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 1-4 of 4 results.

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

Views

Author

Gus Wiseman, Sep 26 2024

Keywords

Comments

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

Examples

			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, ...

Crossrefs

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.

Programs

  • Mathematica
    Total/@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

Views

Author

Gus Wiseman, Sep 27 2024

Keywords

Examples

			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, ...

Crossrefs

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.

Programs

  • Mathematica
    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]}]

Formula

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

Views

Author

Gus Wiseman, Sep 27 2024

Keywords

Examples

			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, ...

Crossrefs

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.

Programs

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

A380595 a(n) is the first nonsquarefree number k such that the n consecutive nonsquarefree numbers starting with k are in arithmetic progression.

Original entry on oeis.org

4, 4, 16, 28, 28, 5050, 6348, 144946, 3348550, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568
Offset: 1

Views

Author

Robert Israel, Jan 27 2025

Keywords

Comments

Since multiples of 4 are not squarefree, the common difference of the arithmetic progression will be 1, 2 or 4 in each case.
For an arithmetic progression of length 10 or more with initial term even and common difference 2 or 4, there would be an odd multiple of 9 between the first and last term. Since multiples of 9 are not squarefree, these could not be consecutive nonsquarefree numbers. Thus for n >= 10, the common difference must be 1, and a(n) = A045882(n).

Examples

			a(2) = 4 because the 2 nonsquarefree numbers starting with 4 are 4, 6, forming an arithmetic progression with difference 2.
a(3) = 16 because the 3 nonsquarefree numbers starting with 16 are 16, 18, 20, forming an arithmetic progression with difference 2.
a(4) = a(5) = 28 because the 5 nonsquarefree numbers starting with 28 are 28, 32, 36, 40, 44, forming an arithmetic progression with difference 4.
a(6) = 5050 because the 6 nonsquarefree numbers starting with 5050 are 5050, 5052, 5054, 5056, 5058, 5060, forming an arithmetic progression with difference 2.
a(7) = 6348 because the 7 nonsquarefree numbers starting with 6348 are 6348, 6350, 6352, 6354, 6356, 6358, 6360, forming an arithmetic progression with difference 2.
a(8) = 144946, because the 8 nonsquarefree numbers starting with 144946 are 144946, 144948, 144950, 144952, 144954, 144956, 144958, 144960, forming an arithmetic progression with difference 2.
a(9) = 3348550, because the 9 nonsquarefree numbers starting with 3348550 are 3348550, 3348552, 3348554, 3348556, 3348558, 3348560, 3348562, 3348564, 3348566, forming an arithmetic progression with difference 2.

Crossrefs

Cf. A013929, A045882, A376267.

Programs

  • Maple
    nsf:= remove(numtheory:-issqrfree, [$4..4*10^6]):
S:= nsf[2..-1]-nsf[1..-2]: nS:= nops(S):
R:= NULL:
for m from 1 do
  found:= false;
  for t from 1 to nS +1-m do
    if nops(convert(S[t..t+m-1],set))=1 then R:= R,nsf[t]; found:= true; break fi
  od;
  if not found then break fi;
od:
R;

Extensions

a(1) = 4 prepended by David A. Corneth, Jan 28 2025
Showing 1-4 of 4 results.

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