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 1-3 of 3 results.

A376307 Run-sums of the sequence of first differences of squarefree numbers.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Sep 21 2024

Keywords

Examples

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with runs:
  (1,1),(2),(1,1),(3),(1),(2),(1,1),(2,2,2),(1,1),(3,3),(1,1),(2),(1,1), ...
with sums A376307 (this sequence).

Crossrefs

Run-sums of first differences of A005117.
Before taking run-sums we had A076259, ones A375927.
For the squarefree numbers themselves we have A373413.
For prime instead of squarefree numbers we have A373822, halved A373823.
For compression instead of run-sums we have A376305, ones A376342.
For run-lengths instead of run-sums we have A376306.
For prime-powers instead of squarefree numbers we have A376310.
For positions of first appearances instead of run-sums we have A376311.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed or anti-run compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A373197, A373198, A375707.

Programs

  • Mathematica
    Total/@Split[Differences[Select[Range[100],SquareFreeQ]]]

A373822 Sum of the n-th maximal run of first differences of odd primes.

Original entry on oeis.org

4, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 12, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 12, 4, 12, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 2, 10, 18, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 12, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4
Offset: 1

Views

Author

Gus Wiseman, Jun 22 2024

Keywords

Comments

Run-sums of A001223. For run-lengths instead of run-sums we have A333254.

Examples

			The odd primes are
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with sums a(n).

Crossrefs

Run-sums of A001223.
For run-lengths we have A333254, run-lengths of run-lengths A373821.
Dividing by two gives A373823.
A000040 lists the primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Programs

  • Mathematica
    Total/@Split[Differences[Select[Range[3,1000],PrimeQ]]]

A376310 Run-sums of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2
Offset: 1

Views

Author

Gus Wiseman, Sep 22 2024

Keywords

Examples

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

Crossrefs

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

Programs

  • Mathematica
    Total/@Split[Differences[Select[Range[100],PrimePowerQ]]]
Showing 1-3 of 3 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.