			a(10) = #{10, 6+4} = 2;
a(11) = #{ } = 0;
a(12) = #{12, 8+4} = 2;
a(13) = #{9+4} = 1;
a(14) = #{14, 10+4, 8+6} = 3;
a(15) = #{15, 9+6} = 2;
a(16) = #{16, 12+4, 10+6} = 3;
a(17) = #{9+8} = 1;
a(18) = #{18, 14+4, 12+6, 10+8, 8+6+4} = 5;
a(19) = #{15+4, 10+9, 9+6+4} = 3;
a(20) = #{20, 16+4, 14+6, 12+8, 10+6+4} = 5.

Links

Crossrefs

Cf. A023895, A096258, A000586, A000009.

Programs

  • Haskell
    a204389 = p a002808_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       b(n, i-1)+ `if`(i>n or isprime(i), 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, May 29 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i>n || PrimeQ[i], 0, b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

Formula

G.f.: (1/(1 + x))*Product_{k>=1} (1 + x^k)/(1 + x^prime(k)). - Ilya Gutkovskiy, Dec 31 2016
G.f.: product_(i>=1) (1+x^A002808(i)). - R. J. Mathar, Mar 01 2023

A379315 Number of strict integer partitions of n with a unique 1 or prime part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 3, 1, 3, 2, 7, 3, 7, 4, 10, 7, 15, 7, 17, 13, 23, 16, 31, 20, 37, 31, 48, 38, 62, 48, 76, 68, 93, 80, 119, 105, 147, 137, 175, 166, 226, 208, 267, 263, 326, 322, 407, 391, 481, 492, 586, 591, 714, 714, 849, 884, 1020, 1050, 1232, 1263
Offset: 0

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Comments

The "old" primes are listed by A008578.

Examples

			The a(10) = 2 through a(15) = 10 partitions:
  (8,2)  (11)     (9,3)    (13)     (9,5)    (8,7)
  (9,1)  (6,5)    (10,2)   (7,6)    (12,2)   (10,5)
         (7,4)    (6,4,2)  (8,5)    (8,4,2)  (11,4)
         (8,3)             (10,3)   (9,4,1)  (12,3)
         (9,2)             (12,1)            (14,1)
         (10,1)            (6,4,3)           (6,5,4)
         (6,4,1)           (8,4,1)           (8,4,3)
                                             (8,6,1)
                                             (9,4,2)
                                             (10,4,1)

Links

Crossrefs

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
For a unique composite part we have A379303, non-strict A379302 (ranks A379301).
Considering 1 nonprime gives A379305, non-strict A379304 (ranks A331915).
For squarefree instead of old prime we have A379309, non-strict A379308 (ranks A379316).
Ranked by A379312 /\ A005117 = squarefree positions of 1 in A379311.
The non-strict version is A379314.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A376682 gives k-th differences of old primes.
Cf. A000070, A023895, A034891, A036497, A038348, A095195, A175804, A204389, A257994, A302540.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
  • PARI
    seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k)) * prod(k=4, n, 1 + if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024

A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, 39, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 60, 61, 62, 65, 66, 67, 68, 69, 72, 74, 76, 78, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 99, 100, 101, 103, 105, 106, 107, 108
Offset: 1

Views

Author

Gus Wiseman, Oct 05 2024

Keywords

Comments

These are points at which the second differences (A073445) are zero.

Examples

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with zeros at (A376602):
  1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, ...

Links

Crossrefs

Partitions into composite numbers are counted by A023895, factorizations A050370.
For prime instead of composite we have A064113.
These are the positions of zeros in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For concavity in primes we have A258025/A258026, weak A333230/A333231.
For upward concavity (instead of zero) we have A376651, downward A376652.
The complement is A376603.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376603 (nonzero curvature), A376651 (concave-up), A376652 (concave-down).
For inflection and undulation points: A064113 (prime), A376588 (non-perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A000040, A036263, A251092, A333214, A333216, A333254.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],CompositeQ],2],0]

A379301 Positive integers whose prime indices include a unique composite number.

Original entry on oeis.org

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131
Offset: 1

Views

Author

Gus Wiseman, Dec 25 2024

Keywords

Comments

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.

Examples

			The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.

