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 10 results.

A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 4, 2, 8, 7, 14, 14, 21, 18, 33, 32, 50, 54, 72, 67, 103, 110, 145, 155, 201, 196, 271, 293, 372, 400, 493, 512, 647, 704, 858, 924, 1115, 1167, 1436, 1560, 1854, 2022, 2368, 2510, 3005, 3255, 3804, 4144, 4792, 5116, 5989, 6514, 7486
Offset: 1

Views

Author

Gus Wiseman, Apr 19 2018

Keywords

Examples

			The a(14) = 7 partitions are (932), (8321), (7421), (653), (6521), (6431), (5432).

Crossrefs

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303139.

Programs

  • Mathematica
    Table[Select[IntegerPartitions[n],UnsameQ@@#&&!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,20}]

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Crossrefs

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A303283 Squarefree numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

42, 78, 105, 114, 130, 174, 182, 195, 210, 222, 230, 231, 258, 266, 285, 318, 345, 357, 366, 370, 390, 406, 426, 429, 435, 455, 462, 470, 474, 483, 494, 518, 534, 546, 555, 570, 598, 602, 606, 610, 627, 638, 642, 645, 651, 663, 665, 678, 690, 705, 714, 715
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of strict integer partitions whose Heinz numbers belong to this sequence begins (4,2,1), (6,2,1), (4,3,2), (8,2,1), (6,3,1), (10,2,1), (6,4,1), (6,3,2), (4,3,2,1), (12,2,1), (9,3,1), (5,4,2), (14,2,1), (8,4,1), (8,3,2), (16,2,1), (9,3,2), (7,4,2), (18,2,1), (12,3,1), (6,3,2,1).

Crossrefs

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303282.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],SquareFreeQ[#]&&!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

Original entry on oeis.org

1, 1, 1, 2, 2, 8, 4, 28, 18, 120, 60, 888, 252, 5220, 1860, 22224, 9552, 311088, 59616, 2473056, 565920, 13627008, 4051872, 235039392, 33805440, 1932037632, 465239808, 20604487680, 4294865664, 386228795904, 35413136640
Offset: 0

Views

Author

Gus Wiseman, Jan 08 2019

Keywords

Comments

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

Examples

			The a(5) = 8 set partitions:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
   {{1,2},{3,4,5}}
   {{1,4},{2,3,5}}
   {{1,2,3},{4,5}}
   {{1,2,5},{3,4}}
   {{1,3,4},{2,5}}
   {{1,4,5},{2,3}}

Crossrefs

Cf. A000110, A051424, A084422, A085945, A186974, A187106, A302569, A302696, A303139, A303140, A320424, A320426, A320430, A320768.

Programs

  • Mathematica
    spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],CoprimeQ@@#&],Range[n]]],{n,10}]

Extensions

a(17)-a(18) from Alois P. Heinz, Jan 17 2019
a(19)-a(30) from Christian Sievers, Nov 28 2024

A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 0, 2, 0, 1
Offset: 1

Views

Author

Gus Wiseman, Apr 19 2018

Keywords

Examples

			Triangle begins:
01:   1
02:   0  1
03:   1  0  1
04:   1  0  0  1
05:   2  0  0  0  1
06:   2  1  0  0  0  1
07:   4  0  0  0  0  0  1
08:   4  1  0  0  0  0  0  1
09:   6  0  1  0  0  0  0  0  1
10:   7  2  0  0  0  0  0  0  0  1
11:  11  0  0  0  0  0  0  0  0  0  1
12:  10  2  1  1  0  0  0  0  0  0  0  1
13:  17  0  0  0  0  0  0  0  0  0  0  0  1
14:  17  4  0  0  0  0  0  0  0  0  0  0  0  1
15:  23  0  2  0  1  0  0  0  0  0  0  0  0  0  1
The strict partitions counted in row 12 are the following.
T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
T(12,2) = 2:  (10,2) (6,4,2)
T(12,3) = 1:  (9,3)
T(12,4) = 1:  (8,4)
T(12,12) = 1: (12)

Crossrefs

First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.
Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#===k&]],{n,15},{k,n}]

Formula

If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.

A320430 Number of set partitions of [n] where the elements of each non-singleton block are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 5, 10, 37, 60, 295, 658, 2621, 5368, 38535, 66506, 551529, 1234264, 5004697, 13721836, 143935131, 256835337, 2971237021, 6485081140, 35162930303, 95872321543, 1315397878401, 2399236456202, 25866803180347, 72374386475590, 563368417647305, 1479943119911866
Offset: 0

Views

Author

Gus Wiseman, Jan 08 2019

Keywords

Comments

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1.

Examples

			The a(4) = 10 set partitions: 1|2|3|4, 14|2|3, 13|2|4, 12|3|4, 1|23|4, 1|2|34, 134|2, 123|4, 14|23, 12|34.

Links

Crossrefs

Cf. A000110, A000258, A008277, A051424, A085945, A186974, A187106, A302569, A303139, A320424, A320426, A320423, A333517.

