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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A376595 Points of nonzero curvature in the sequence of nonsquarefree numbers (A013929).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 14, 15, 16, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 38, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 77, 78, 79, 80, 84, 85, 86, 87, 89, 90, 91
Offset: 1

Views

Author

Gus Wiseman, Oct 04 2024

Keywords

Comments

These are points at which the second differences (A376593) are nonzero.

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, 54, ...
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, 3, ...
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, 3, ...
with nonzeros (A376594) at:
  1, 2, 3, 4, 6, 7, 8, 9, 10, 14, 15, 16, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, ...

Links

Crossrefs

The first differences were A078147.
These are the nonzeros of A376593.
The complement is A376594.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A114374 counts integer partitions into nonsquarefree numbers.
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376598 (prime-power), A376601 (non-prime-power).
For nonsquarefree numbers: A078147 (first differences), A376593 (second differences), A376594 (inflections and undulations).
Cf. A007674, A036263, A053797, A053806, A061398, A120992, A373198, A375707, A376306, A376312, A376590.

Programs

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

A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, 38, 40, 42, 46, 49, 53, 54, 60, 61, 63, 65, 67, 69, 71, 72, 78, 79, 81, 82, 83, 84, 85, 89, 91, 99, 101, 105, 106, 110, 111, 113, 115, 117, 118, 122, 124, 132, 134, 136, 138, 148
Offset: 1

Views

Author

Gus Wiseman, Oct 05 2024

Keywords

Comments

These are points at which the second differences (A376599) are nonzero.
Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, subtract 1 and shift left.

Examples

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with nonzero terms (A376601) at:
  1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, ...

Links

Crossrefs

For first differences we had A375735, ones A375713(n) - 1.
These are the nonzeros of A376599.
The complement is A376600.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A024619/A361102 list non-prime-powers inclusive.
A057820 gives first differences of prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376600 (inflections and undulations).
For nonzero curvature: A333214 (prime), A376603 (composite), A376588 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Programs

  • Mathematica
    Join@@Position[Sign[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2]],1|-1]
Previous Showing 11-12 of 12 results.

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