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 40 results. Next

A276625 Finitary numbers. Matula-Goebel numbers of rooted identity trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 39, 41, 47, 55, 58, 62, 65, 66, 78, 79, 82, 87, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 143, 145, 155, 158, 165, 167, 174, 179, 186, 195, 202, 205, 211, 218, 226, 235, 237, 246, 254, 257, 271, 274, 282, 286, 290, 293, 303, 310, 313, 317, 319, 327, 330
Offset: 1

Views

Author

Gus Wiseman, Sep 29 2016

Keywords

Comments

For any positive integer n the following are equivalent:
(1) n is a finitary number.
(2) prime(n) is a finitary number.
(3) n is a product of distinct finitary prime numbers.
These conditions are necessary and sufficient to define an infinite set of positive integers but do not specify how this set should be enumerated or indexed (is there a more natural way? viz. A215366) so here they are listed in increasing order of the corresponding Matula-Goebel numbers. The following comment is from A007097.
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012
Notes on use of the word "finitary": It is possible to have a finite set containing an infinite set. For example {{1,2,3...}} contains only one element. In contrast, a finitary set is a finite set whose elements are also required to be finitary sets. There are also no multisets allowed in finitary sets, although you can have repeated elements. For example {{{}},{{},{{}}}} is still considered a finitary set even though the multiset union {{},{},{{}}} is not a set. The finitary numbers of A276625 refer to multisets (trees) that don't involve any proper multisets (i.e. only sets). This is in addition to the (somewhat redundant) meaning of finitary sets as described in this comment on A004111 "There is a natural correspondence between rooted identity trees and finitary sets (sets whose transitive closure is finite); each node represents a set, with the children of that node representing the members of that set. When the set corresponding to an identity tree is written out using braces, there is one set of braces for each node of the tree; thus a(n) is also the number of sets that can be made using n pairs of braces. - Franklin T. Adams-Watters, Oct 25 2011" - Gus Wiseman, Oct 03 2016

Examples

			This sequence is proposed to be a canonical representation for rooted identity trees. The first thirty terms are the following.
1  ()           26 (()(()(())))     62  (()((((())))))
2  (())         29 ((()((()))))     65  (((()))(()(())))
3  ((()))       30 (()(())((())))   66  (()(())(((()))))
5  (((())))     31 (((((())))))     78  (()(())(()(())))
6  (()(()))     33 ((())(((()))))   79  ((()(((())))))
10 (()((())))   39 ((())(()(())))   82  (()((()(()))))
11 ((((()))))   41 (((()(()))))     87  ((())(()((()))))
13 ((()(())))   47 (((())((()))))   93  ((())((((())))))
15 ((())((()))) 55 (((()))(((())))) 94  (()((())((()))))
22 (()(((())))) 58 (()(()((()))))   101 ((()(()(()))))
We build the sequence as follows: The empty product is 1, so by (3) 1 is finitary. So is prime(1) = 2 by (2), so is prime(2) = 3 by (2), so is prime(3) = 5 by (2), so is 2*3 = 6 by (3), and so on. - _N. J. A. Sloane_, Oct 03 2016

Links

Crossrefs

Cf. A000040 (prime numbers), A000720 (PrimePi).
Cf. A004111 (identity trees), A116540 (set multipartitions). Contained in A005117 (squarefree numbers). Contains A076146 (ordinal numbers), A007097 (rooted paths), A277098 (finitary primes).
Cf. A206497 (automorphism group sizes), A348066 (reduce to identity tree).

Programs

  • Mathematica
    primeMS[n_Integer?Positive]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_Integer?Positive]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
fin[n_Integer?Positive]:=If[n===1,1,Block[{x=fin[n-1]+1},While[Not[finitaryQ[x]],x++];x]];
Array[fin,200]

Formula

a(n) = primePi(A277098(n)).

A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

Original entry on oeis.org

2, 4, 5, 8, 10, 13, 16, 17, 20, 23, 25, 26, 31, 32, 34, 37, 40, 43, 46, 47, 50, 52, 61, 62, 64, 65, 67, 68, 73, 74, 79, 80, 85, 86, 89, 92, 94, 100, 103, 104, 107, 109, 113, 115, 122, 124, 125, 128, 130, 134, 136, 137, 146, 148, 149, 151, 155, 158, 160, 163
Offset: 1

Views

Author

Gus Wiseman, Aug 16 2018

Keywords

Comments

A self-describing sequence.
The prime indices of m are the numbers k such that prime(k) divides m.
The sequence is monotonically increasing, since once a number is rejected it stays rejected. Sequence is closed under multiplication for a similar reason. - N. J. A. Sloane, Aug 26 2018

Examples

			After the initial term 2, the next term cannot be 3 because 3 has prime index 2, and 2 is already in the sequence. The next term could be 10, which has prime indices 1 and 3, but 4 (with prime index 1) is smaller. So a(2) = 4.

Links

Crossrefs

Cf. A000002, A001462, A079000, A079254, A214577, A276625, A277098, A280996, A303431.
For first differences see A317963, for primes see A317964.