Crossrefs

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
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.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- 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.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

Programs

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

A379304 Number of integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014
Offset: 0

Views

Author

Gus Wiseman, Dec 27 2024

Keywords

Examples

			The a(2) = 1 through a(9) = 9 partitions:
  (2)  (3)   (31)   (5)     (42)     (7)       (62)       (54)
       (21)  (211)  (311)   (51)     (43)      (71)       (63)
                    (2111)  (3111)   (421)     (431)      (621)
                            (21111)  (511)     (4211)     (711)
                                     (31111)   (5111)     (4311)
                                     (211111)  (311111)   (42111)
                                               (2111111)  (51111)
                                                          (3111111)
                                                          (21111111)

Crossrefs

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A331915 = positions of one in A257994.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379305.
For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
Considering 1 prime gives A379314 (strict A379315), ranks A379312.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,_?PrimeQ]==1&]],{n,0,30}]

A379305 Number of strict integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866
Offset: 0

Views

Author

Gus Wiseman, Dec 27 2024

Keywords

Examples

			The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
  (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)
       (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)
                              (421)  (431)  (621)  (631)  (74)   (B1)
                                                          (83)   (642)
                                                          (92)   (651)
                                                          (821)  (741)
                                                                 (831)
                                                                 (921)

Crossrefs

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.
For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).
The non-strict version is A379304.
For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).
Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?PrimeQ]==1&]],{n,0,30}]

A379308 Number of integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 3, 5, 5, 1, 6, 9, 9, 2, 10, 14, 18, 6, 18, 24, 30, 11, 28, 39, 47, 24, 48, 63, 76, 41, 74, 95, 118, 65, 120, 149, 181, 107, 181, 221, 266, 169, 266, 335, 398, 262, 394, 487, 578, 391, 578, 697, 844, 592, 834, 997, 1198, 867
Offset: 0

Views

Author

Gus Wiseman, Dec 26 2024

Keywords

Examples

			The a(1) = 1 through a(11) = 5 partitions:
  (1)  (2)  (3)  .  (5)    (6)    (7)    .  (5,4)    (10)     (11)
                    (4,1)  (4,2)  (4,3)     (8,1)    (6,4)    (7,4)
                                            (4,4,1)  (8,2)    (8,3)
                                                     (9,1)    (9,2)
                                                     (4,4,2)  (4,4,3)

Crossrefs

If all parts are squarefree we have A073576 (strict A087188), ranks A302478.
If no parts are squarefree we have A114374 (strict A256012), ranks A379307.
For composite instead of squarefree we have A379302 (strict A379303), ranks A379301.
For prime instead of squarefree we have A379304, (strict A379305), ranks A331915.
The strict case is A379309.
For old prime instead of squarefree we have A379314, (strict A379315), ranks A379312.
Ranked by A379316, positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A013928, A023895, A034891, A072284, A073247, A120327, A175804, A376657, A377430.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

A379309 Number of strict integer partitions of n with a unique squarefree part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315
Offset: 0

Views

Author

Gus Wiseman, Dec 27 2024

Keywords

Examples

			The a(9) = 2 through a(15) = 7 partitions:
  (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
  (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
         (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
         (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
                                       (12,2)   (12,3)
                                       (8,4,2)  (8,4,3)
                                       (9,4,1)  (9,4,2)

Crossrefs

If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).
If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).
For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).
For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).
The non-strict version is A379308, ranks A379316.
For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).
Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]
  • PARI
    lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025

Extensions

More terms from Jinyuan Wang, Feb 21 2025

A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163
Offset: 1

Views

Author

Gus Wiseman, Oct 05 2024

Keywords

Comments

These are points at which the second differences (A073445) are nonzero.

Examples

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with nonzero terms at (A376603):
  2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...

Links

Crossrefs

Partitions into composite numbers are counted by A023895, factorizations A050370.
These are the positions of nonzero terms in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For prime instead of composite we have A333214.
The complement is A376602.
For upward concavity (instead of nonzero) we have A376651, downward A376652.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).
For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
Cf. A000961, A064113, A246655, A251092, A258025, A333254.

Programs

  • Mathematica
    Join@@Position[Sign[Differences[Select[Range[100],CompositeQ],2]],1|-1]
Showing 1-10 of 31 results. Next

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