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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A375739 Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 5, 6, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88
Offset: 1

Views

Author

Gus Wiseman, Sep 10 2024

Keywords

Comments

Non-perfect-powers (A007916) are numbers with no proper integer roots.
An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.
Also non-perfect-powers x such that x + 1 is also a non-perfect-power.

Examples

			The initial anti-runs are the following, whose maxima are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

Crossrefs

For nonprime numbers we have A068780, runs A006093 with 2 removed.
For squarefree numbers we have A007674, runs A373415.
For nonsquarefree numbers we have A068781, runs A072284 minus 1 and shifted.
For prime-powers we have A006549, runs A373674.
For non-prime-powers we have A255346, runs A373677.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738
- last: A375739 (this)
- sum: A375737
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A045542, A046933, A053797, A216765, A373576, A373679, A375714.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Max/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100],radQ[#]&&radQ[#+1]&]

A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84
Offset: 1

Views

Author

Gus Wiseman, Oct 03 2024

Keywords

Comments

These are points at which the second differences (A376562) are zero.
Non-perfect-powers (A007916) are numbers without a proper integer root.

Examples

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with zeros at (A376588):
  3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, ...

Links

Crossrefs

The version for A000002 is empty.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros in A376562, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376589 (nonzero curvature).
For second differences: A064113 (prime), A376602 (composite), {} (perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power inclusive), A376600 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A294068, A336416, A361102, A376599, A376590.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ],2],0]

A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, 106, 107, 109, 110, 111, 112, 126, 127, 150, 151, 176, 177, 195, 196, 203, 204, 220, 221, 232, 233, 264, 265, 298, 299, 316, 317, 333, 334, 371, 372, 411, 412, 453, 454, 480, 481, 496
Offset: 1

Views

Author

Gus Wiseman, Oct 03 2024

Keywords

Comments

These are points at which the second differences (A376562) are nonzero.
Non-perfect-powers (A007916) are numbers without a proper integer root.

Examples

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with nonzeros at (A376589):
  1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, ...

Links

Crossrefs

For first differences we had A375706, ones A375740, complement A375714.
These are the positions of nonzeros in A376562, complement A376588.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376588 (inflection and undulation points).
For second differences: A064113 (prime), A376602 (composite), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A182853, A294068, A333254, A336416, A376599.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Sign[Differences[Select[Range[1000],radQ],2]],1|-1]

A378363 Greatest number <= n that is 1 or not a perfect-power.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 24, 24, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67
Offset: 1

Views

Author

Gus Wiseman, Nov 24 2024

Keywords

Comments

Perfect-powers (A001597) are 1 and numbers with a proper integer root, complement A007916.

Examples

			In the non-perfect-powers ... 5, 6, 7, 10, 11 ... the greatest term <= 8 is 7, so a(8) = 7.

Crossrefs

The union is A007916, complement A001597.
The version for prime numbers is A007917 or A151799, opposite A159477.
The version for prime-powers is A031218, opposite A000015.
The version for squarefree numbers is A067535, opposite A070321.
The version for perfect-powers is A081676, opposite A377468.
The version for composite numbers is A179278, opposite A113646.
Terms appearing multiple times are A375704, opposite A375703.
The run-lengths are A375706.
Terms appearing only once are A375739, opposite A375738.
The version for nonsquarefree numbers is A378033, opposite A120327.
The opposite version is A378358.
Subtracting n gives A378364, opposite A378357.
The version for non-prime-powers is A378367 (subtracted A378371), opposite A378372.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289.
A007916 lists the non-perfect-powers, differences A375706.
A069623 counts perfect-powers <= n.
A076411 counts perfect-powers < n.
A131605 lists perfect-powers that are not prime-powers.
A377432 counts perfect-powers between primes, zeros A377436.
Cf. A007918, A014210, A014234, A045542, A052410, A065514, A076412, A162966, A188951, A345531.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Table[NestWhile[#-1&,n,#>1&&perpowQ[#]&],{n,100}]
  • Python
    from sympy import mobius, integer_nthroot
def A378363(n):
    def f(x): return int(1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = n-f(n)
    m, k = a, f(a)+a
    while m != k: m, k = k, f(k)+a
    return m # Chai Wah Wu, Nov 26 2024
Previous Showing 11-14 of 14 results.

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