Programs

  • Maple
    A:= NULL:
P:= {}:
for n  from 2 to 1000 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    A:= A, n;
    P:= P union {ithprime(n)};
  fi
od:
A; # Robert Israel, Aug 26 2018
  • Mathematica
    gaQ[n_]:=Or[n==0,And@@Cases[FactorInteger[n],{p_,k_}:>!gaQ[PrimePi[p]]]];
Select[Range[100],gaQ]

Extensions

Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019

A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 19, 21, 27, 29, 33, 37, 39, 43, 47, 49, 53, 57, 59, 61, 63, 71, 77, 79, 81, 83, 87, 89, 91, 97, 99, 101, 107, 111, 113, 117, 121, 127, 129, 131, 133, 139, 141, 143, 147, 149, 151, 159, 163, 169, 171, 173, 177, 179, 181, 183, 189, 193, 197
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:
   1: {}
   3: {2}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  19: {8}
  21: {2,4}
  27: {2,2,2}
  29: {10}
  33: {2,5}
  37: {12}
  39: {2,6}
  43: {14}
  47: {15}
  49: {4,4}
  53: {16}
  57: {2,8}
  59: {17}
  61: {18}
  63: {2,2,4}

Crossrefs

Complement of A324694. Prime indices are A304360.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098.
Cf. A324696, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

Programs

  • Mathematica
    aQ[n_]:=And@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>!aQ[PrimePi[p]]];
Select[Range[100],aQ]

A324694 Lexicographically earliest sequence of positive integers divisible by prime(m) for some m not already in the sequence.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95
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:
   2: {1}
   4: {1,1}
   5: {3}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  26: {1,6}
  28: {1,1,4}
  30: {1,2,3}

Crossrefs

Complement of A324695.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324696, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

Programs

  • Mathematica
    aQ[n_]:=!And@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>aQ[PrimePi[p]]];
Select[Range[100],aQ]

A302494 Products of distinct primes of squarefree index.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 143, 145, 146, 149, 155, 157, 158, 163
Offset: 1

Views

Author

Gus Wiseman, Apr 08 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
05: {{2}}
06: {{},{1}}
10: {{},{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
22: {{},{3}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
39: {{1},{1,2}}

Crossrefs

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A270995, A275024, A276625, A277098, A279785, A281113, A296120, A301754, A302242, A302243.

Programs

  • Mathematica
    Select[Range[100],Or[#===1,SquareFreeQ[#]&&And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]
  • PARI
    is(n) = if(bigomega(n)!=omega(n), return(0), my(f=factor(n)[, 1]~); for(k=1, #f, if(!issquarefree(primepi(f[k])) && primepi(f[k])!=1, return(0)))); 1 \\ Felix Fröhlich, Apr 10 2018

A324696 Lexicographically earliest sequence containing 1 and all numbers divisible by prime(m) for some m not already in the sequence.

Original entry on oeis.org

1, 3, 6, 7, 9, 11, 12, 14, 15, 18, 19, 21, 22, 24, 27, 28, 29, 30, 33, 35, 36, 38, 39, 41, 42, 44, 45, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 63, 66, 69, 70, 71, 72, 75, 76, 77, 78, 81, 82, 83, 84, 87, 88, 90, 91, 93, 95, 96, 97, 98, 99, 101, 102, 105
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:
   1: {}
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}

Crossrefs

Complement of A304360.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098.
Cf. A324694, A324695, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

Programs

  • Mathematica
    aQ[n_]:=n==1||Or@@Cases[FactorInteger[n],{p_,k_}:>!aQ[PrimePi[p]]];
Select[Range[100],aQ]

A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 43, 89, 185, 391, 840, 1822, 3975, 8727, 19280, 42841, 95661, 214490
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2018

Keywords

Examples

			The a(7) = 11 locally disjoint rooted identity trees:
((((((o))))))
((((o(o)))))
(((o((o)))))
((o(((o)))))
((o(o(o))))
(((o)((o))))
(o((((o)))))
(o((o(o))))
(o(o((o))))
((o)(((o))))
(o(o)((o)))

Links

Crossrefs

Cf. A000081, A004111, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316473, A316474, A316494.

Programs

  • Mathematica
    strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&(Intersection@@#=!={})&]=={}&]];
Table[Length[strut[n]],{n,20}]

A324704 Lexicographically earliest sequence containing 1 and all numbers > 2 divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84
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}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}

Crossrefs

Complement of A324698.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

Programs

A324698 Lexicographically earliest sequence containing 2 and all numbers > 1 whose prime indices already belong to the sequence.

Original entry on oeis.org

2, 3, 5, 9, 11, 15, 23, 25, 27, 31, 33, 45, 47, 55, 69, 75, 81, 83, 93, 97, 99, 103, 115, 121, 125, 127, 135, 137, 141, 155, 165, 197, 207, 211, 225, 235, 243, 249, 253, 257, 275, 279, 291, 297, 309, 341, 345, 347, 363, 375, 379, 381, 405, 411, 415, 419, 423
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:
   2: {1}
   3: {2}
   5: {3}
   9: {2,2}
  11: {5}
  15: {2,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  45: {2,2,3}
  47: {15}
  55: {3,5}
  69: {2,9}
  75: {2,3,3}
  81: {2,2,2,2}
  83: {23}
  93: {2,11}
  97: {25}
  99: {2,2,5}

Crossrefs

Complement of A324704.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324699, A324700, A324701, A324702, A324703, A324705.

Programs

A317710 Uniform tree numbers. Matula-Goebel numbers of uniform rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 62, 64, 65, 66, 67, 69, 70, 73, 77, 78, 79, 81, 82, 83, 85, 86, 87, 91, 93, 94, 95, 97
Offset: 1

Views

Author

Gus Wiseman, Aug 05 2018

Keywords

Comments

A positive integer n is a uniform tree number iff either n = 1 or n is a power of a squarefree number whose prime indices are also uniform tree numbers. A prime index of n is a number m such that prime(m) divides n.

Links

Crossrefs

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317589.
Cf. A317705, A317707, A317708, A317709, A317711 (complement), A317712, A317717, A317718.

Programs

  • Mathematica
    rupQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100],rupQ]
Showing 1-10 of 40 results. Next

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