A376306 -id:A376306 - cp's OEIS frontend

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

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

Gus Wiseman, Oct 04 2024

nonn

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

			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, ...

Gus Wiseman, Points of nonzero curvature in the nonsquarefree numbers.

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.

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

A376264 Run-sums of first differences (A078147) of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, 4, 2, 2, 16, 1, 3, 1, 3, 2, 2, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, 4, 1, 2, 1, 3, 1, 12, 1, 3, 4, 4, 4, 3, 1, 16, 1, 3, 4, 4, 4, 2, 3, 3, 4, 8, 1, 3, 4, 4, 3, 1, 3, 1, 8, 1, 3, 4, 1, 3, 4

Offset: 1

Gus Wiseman, Sep 26 2024

nonn

Does the image include all positive integers? I have only verified this up to 8.

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
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, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, ...

Before taking run-sums we had A078147.
For nonprime instead of nonsquarefree numbers we have A373822.
Positions of first appearances are A376265, sorted A376266.
For run-lengths instead of run-sums we have A376267.
For squarefree instead of nonsquarefree we have A376307.
For prime-powers instead of nonsquarefree numbers we have A376310.
For compression instead of run-sums we have A376312.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
Cf. A053797, A053806, A061398, A072284, A120992, A373197, A373413, A375707, A376305, A376306, A376311.

Total/@Split[Differences[Select[Range[1000],!SquareFreeQ[#]&]]]//Most

A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Sep 27 2024

nonn

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
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, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with lengths (A376267):
  1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...

Robert Israel, Table of n, a(n) for n = 1..10000

For prime instead of nonsquarefree numbers we have A333254.
For run-sums instead of run-lengths we have A376264.
For squarefree instead of nonsquarefree we have A376306.
For prime-powers instead of nonsquarefree numbers we have A376309.
For compression instead of run-lengths we have A376312.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
Cf. A007674, A053797, A053806, A072284, A112925, A120992, A373198, A375707, A376305, A376307, A376311, A380595.

nsf:= remove(numtheory:-issqrfree, [$4..1000]):
S:= nsf[2..-1]-nsf[1..-2]:
R:= NULL: x:= 4: t:= 1:
for i from 2 to nops(S) do
  if S[i] = x then t:= t+1
  else R:= R,t; x:= S[i]; t:= 1
  fi
od:
R; # Robert Israel, Jan 27 2025

Length/@Split[Differences[Select[Range[1000], !SquareFreeQ[#]&]]]//Most

A376265 Position of first appearance of n in A376264 (run-sums of first differences of nonsquarefree numbers), or 0 if there are none.

Original entry on oeis.org

2, 8, 3, 1, 6222, 14, 308540, 18

Offset: 1

Gus Wiseman, Sep 27 2024

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
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, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, ...
with first appearances at (A376265):
  2, 8, 3, 1, 6222, 14, 308540, 18, ...

This is the position of first appearance of n in A376264.
The sorted version is A376266.
For run-lengths instead of firsts of run-sums we have A376267.
For compression instead of firsts of run-sums we have A376312.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A333254 lists run-lengths of differences between consecutive primes.
A376305 gives run-compression of first differences of squarefree numbers.
A376307 gives run-sums of first differences of squarefree numbers.
Cf. A037201, A053797, A053806, A120992, A373198, A373413, A373822, A375707, A376306, A376310, A376311.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Total/@Split[Differences[Select[Range[10000],!SquareFreeQ[#]&]]]//Most;
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A376264(a(n)) = n.

A376266 Sorted positions of first appearances in A376264 (run-sums of first differences of nonsquarefree numbers).

Original entry on oeis.org

1, 2, 3, 8, 10, 14, 18, 53, 1437, 6222, 40874

Offset: 1

Gus Wiseman, Sep 27 2024

			The sequence of nonsquarefree numbers (A013929) is:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...
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, ...
with runs:
  (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...
with sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, ...
with first appearances at (A376266):
  1, 2, 3, 8, 10, 14, 18, 53, 1437, 6222, 40874, ...

These are the positions of first appearances in A376264.
The unsorted version is A376265.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A333254 lists run-lengths of differences between consecutive primes.
A376267 gives run-lengths of first differences of nonsquarefree numbers.
A376312 gives run-compression of first differences of nonsquarefree numbers.
A376305 gives run-compression of differences of squarefree numbers, ones A376342.
Cf. A037201, A053797, A053806, A120992, A373197, A373198, A373413, A373822, A375707, A376306, A376307, A376521.

q=Total/@Split[Differences[Select[Range[10000], !SquareFreeQ[#]&]]]//Most;
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

cp's OEIS Frontend

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

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A376264 Run-sums of first differences (A078147) of nonsquarefree numbers (A013929).

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A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

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A376265 Position of first appearance of n in A376264 (run-sums of first differences of nonsquarefree numbers), or 0 if there are none.

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Formula

A376266 Sorted positions of first appearances in A376264 (run-sums of first differences of nonsquarefree numbers).

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