Programs

  • Mathematica
    spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Length[#]==1||CoprimeQ@@#&],Range[n]]],{n,10}]

Extensions

a(14)-a(15) from Alois P. Heinz, Jan 08 2019
a(16) from Alois P. Heinz, Mar 26 2020
a(17)-a(24) from Giovanni Resta, Mar 27 2020
a(25)-a(28) from Alois P. Heinz, Aug 03 2023

A320424 Number of set partitions of {1,...,n} where each block's elements are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 4, 13, 31, 140, 480, 2306, 9179, 58209, 249205, 1802970, 9463155, 63813439, 389176317, 3415876088, 20506436732, 195865505549, 1353967583125, 12006363947433, 93067012435816, 1019489483393439, 7779097711766093, 86684751695545733, 766357409555622203
Offset: 0

Views

Author

Gus Wiseman, Jan 08 2019

Keywords

Comments

Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

Examples

			The a(5) = 13 set partitions:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
   {{1},{2,3,4,5}}
   {{1,2},{3,4,5}}
   {{1,3},{2,4,5}}
   {{1,4},{2,3,5}}
   {{1,5},{2,3,4}}
   {{1,2,3},{4,5}}
   {{1,2,4},{3,5}}
   {{1,2,5},{3,4}}
   {{1,3,4},{2,5}}
   {{1,4,5},{2,3}}
    {{1,2,3,4,5}}
For example, {{1},{2,5},{3,4}} belongs to the list because {1} is relatively prime, {2,5} is relatively prime, and {3,4} is relatively prime. On the other hand, {{1},{2,4},{3,5}} is missing from the list because {2,4} is not relatively prime.

Crossrefs

Cf. A000110, A000258, A000837, A008277, A085945, A289508, A289509, A300486, A303139, A320423, A320430.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],And@@(GCD@@#==1&/@#)&]],{n,10}]
  • PARI
    lista(nn) = my(m, t=Mat([[], 1]), v, w, z); print1(1); for(n=1, nn, m=Map(); for(i=1, #t~, v=t[i, 1]; if(n-2+sum(j=1, #v, v[j]>1)Jinyuan Wang, Mar 02 2025

Extensions

a(13)-a(23) from Alois P. Heinz, Jan 08 2019
a(24)-a(26) from Jinyuan Wang, Mar 02 2025

A319299 Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 7, 2, 1, 1, 14, 1, 17, 3, 1, 1, 27, 2, 1, 34, 6, 1, 1, 55, 1, 63, 7, 3, 2, 1, 1, 100, 1, 119, 14, 1, 1, 167, 6, 2, 1, 209, 17, 3, 1, 1, 296, 1, 347, 27, 7, 2, 1, 1, 489, 1, 582, 34, 6, 3, 1, 1, 775, 14, 2, 1, 945, 55, 1, 1, 1254
Offset: 1

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Examples

			Triangle begins:
    1
    1   1
    2   1
    3   1   1
    6   1
    7   2   1   1
   14   1
   17   3   1   1
   27   2   1
   34   6   1   1
   55   1
   63   7   3   2   1   1
  100   1
  119  14   1   1
  167   6   2   1
  209  17   3   1   1
  296   1
  347  27   7   2   1   1
  489   1
  582  34   6   3   1   1

Links

Crossrefs

A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.
Cf. A018783, A027750, A078374, A078392, A289508, A289509, A303139, A305563, A319300.

Programs

  • Maple
    # with table A000837 obtained from that sequence
f:= proc(n) local D,d;
  D:= sort(convert(numtheory:-divisors(n),list),`>`);
  seq(A000837[d],d=D)
end proc:
map(f, [$1..60]); # Robert Israel, Jul 09 2020
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],GCD@@#==k&]],{n,20},{k,Divisors[n]}]

Formula

T(n,k) = A000837(n/A027750(n,k)).

A320435 Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 10, 5, 1, 1, 11, 19, 15, 6, 1, 1, 17, 34, 35, 21, 7, 1, 1, 21, 52, 69, 56, 28, 8, 1, 1, 27, 79, 125, 126, 84, 36, 9, 1, 1, 31, 109, 205, 251, 210, 120, 45, 10, 1, 1, 41, 154, 325, 461, 462, 330, 165, 55, 11, 1, 1, 45, 196
Offset: 1

Views

Author

Gus Wiseman, Jan 08 2019

Keywords

Comments

Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

Examples

			Triangle begins:
    1
    1    1
    1    3    1
    1    5    4    1
    1    9   10    5    1
    1   11   19   15    6    1
    1   17   34   35   21    7    1
    1   21   52   69   56   28    8    1
    1   27   79  125  126   84   36    9    1
    1   31  109  205  251  210  120   45   10    1
    1   41  154  325  461  462  330  165   55   11    1
    1   45  196  479  786  923  792  495  220   66   12    1
    1   57  262  699 1281 1715 1716 1287  715  286   78   13    1
The T(6,2) = 11 sets are: {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {5,6}. Missing from this list are: {2,4}, {2,6}, {3,6}, {4,6}.

Links

Crossrefs

Row sums are A085945.
Second column is A015614.
Cf. A000837, A186974, A187106, A289508, A289509, A300486, A303139, A320424, A320436.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n],{k}],GCD@@#==1&]],{n,10},{k,n}]
  • PARI
    T(n,k) = sum(d=1, n\k, moebius(d)*binomial(n\d, k)) \\ Andrew Howroyd, Jan 19 2023

Formula

T(n,k) = Sum_{d=1..floor(n/k)} mu(d)*binomial(floor(n/d), k). - Andrew Howroyd, Jan 19 2023

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72
Offset: 1

Views

Author

Gus Wiseman, Jan 09 2019

Keywords

Comments

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

Examples

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Links

Crossrefs

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)
  • PARI
    a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

Formula

a(n) = Product_{p prime <= n} floor(log_p(n)).
a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019
Showing 1-10 of 10 results.

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