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.

Previous Showing 11-17 of 17 results.

A325706 Heinz numbers of integer partitions containing all of their distinct multiplicities.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 36, 38, 40, 42, 46, 58, 60, 62, 66, 70, 74, 78, 82, 84, 86, 90, 94, 102, 106, 110, 112, 114, 118, 120, 122, 125, 126, 130, 132, 134, 138, 142, 146, 150, 154, 156, 158, 166, 170, 174, 178, 180, 182, 186, 190, 194, 198
Offset: 1

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers n divisible by the squarefree kernel of their "shadow" A181819(n).
The enumeration of these partitions by sum is given by A325705.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}

Crossrefs

Cf. A056239, A109297, A112798, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325705, A325707, A325755.

Programs

  • Mathematica
    Select[Range[100],#==1||SubsetQ[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]&]

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]

A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 23, 26, 32, 36, 43, 48, 57, 64, 74, 84, 98, 110, 127, 145, 165, 189, 215, 244, 277, 316, 356, 403, 455, 513, 577, 650, 727, 817, 913, 1024, 1143, 1279, 1425, 1592, 1773, 1977, 2198, 2448, 2717
Offset: 0

Views

Author

Gus Wiseman, May 18 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A325708.

Examples

			The initial terms count the following partitions:
   1: (1)
   3: (21)
   4: (211)
   5: (221)
   6: (321)
   6: (2211)
   7: (3211)
   8: (3221)
   8: (32111)
   9: (3321)
   9: (32211)
  10: (4321)
  10: (33211)
  10: (32221)
  10: (322111)
  11: (43211)
  11: (33221)
  11: (332111)
  11: (322211)
  12: (43221)
  12: (432111)
  12: (33321)
  12: (332211)
  12: (3222111)

Crossrefs

Cf. A000009 (partitions covering an initial interval), A055932, A109297, A114639, A114640, A290689, A324753, A325702, A325706, A325708.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A324757 Number of integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 4, 6, 9, 7, 14, 12, 19, 21, 28, 29, 41, 45, 56, 64, 81, 89, 114, 125, 154, 176, 211, 236, 288, 324, 383, 432, 514, 578, 678, 766, 891, 1006, 1176, 1306, 1525, 1711, 1966, 2212, 2538, 2839, 3258, 3646, 4150, 4647, 5288, 5891, 6698, 7472
Offset: 0

Views

Author

Gus Wiseman, Mar 17 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(10) = 9 integer partitions:
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)
            (22)       (33)   (43)  (44)    (54)   (55)
                       (42)   (52)  (422)   (63)   (64)
                       (222)        (2222)  (72)   (73)
                                            (333)  (82)
                                            (522)  (433)
                                                   (442)
                                                   (4222)
                                                   (22222)

Crossrefs

The subset version is A324742, with maximal case A324763. The strict case is A324752. The Heinz number version is A324761. An infinite version is A324695.
Cf. A000720, A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324764, A324742, A324753, A324756.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!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]

A325766 Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 4, 5, 4, 6, 7, 8, 6, 12, 11, 19, 16, 22, 22, 25, 32, 38, 45, 45, 51, 53, 71, 69, 85, 92, 118, 125, 147, 149, 184, 187, 230, 254, 290, 317, 372, 397, 449, 502, 544, 616, 680, 758, 841, 930, 1042, 1151, 1262
Offset: 0

Views

Author

Gus Wiseman, May 19 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A325767.

Examples

			The initial terms count the following partitions:
   1: (1)
   4: (2,1,1)
   5: (2,2,1)
   6: (2,2,1,1)
   7: (3,2,1,1)
   8: (3,2,1,1,1)
   9: (3,2,2,1,1)
  10: (3,2,2,1,1,1)
  11: (3,3,2,2,1)
  11: (3,3,2,1,1,1)
  11: (3,2,2,2,1,1)
  12: (4,3,2,1,1,1)
  13: (4,3,2,2,1,1)
  13: (4,3,2,1,1,1,1)
  13: (3,3,3,2,1,1)
  13: (3,3,2,2,2,1)
  13: (3,3,2,2,1,1,1)
  14: (4,3,2,2,1,1,1)
  14: (3,3,3,2,2,1)
  14: (3,3,3,2,1,1,1)
  14: (3,3,2,2,2,1,1)

Crossrefs

Cf. A000009 (partitions covering an initial interval), A055932, A114639, A114640, A290689, A324753, A325702, A325706, A325707, A325708, A325767.

Programs

  • Mathematica
    submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&submultQ[Sort[Length/@Split[#]],Sort[#]]&]],{n,0,30}]
Previous Showing 11-17 of 17 results.

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