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.

A325702 Number of integer partitions of n containing their multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 2, 1, 3, 3, 8, 7, 10, 13, 17, 19, 28, 35, 38, 51, 67, 81, 100, 128, 157, 195, 233, 285, 348, 427, 506, 613, 733, 873, 1063, 1263, 1503, 1802, 2131, 2537, 3005, 3565, 4171, 4922, 5820, 6775, 8001, 9333, 10860, 12739, 14840, 17206, 20029, 23248
Offset: 0

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A325755.

Examples

			The partition x = (4,3,1,1,1) has multiplicities (3,1,1), which are a submultiset of x, so x is counted under a(10).
The a(1) = 1 through a(11) = 7 partitions:
  (1)  (22)   (221)  (2211)  (3211)  (4211)   (333)    (3322)    (7211)
       (211)         (3111)          (32111)  (5211)   (3331)    (33221)
                                     (41111)  (32211)  (6211)    (52211)
                                                       (42211)   (53111)
                                                       (43111)   (322211)
                                                       (322111)  (332111)
                                                       (421111)  (431111)
                                                       (511111)

Crossrefs

Cf. A000041, A181819, A225486, A290689, A290822, A304360, A323014, A324736, A324748, A324753, A324843, A325254, A325755.

Programs

  • Mathematica
    submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],submultQ[Sort[Length/@Split[#]],#]&]],{n,0,30}]

A324751 Number of strict integer partitions of n containing no prime indices of the parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 2, 4, 5, 5, 6, 8, 8, 12, 10, 14, 13, 18, 19, 26, 25, 30, 34, 39, 40, 51, 55, 60, 71, 77, 90, 97, 111, 123, 136, 153, 170, 179, 216, 230, 264, 282, 322, 345, 385, 423, 470, 513, 573, 629, 686, 755, 834, 910, 1005, 1095, 1194, 1303, 1433
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

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 a(1) = 1 through a(13) = 8 strict integer partitions (A...D = 10...13):
  1   2   3   4    5   6    7    8    9    A    B     C     D
              31       42   43   71   54   64   65    75    76
                       51   52        63   73   83    84    85
                                      72   82   542   93    94
                                           91   731   A2    B2
                                                      B1    643
                                                            751
                                                            931

Crossrefs

The subset version is A324741, with maximal case A324743. The non-strict version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324742, A324748.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421
Offset: 0

Views

Author

Gus Wiseman, Mar 13 2019

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.
Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

Examples

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

Links

Crossrefs

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,10}]
  • PARI
    pset(n)={my(b=0, f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 15 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A324753 Number of integer partitions of n containing all prime indices of their parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 8, 14, 16, 23, 29, 40, 49, 66, 81, 109, 133, 172, 211, 274, 332, 419, 511, 640, 775, 965, 1165, 1434, 1730, 2109, 2530, 3083, 3683, 4447, 5308, 6375, 7573, 9062, 10730, 12786, 15104, 17909, 21095, 24937, 29284, 34488, 40421, 47450
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

Keywords

Comments

These could be described as transitive integer partitions.
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 a(1) = 1 through a(8) = 8 integer partitions:
  (1)  (11)  (21)   (211)   (41)     (321)     (421)      (3221)
             (111)  (1111)  (221)    (411)     (2221)     (4211)
                            (2111)   (2211)    (3211)     (22211)
                            (11111)  (21111)   (4111)     (32111)
                                     (111111)  (22111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

Crossrefs

The subset version is A324736. The strict case is A324748. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A000837, A001462, A007097, A051424, A112798, A276625, A279861, A290689, A290760.
Cf. A324697, A324737.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 15, 17, 31, 35, 57, 70, 111, 136, 213, 265, 405, 517, 763, 987, 1458, 1893, 2736, 3611, 5161, 6836, 9702
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Comments

A subset of totally transitive rooted trees (A318185).

Examples

			The a(1) = 1 through a(8) = 8 rooted trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
                (o(o))  (oo(o))  (oo(oo))   (ooo(oo))   (ooo(ooo))
                                 (ooo(o))   (oooo(o))   (oooo(oo))
                                 (o(o)(o))  (oo(o)(o))  (ooooo(o))
                                                        (oo(o)(oo))
                                                        (ooo(o)(o))
                                                        (o(o)(o)(o))
                                                        (o(o)(o(o)))

Crossrefs

The Matula-Goebel numbers of these trees are given by A324842.
Cf. A000081, A279861, A290689, A290822, A318185.
Cf. A324704, A324736, A324748, A324753, A324847, A324848, A324854.

Programs

  • Mathematica
    submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[submultQ[b,#],{b,#}]&];
Table[Length[rallt[n]],{n,10}]

A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 84, 168, 216, 432, 648, 1296, 2448, 4896, 6528, 13056, 19584, 39168, 77760, 155520, 229248, 458496, 790272, 1580544, 3128832, 6257664, 9386496, 18772992, 24081408, 48162816, 95938560, 191877120, 378335232, 756670464, 1135005696, 2270011392
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2019

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.
Also the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.

Examples

			The a(1) = 1 through a(6) = 16 subsets:
  {}  {}   {}     {}       {}         {}
      {2}  {3}    {3}      {4}        {4}
           {2,3}  {4}      {5}        {5}
                  {2,3}    {3,5}      {6}
                  {3,4}    {4,5}      {3,5}
                  {2,3,4}  {2,3,5}    {4,5}
                           {3,4,5}    {4,6}
                           {2,3,4,5}  {5,6}
                                      {2,3,5}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}
                                      {2,3,4,5}
                                      {2,3,5,6}
                                      {3,4,5,6}
                                      {2,3,4,5,6}
An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.

