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}]

A175599 The difference between maximal run length and minimal run length in binary representation of n.

Original entry on oeis.org

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

Views

Author

Juri-Stepan Gerasimov, Jul 23 2010

Keywords

Formula

a(n)=A043276(n)-A175597(n).

Extensions

Terms checked by D. S. McNeil, Nov 26 2010

A175611 Primes p such that A175599(p)=0.

Original entry on oeis.org

2, 3, 5, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 24 2010

Keywords

Comments

Primes p such that A043276(p)=A175597(p).

Links

Crossrefs

Cf. A000668, A002450.

Extensions

a(11)-a(14) from Charles R Greathouse IV, Jul 22 2016

A175665 The product of maximal run and minimal run lengths in the binary representation of n.

Original entry on oeis.org

1, 1, 4, 2, 1, 2, 9, 3, 2, 1, 2, 4, 2, 3, 16, 4, 3, 2, 2, 2, 1, 2, 3, 6, 2, 2, 2, 6, 3, 4, 25, 5, 4, 3, 3, 2, 2, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 8, 3, 2, 4, 2, 2, 2, 3, 9, 3, 3, 3, 8, 4, 5, 36, 6, 5, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 3, 4, 5, 10, 4, 3, 6, 2, 2, 2, 6, 3
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 04 2010

Keywords

Crossrefs

Cf. A175599.

Formula

a(n)=A043276(n)*A175597(n).

Extensions

Corrected by D. S. McNeil, Nov 26 2010

A175681 The sum of maximal run and minimal run lengths in binary representation of n.

Original entry on oeis.org

2, 2, 4, 3, 2, 3, 6, 4, 3, 2, 3, 4, 3, 4, 8, 5, 4, 3, 3, 3, 2, 3, 4, 5, 3, 3, 3, 5, 4, 5, 10, 6, 5, 4, 4, 3, 3, 3, 4, 4, 3, 2, 3, 3, 3, 4, 5, 6, 4, 3, 4, 3, 3, 3, 4, 6, 4, 4, 4, 6, 5, 6, 12, 7, 6, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 4, 5, 5, 4, 3, 3, 3, 2, 3, 4, 4, 3, 3, 3, 4, 4, 5, 6, 7, 5, 4, 5, 3, 3, 3, 5, 3, 3
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 08 2010

Keywords

Crossrefs

Cf. A175599, A175665.

Formula

a(n)=A043276(n)+A175597(n).
Showing 1-5 of 5 results.

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