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.

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A324700 Lexicographically earliest sequence containing 0 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

Original entry on oeis.org

0, 2, 4, 5, 8, 10, 11, 13, 16, 20, 22, 23, 25, 26, 31, 32, 37, 40, 43, 44, 46, 50, 52, 55, 59, 62, 64, 65, 73, 74, 80, 83, 86, 88, 89, 92, 100, 101, 103, 104, 110, 115, 118, 121, 124, 125, 128, 130, 131, 137, 143, 146, 148, 155, 160, 163, 166, 169, 172, 176
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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:
   0
   2: {1}
   4: {1,1}
   5: {3}
   8: {1,1,1}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}
  40: {1,1,1,3}
  43: {14}
  44: {1,1,5}

Crossrefs

Prime indices are A324701.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324702, A324703, A324704, A324705.

Programs

Formula

a(n) = A324701(n) - 1.

A324701 Lexicographically earliest sequence containing 1 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Original entry on oeis.org

1, 3, 5, 6, 9, 11, 12, 14, 17, 21, 23, 24, 26, 27, 32, 33, 38, 41, 44, 45, 47, 51, 53, 56, 60, 63, 65, 66, 74, 75, 81, 84, 87, 89, 90, 93, 101, 102, 104, 105, 111, 116, 119, 122, 125, 126, 129, 131, 132, 138, 144, 147, 149, 156, 161, 164, 167, 170, 173, 177
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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.

Crossrefs

Prime indices of A324700.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324702, A324703, A324704, A324705.

Programs

Formula

a(n) = A324700(n) + 1.

A324702 Lexicographically earliest sequence containing 2 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

Original entry on oeis.org

2, 5, 13, 25, 43, 65, 101, 125, 169, 193, 215, 317, 325, 505, 557, 559, 625, 701, 845, 965, 1013, 1075, 1181, 1313, 1321, 1585, 1625, 1849, 2111, 2161, 2197, 2509, 2525, 2785, 2795, 3125, 3505, 3617, 4049, 4057, 4121, 4225, 4343, 4639, 4825, 5065, 5297, 5375
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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 2 and numbers whose prime indices belong to A324703.

Examples

			The sequence of terms together with their prime indices begins:
    2: {1}
    5: {3}
   13: {6}
   25: {3,3}
   43: {14}
   65: {3,6}
  101: {26}
  125: {3,3,3}
  169: {6,6}
  193: {44}
  215: {3,14}
  317: {66}
  325: {3,3,6}
  505: {3,26}
  557: {102}
  559: {6,14}
  625: {3,3,3,3}
  701: {126}
  845: {3,6,6}
  965: {3,44}

Crossrefs

Prime indices > 1 are A324703.
Cf. A000002, A000720, A001222, A001462, A007097, A045965, A055396, A061395, A064989, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324700, A324701, A324704, A324705.

Programs

Formula

a(n) = A324703(n) - 1.

A324703 Lexicographically earliest sequence containing 3 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Original entry on oeis.org

3, 6, 14, 26, 44, 66, 102, 126, 170, 194, 216, 318, 326, 506, 558, 560, 626, 702, 846, 966, 1014, 1076, 1182, 1314, 1322, 1586, 1626, 1850, 2112, 2162, 2198, 2510, 2526, 2786, 2796, 3126, 3506, 3618, 4050, 4058, 4122, 4226, 4344, 4640, 4826, 5066, 5298, 5376
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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.

Crossrefs

Prime indices of A324702.
Cf. A000002, A000720, A001222, A001462, A007097, A045965, A055396, A061395, A064989, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324700, A324701, A324704, A324705.

Programs

Formula

a(n) = A324702(n) + 1.

A324705 Lexicographically earliest sequence containing 1 and all composite numbers divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106
Offset: 1

Views

Author

Gus Wiseman, Mar 11 2019

Keywords

Comments

A self-describing sequence, similar to A304360.
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:
   1: {}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  34: {1,7}
  35: {3,4}
  36: {1,1,2,2}

Crossrefs

Complement of A324697.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324698, A324699, A324700, A324701, A324702, A324703, A324704.