Links

Crossrefs

Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.
Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[2,n]],Function[set,SubsetQ[set,Select[Range[2,n],SubsetQ[set,PrimePi/@First/@FactorInteger[#]]&]]]]],{n,10}]
  • PARI
    pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k, b)->if(k>#p, 1, my(t=self()(k+1, b+(1<Andrew Howroyd, Aug 24 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

A324854 Lexicographically earliest sequence containing 1 and all positive integers > 2 whose prime indices already belong to the sequence.

Original entry on oeis.org

1, 4, 7, 8, 14, 16, 17, 19, 28, 32, 34, 38, 43, 49, 53, 56, 59, 64, 67, 68, 76, 86, 98, 106, 107, 112, 118, 119, 128, 131, 133, 134, 136, 139, 152, 163, 172, 191, 196, 212, 214, 224, 227, 236, 238, 241, 256, 262, 263, 266, 268, 272, 277, 278, 289, 301, 304
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

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.
A multiplicative semigroup: if x and y are in the sequence then so is x*y. - Robert Israel, Mar 19 2019

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  38: {1,8}
  43: {14}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  59: {17}
  64: {1,1,1,1,1,1}
  67: {19}
  68: {1,1,7}

Links

Crossrefs

Cf. A000002, A000720, A001462, A079254, A112798, A276625, A290822, A304360.
Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324855.

Programs

  • Maple
    S:= {1}:
for n from 3 to 400 do
  if map(numtheory:-pi, numtheory:-factorset(n)) subset S then
    S:= S union {n}
  fi
od:
sort(convert(S,list)); # Robert Israel, Mar 19 2019
  • Mathematica
    aQ[n_]:=Switch[n,1,True,2,False,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A324752 Number of strict integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 1, 4, 4, 4, 5, 6, 7, 10, 9, 12, 12, 16, 17, 22, 22, 26, 31, 35, 37, 46, 50, 55, 66, 70, 82, 90, 101, 114, 127, 143, 159, 172, 202, 215, 246, 267, 301, 327, 366, 402, 447, 491, 545, 600, 655, 722, 795, 875, 964, 1050, 1152, 1259, 1383
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

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 a(2) = 1 through a(17) = 12 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8  9   A   B    C   D    E    F    G    H
              42  43     54  64  65   75  76   86   87   97   98
                  52     63  73  83   84  85   95   96   A6   A7
                         72  82  542  93  94   A4   A5   C4   B6
                                      A2  B2   B3   B4   D3   C5
                                          643  752  C3   E2   D4
                                               842  D2   763  E3
                                                    654  943  854
                                                    843  A42  863
                                                    852       872
                                                              A52
                                                              B42
An example for n = 60 is (19,14,13,7,5,2), with prime indices:
  19: {8}
  14: {1,4}
  13: {6}
   7: {4}
   5: {3}
   2: {1}
None of these prime indices {1,3,4,6,8} belong to the partition, as required.

Crossrefs

The subset version is A324742, with maximal case is A324763. The non-strict version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324743, A324748, A324751, A324756, A324758.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 32, 36, 48, 54, 56, 64, 72, 78, 84, 96, 108, 112, 128, 144, 152, 156, 162, 168, 192, 196, 216, 224, 234, 252, 256, 288, 304, 312, 324, 336, 384, 392, 432, 444, 448, 456, 468, 486, 504, 512, 576, 588, 608, 624, 648, 672, 702
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Examples

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))

Crossrefs

A subsequence of A120383.
Cf. A000081, A007097, A112798, A279861, A290689, A290760, A290822, A318186.
Cf. A324704, A324736, A324748, A324753, A324843, A324847, A324848, A324854.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[Divisible[n,x],{x,primeMS[n]}],And@@qaQ/@primeMS[n]];
Select[Range[1000],qaQ]

A324855 Lexicographically earliest sequence containing 2 and all squarefree numbers > 2 whose prime indices already belong to the sequence.

Original entry on oeis.org

2, 3, 5, 11, 15, 31, 33, 47, 55, 93, 127, 137, 141, 155, 165, 211, 235, 257, 341, 381, 411, 465, 487, 517, 633, 635, 685, 705, 709, 771, 773, 811, 907, 977, 1023, 1055, 1285, 1297, 1397, 1457, 1461, 1483, 1507, 1551, 1621, 1705, 1905, 2055, 2127, 2293, 2319
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

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 sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   15: {2,3}
   31: {11}
   33: {2,5}
   47: {15}
   55: {3,5}
   93: {2,11}
  127: {31}
  137: {33}
  141: {2,15}
  155: {3,11}
  165: {2,3,5}
  211: {47}
  235: {3,15}
  257: {55}
  341: {5,11}
  381: {2,31}

Links

Crossrefs

Cf. A000002, A000720, A001462, A079254, A109298, A112798, A276625, A290822.
Cf. A324697, A324698, A324736, A324748, A324753, A324843, A324850, A324854.
Contains A007097 except for 1.

Programs

  • Maple
    S:= {2}: count:= 1:
for n from 3 by 2 while count < 100 do
  F:= ifactors(n)[2];
  if max(map(t -> t[2],F))=1 and {seq(numtheory:-pi(t[1]),t=F)} subset S then
     S:= S union {n}; count:= count+1;
  fi
od:
sort(convert(S,list)); # Robert Israel, Mar 22 2019
  • Mathematica
    aQ[n_]:=Switch[n,1,False,2,True,?(!SquareFreeQ[#]&),False,,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];
Select[Range[1000],aQ]
Showing 1-10 of 10 results.

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