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 1-5 of 5 results.

A353931 Least run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 2, 6, 1, 2, 4, 7, 1, 8, 2, 2, 1, 9, 2, 6, 1, 6, 2, 10, 1, 11, 5, 2, 1, 3, 2, 12, 1, 2, 3, 13, 1, 14, 2, 3, 1, 15, 2, 8, 1, 2, 2, 16, 1, 3, 3, 2, 1, 17, 2, 18, 1, 4, 6, 3, 1, 19, 2, 2, 1, 20, 3, 21, 1, 2, 2, 4, 1, 22, 3, 8, 1
Offset: 1

Views

Author

Gus Wiseman, Jun 07 2022

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.
Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Examples

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.

Crossrefs

Positions of first appearances are A008578.
For run-lengths instead of run-sums we have A051904, greatest A051903.
For run-sums and binary expansion we have A144790, greatest A038374.
For run-lengths and binary expansion we have A175597, greatest A043276.
Distinct run-sums are counted by A353835, weak A353861.
The greatest run-sum is given by A353862.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

Programs

  • Mathematica
    Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A353862 Greatest run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 2, 6, 4, 3, 4, 7, 4, 8, 3, 4, 5, 9, 3, 6, 6, 6, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 3, 13, 4, 14, 5, 4, 9, 15, 4, 8, 6, 7, 6, 16, 6, 5, 4, 8, 10, 17, 3, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 4, 21, 12, 6, 8, 5, 6, 22, 4, 8
Offset: 1

Views

Author

Gus Wiseman, May 23 2022

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 run-sum of a sequence is the sum of any maximal consecutive constant subsequence.

Examples

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 4.

Crossrefs

Positions of first appearances are A008578.
For binary expansion we have A038374, least A144790.
For run-lengths instead of run-sums we have A051903.
Distinct run-sums are counted by A353835, weak A353861.
The least run-sum is given by A353931.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A182857, A316413, A353836, A353839, A353848, A353852, A353866.

Programs

  • Mathematica
    Table[Max@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A144795 A positive integer n is included if every 1 in binary n is next to at least one other 1.

Original entry on oeis.org

3, 6, 7, 12, 14, 15, 24, 27, 28, 30, 31, 48, 51, 54, 55, 56, 59, 60, 62, 63, 96, 99, 102, 103, 108, 110, 111, 112, 115, 118, 119, 120, 123, 124, 126, 127, 192, 195, 198, 199, 204, 206, 207, 216, 219, 220, 222, 223, 224, 227, 230, 231, 236, 238, 239, 240, 243, 246
Offset: 1

Views

Author

Leroy Quet, Sep 21 2008

Keywords

Comments

n is included if A144790(n) >= 2.
A173024 is a subsequence. - Reinhard Zumkeller, Feb 07 2010

Links

Crossrefs

Cf. A144790, A144794.
Cf. A004760, A173022, A005251, A173025. - Reinhard Zumkeller, Feb 07 2010
Complement of A377169.

Programs

  • Maple
    isA144795 := proc(n) local bind,i ; bind := convert(n,base,2) ; for i from 1 to nops(bind) do if i = 1 then if op(i,bind) = 1 and op(i+1,bind) = 0 then RETURN(false) : fi; elif i = nops(bind) then if op(i,bind) = 1 and op(i-1,bind) = 0 then RETURN(false) : fi; else if op(i,bind) = 1 and op(i-1,bind) = 0 and op(i+1,bind) = 0 then RETURN(false) : fi; fi; od: RETURN(true) ; end: for n from 3 to 400 do if isA144795(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 29 2008
  • Mathematica
    Select[Range@ 250, AllTrue[Map[Length, Select[Split@ IntegerDigits[#, 2], First@ # == 1 &]], # > 1 &] &] (* Michael De Vlieger, Aug 20 2017 *)

Extensions

Extended by R. J. Mathar, Sep 29 2008

A144789 Consider the runs of 0's in the binary representation of n, each of these runs being on the edge of the binary representation n and/or being bounded by 1's. a(n) = the length of the shortest such run (with positive length) of 0's in binary n. a(n) = 0 if there are no runs of 0's in binary n.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 2, 1, 1, 0, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 5, 4, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 6, 5, 1, 4, 2, 1, 1, 3, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 1, 1, 2, 1, 1
Offset: 1

Views

Author

Leroy Quet, Sep 21 2008

Keywords

Examples

			20 in binary is 10100. The runs of 0's are as follows: 1(0)1(00). The shortest of these runs contains exactly one 0's So a(20) = 1.

Crossrefs

Cf. A087117, A144790.

Programs

  • Maple
    A007814 := proc(n) local nshf,a ; a := 0 ; nshf := n ; while nshf mod 2 = 0 do nshf := nshf/2 ; a := a+1 ; od: a ; end: A144789 := proc(n) option remember ; local lp2,lp2sh,bind ; bind := convert(n,base,2) ; if add(i,i=bind) = nops(bind) then RETURN(0) ; fi; lp2 := A007814(n) ; if lp2 = 0 then A144789(floor(n/2)) ; else lp2sh := A144789(n/2^lp2) ; if lp2sh = 0 then lp2 ; else min(lp2,lp2sh) ; fi; fi; end: for n from 1 to 140 do printf("%d,",A144789(n)) ; od: # R. J. Mathar, Sep 29 2008
  • Mathematica
    Table[Min[Length/@Select[Split[IntegerDigits[n,2]],MemberQ[#,0]&]],{n,120}]/.\[Infinity]->0 (* Harvey P. Dale, Jul 24 2016 *)

Extensions

Extended by R. J. Mathar, Sep 29 2008

A144793 Consider the runs of 0's in the binary representation of n, each of these runs being on the edge of the binary representation n and/or being bounded by 1's. Consider also the runs of 1's in the binary representation of n, each of these runs being on the edge of the binary representation n and/or being bounded by 0's. A positive integer n is included in this sequence if the length of the shortest such run of 0's in binary n equals the length of the shortest such run of 1's in binary n.

Original entry on oeis.org

2, 5, 10, 11, 12, 13, 18, 20, 21, 22, 23, 26, 29, 34, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 56, 58, 61, 66, 69, 70, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 98, 101, 103, 104, 105, 106, 107, 109, 114, 115, 116, 117
Offset: 1

Views

Author

Leroy Quet, Sep 21 2008, Oct 07 2008

Keywords

Comments

This sequence contains those positive integers m where A144789(m) = A144790(m).

Examples

			1564 in binary is 11000011100. The runs of 0's are like this: 11(0000)111(00). The runs of 1's are like this: (11)0000(111)00. The shortest run of 0's contains two 0's. The shortest run of 1's contains two 1's. Since both the shortest run of 0's and the shortest run of 1's are of the same length, 1564 is included in this sequence.

Crossrefs

Cf. A090050, A144789, A144790.

Extensions

Extended by Ray Chandler, Nov 04 2008
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.