Programs

  • Mathematica
    aQ[n_]:=Switch[n,1,True,?PrimeQ,False,,!And@@Cases[FactorInteger[n],{p_,k_}:>!aQ[PrimePi[p]]]];
Select[Range[200],aQ]

A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 100, 112, 125, 180, 250, 252, 300, 352, 360, 392, 396, 405, 450, 468, 504, 540, 588, 600, 612, 675, 684, 720, 756, 792, 828, 832, 882, 900, 936, 1008, 1044, 1116, 1125, 1176, 1188, 1200, 1224, 1332, 1350, 1368, 1372, 1404, 1440, 1452, 1476
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2023

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.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    9: {2,2}
   12: {1,1,2}
   18: {1,2,2}
   40: {1,1,1,3}
  100: {1,1,3,3}
  112: {1,1,1,1,4}
  125: {3,3,3}
  180: {1,1,2,2,3}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  300: {1,1,2,3,3}
  352: {1,1,1,1,1,5}
  360: {1,1,1,2,2,3}
For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.

Crossrefs

Without taking median we have A109298, unordered A109297.
For mean instead of median we have A324570, counted by A114638.
For indices instead of multiplicities we have A360249, counted by A360245.
For indices instead of distinct indices we have A360454, counted by A360456.
These partitions are counted by A360455.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranks A359889.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A325347 = partitions with integer median, strict A359907, ranks A359908.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median.
A360005 gives median of prime indices (times two).
Cf. A000975, A359890, A359903, A360068, A360244, A360247, A360248.

Programs

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

A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 2916, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10368, 11664, 12500, 13824, 15552, 15625, 16384
Offset: 1

Views

Author

James Sellers, Apr 22 2000

Keywords

Comments

A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). [Reinhard Zumkeller, Apr 28 2012]
Heinz numbers of integer partitions where the multiplicity of each part k is at least prime(k). These partitions are counted by A325132. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

Examples

			8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
From _Gus Wiseman_, Apr 02 2019: (Start)
The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}
(End)

Links

Crossrefs

Cf. A048103, A001694, A036966.
Cf. A100716.
Cf. A056239, A109298, A112798, A115584, A117144, A124010, A276078, A324525, A324571, A325127, A325128, A325130, A325131, A325132.

Programs

  • Haskell
    a054744 n = a054744_list !! (n-1)
a054744_list = filter (\x -> and $
   zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012
  • Mathematica
    Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=p]&] (* Gus Wiseman, Apr 02 2019 *)

Formula

If n = Product p_i^e_i then p_i<=e_i for all i.
Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/(p^(p-1)*(p-1)) = 1.58396891058853238595.... - Amiram Eldar, Oct 24 2020

A324588 Heinz numbers of integer partitions of n into perfect squares (A001156).

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 23, 28, 32, 46, 49, 53, 56, 64, 92, 97, 98, 106, 112, 128, 151, 161, 184, 194, 196, 212, 224, 227, 256, 302, 311, 322, 343, 368, 371, 388, 392, 419, 424, 448, 454, 512, 529, 541, 604, 622, 644, 661, 679, 686, 736, 742, 776, 784, 827, 838
Offset: 1

Views

Author

Gus Wiseman, Mar 08 2019

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A011757.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
  14: {1,4}
  16: {1,1,1,1}
  23: {9}
  28: {1,1,4}
  32: {1,1,1,1,1}
  46: {1,9}
  49: {4,4}
  53: {16}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  92: {1,1,9}
  97: {25}
  98: {1,4,4}

Crossrefs

Cf. A001156, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A118914, A276078.
Cf. A109298, A324524, A324525, A324571, A324572, A324587.

Programs

  • Mathematica
    Select[Range[100],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Sqrt[PrimePi[p]]]]&]

A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141
Offset: 1

Views

Author

Gus Wiseman, Apr 01 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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears fewer than k times. Such partitions are counted by A087153.
The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) = 0.44070243286030291209... - Amiram Eldar, Feb 02 2021

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}
  47: {15}
  49: {4,4}

Links

Crossrefs

Cf. A056239, A062457, A087153, A109298, A112798, A115584, A118914, A276078.
Cf. A324524, A324525, A324571, A325127, A325130.

Programs

  • Mathematica
    Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k

A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125
Offset: 1

Views

Author

Gus Wiseman, May 18 2025

Keywords

Comments

First differs from A383532 in having 325.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

Examples

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}

Crossrefs

Ranking sequences are shown in parentheses below.
These partitions are counted by A383508.
A048767 is the Look-and-Say transform.
A048768 gives Look-and-Say fixed points, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A047966, A051903, A109298, A212166, A238745, A325368, A351592, A384006.

Programs

  • Mathematica
    disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&]
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