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 1-7 of 7 results.

A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

Original entry on oeis.org

-3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, 0, -1, -2, 3, -3, 2, 1, -2, 0, 2, -2, -1, 0, 3, 0, 0, 0, -3, 2, -2, 2, -2, 0, 1, 2, -1, -2, 3, 0, -1, -2, 1, 0, -1, 2, 1, -2, 0, 2, -3, 1, -1, 2, -2, 3, 0, 0, -3, 2, 1, -2, 0, 2
Offset: 1

Views

Author

Gus Wiseman, Oct 01 2024

Keywords

Comments

The range is {-3, -2, -1, 0, 1, 2, 3}.

Examples

			The nonsquarefree numbers (A013929) are:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, ...
with first differences (A078147):
  4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, ...
with first differences (A376593):
  -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, ...

Crossrefs

The version for A000002 is A376604, first differences of A054354.
The first differences were A078147.
Zeros are A376594, complement A376595.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A064113 lists positions of adjacent equal prime gaps.
A114374 counts partitions into nonsquarefree numbers.
A246655 lists prime-powers exclusive, inclusive A000961.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376594 (inflections and undulations), A376595 (nonzero curvature).
Cf. A061398, A053797, A053806, A120992, A182853, A251092, A373198, A375707, A376305, A376306, A376311, A376312, A376655.

Programs

  • Mathematica
    Differences[Select[Range[100],!SquareFreeQ[#]&],2]
  • Python
    from math import isqrt
from sympy import mobius, factorint
def A376593(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    k = next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values()))
    return next(i for i in range(1-k,5-k) if any(d>1 for d in factorint(m+(k<<1)+i).values())) # Chai Wah Wu, Oct 02 2024

A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

Original entry on oeis.org

0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 3, -2, 0, 0, -1, 0, 1, -1, 2, -2, 0, 1, -1, 0, 1, -1, 2, -2, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 1, 2, -3, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 2, -2, 3, -2, -1
Offset: 1

Views

Author

Gus Wiseman, Oct 01 2024

Keywords

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...

Crossrefs

The version for A000002 is A376604, first differences of A054354.
The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
Zeros are A376591, complement A376592.
Sorted positions of first appearances are A376655.
A000040 lists the prime numbers, differences A001223.
A001597 lists perfect-powers, complement A007916.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree numbers: A076259 (first differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

Programs

  • Mathematica
    Differences[Select[Range[100],SquareFreeQ],2]
  • Python
    from math import isqrt
from sympy import mobius
def A376590(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    a = iterfun(f,n)
    b = iterfun(lambda x:f(x)+1,a)
    return a+iterfun(lambda x:f(x)+2,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A376591 Inflection and undulation points in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, 62, 65, 68, 71, 79, 84, 87, 96, 98, 99, 101, 103, 107, 110, 113, 118, 120, 121, 123, 128, 131, 134, 137, 142, 144, 145, 147, 152, 153, 155, 158, 163, 165, 166, 172, 175, 179, 184
Offset: 1

Views

Author

Gus Wiseman, Oct 04 2024

Keywords

Comments

These are points at which the second differences (A376590) are zero.

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with zeros at (A376591):
 1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, ...

Links

Crossrefs

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the zeros of A376590.
The complement is A376592.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376592 (nonzero curvature).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A376593, A376655.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],SquareFreeQ],2],0]

A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95
Offset: 1

Views

Author

Gus Wiseman, Oct 04 2024

Keywords

Comments

These are points at which the second differences (A376590) are nonzero.

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with nonzeros at (A376591):
  2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, ...

Links

Crossrefs

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the nonzeros of A376590.
The complement is A376591.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376591 (inflection and undulation points).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A373198, A376655.

Programs

  • Mathematica
    Join@@Position[Sign[Differences[Select[Range[100], SquareFreeQ],2]],1|-1]

A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

Original entry on oeis.org

1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, 206, 207, 216, 244, 245, 304, 325, 327, 402, 466, 484, 605, 801, 880, 939, 1033, 1055, 1077, 1234, 1281, 1721, 1890, 1891, 1906, 1940, 1960, 1962, 2257, 2290, 2410, 2880, 3150
Offset: 1

Views

Author

Gus Wiseman, Oct 06 2024

Keywords

Examples

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with first appearances (A376653):
  1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, ...

Crossrefs

For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
These are the sorted positions of first appearances in A376596.
The exclusive version is a(n) - 1 = A376654(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A053707, A069623, A093555, A174965, A251092, A333254, A361102.

Programs

  • Mathematica
    q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149
Offset: 1

Views

Author

Gus Wiseman, Oct 06 2024

Keywords

Examples

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

Crossrefs

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

Programs

  • Mathematica
    q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A376656 Sorted positions of first appearances in the second differences (A036263) of consecutive primes (A000040).

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, 242, 262, 294, 296, 428, 429, 446, 708, 756, 834, 1005, 1182, 1229, 1663, 1830, 1831, 1846, 1879, 2191, 2224, 2343, 2809, 3077, 3086, 3384, 3385, 3427, 3643, 3644, 3793, 3795, 4230
Offset: 1

Views

Author

Gus Wiseman, Oct 07 2024

Keywords

Comments

The prime numbers are (A000040):
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...
with first differences (A001223):
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, ...
with second differences (A036263):
1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, ...
with sorted first appearances at (A376656):
1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, ...

Crossrefs

These are the sorted positions of first appearances in A036263.
For first differences we had A373400(n) + 1, except initial terms.
For prime-powers instead of prime numbers we have A376653/A376654.
For squarefree instead of prime numbers we have A376655, sorted firsts of A376590.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
Cf. A064113, A175632, A251092, A258025, A258026, A333214, A373402, A376305, A376311.

Programs

  • Mathematica
    q=Differences[Select[Range[1000],PrimeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
Showing 1-7 of 7 results.

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