A007674 -id:A007674 - cp's OEIS frontend

A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

1, 4, 7, 11, 12, 13, 14, 22, 25, 26, 29, 32, 35, 39, 40, 41, 42, 50, 53, 54, 61, 64, 70, 71, 72, 75, 78, 81, 82, 83, 84, 87, 90, 98, 99, 102, 109, 110, 117, 120, 123, 124, 127, 135, 139, 140, 144, 151, 154, 155, 156, 157, 160, 163, 168, 169, 170, 173, 176, 179

Offset: 1

Gus Wiseman, Sep 09 2024

nonn

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.

For prime numbers we have A029709.
Positions of 4's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],4]

Complement of A375709 U A375710 U A375711.

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

A215726 Numbers k such that the k-th triangular number is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 19, 20, 21, 22, 28, 29, 30, 33, 34, 37, 38, 41, 42, 43, 46, 51, 52, 57, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 73, 76, 77, 78, 82, 83, 84, 85, 86, 91, 92, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 115, 118, 122, 123

Offset: 1

Zak Seidov, Aug 22 2012

nonn

The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Granville and Ramaré, 1996). - Amiram Eldar, Feb 17 2021

			14 is a term because A000217(14) = 14*15/2 = 105 = 3*5*7.

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 184.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
Andrew Granville and Olivier Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika, Vol. 43, No. 1 (1996), pp. 73-107; alternative link.

A007674 is a subsequence.

Cf. A000217, A005117, A061304, A065474.

Select[Range[123],SquareFreeQ[#(#+1)/2]&]
Position[Accumulate[Range[150]],?(SquareFreeQ[#]&)]//Flatten//Rest (* _Harvey P. Dale, Jul 07 2020 *)

is(n)=issquarefree(n/gcd(n,2))&&issquarefree((n+1)/gcd(n+1,2)) \\ Charles R Greathouse IV, Jun 06 2013

Numbers k such that A000217(k) is squarefree. [corrected by Zak Seidov, Jun 05 2013]

A369020 Numbers k such that k and k+1 have the same maximal exponent in their prime factorization.

Original entry on oeis.org

2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 99, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Jan 12 2024

Differs from A358817 by having the terms 99, 165, 166, ..., which are not in A358817, and not having the terms 1, 440, 1331, 1575, ..., which are in A358817.
Numbers k such that A051903(k) = A051903(k+1).
If k is a term then k*(k+1) is a term of A362605.
The asymptotic density of this sequence is d(2) + Sum_{k>=2} (d(k) + d(k+1) - 2 * d2(k)) = 0.36939178586283962461..., where d(k) = Product_{p prime} (1 - 2/p^k) and d2(k) = Product_{p prime} (1 - 1/p^k - 1/p^(k+1)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A358817, A362605, A369022.

Subsequences: A007674, A071318, A369021.

emax[n_] := emax[n] = Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; Select[Range[200], emax[#] == emax[# + 1] &]

emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));
lista(kmax) = {my(e1 = 0, e2); for(k = 2, kmax, e2 = emax(k); if(e1 == e2, print1(k-1, ", ")); e1 = e2);}

A069977 Numbers k such that k and k+2 are squarefree.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 19, 21, 29, 31, 33, 35, 37, 39, 41, 51, 53, 55, 57, 59, 65, 67, 69, 71, 77, 83, 85, 87, 89, 91, 93, 95, 101, 103, 105, 107, 109, 111, 113, 127, 129, 131, 137, 139, 141, 143, 149, 155, 157, 159, 161, 163, 165, 177, 179, 181, 183, 185, 191, 193

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 01 2002

nonn

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Bin Chen, On almost-prime $k$-tuples, arXiv preprint (2023). arXiv:2301.05044 [math.NT], 2023.
Leon Mirsky, On the frequency of pairs of square-free numbers with a given difference, Bull. Amer. Math. Soc., Vo. 55, No. 10 (1949), pp. 936-939.

Cf. A007674, A065474.

f[n_] := Union[ Transpose[ FactorInteger[n]] [[ -1]]] [[ -1]]; Select[ Range[200], f[ # ] < 2 && f[ # + 2] < 2 & ]
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,200}],{1,,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Oct 06 2017 *)

is(n)=n%2 && issquarefree(n) && issquarefree(n+2) \\ Charles R Greathouse IV, May 15 2016

Edited and extended by Robert G. Wilson v, May 04 2002

1 prepended by Vladimir Joseph Stephan Orlovsky, Mar 30 2011

A075424 a(n) = A075423(A075423(n)).

Original entry on oeis.org

0, 1, 0, 1, 4, 5, 0, 1, 2, 9, 4, 5, 12, 13, 0, 1, 4, 5, 2, 9, 20, 21, 4, 1, 4, 1, 12, 13, 28, 29, 0, 1, 32, 33, 4, 5, 36, 37, 2, 9, 40, 41, 20, 13, 14, 45, 4, 5, 2, 9, 4, 25, 4, 5, 12, 13, 56, 57, 28, 29, 60, 9, 0, 1, 64, 65, 32, 33, 68, 69, 4, 5, 72, 13, 36, 37, 76, 77, 2, 1, 2, 81, 40, 41

Offset: 2

Reinhard Zumkeller, Sep 15 2002

From Robert Israel, Apr 11 2019: (Start)
a(n) == n (mod 2).
a(n) = 0 iff n is in A000079.
a(n) = n-2 iff n-1 is in A007674. (End)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000079, A007674, A075423, A075425.

A075423:= n -> convert(numtheory:-factorset(n),`*`)-1:
map(A075423@@2, [$2..100]); # Robert Israel, Apr 11 2019

A172187 Numbers k such that k and k+1 are squarefree but 2*k+1 is not.

Original entry on oeis.org

13, 22, 37, 58, 73, 85, 94, 122, 130, 137, 157, 166, 181, 193, 202, 229, 237, 238, 253, 262, 265, 301, 302, 310, 318, 346, 373, 382, 409, 418, 433, 437, 445, 454, 462, 465, 481, 514, 517, 526, 537, 541, 553, 562, 589, 598, 634, 661, 662, 670, 697, 706, 733

Offset: 1

Artur Jasinski, Jan 28 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A007674 and A172186.

Select[Range[750], And @@ SquareFreeQ /@ {#, # + 1} && !SquareFreeQ[2# + 1] &] (* Amiram Eldar, Mar 22 2020 *)

A194002 Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.

Original entry on oeis.org

1, 65, 137, 209, 217, 281, 353, 433, 641, 713, 785, 793, 857, 937, 1001, 1217, 1289, 1361, 1433, 1505, 1577, 1657, 1793, 1865, 1937, 2081, 2089, 2233, 2305, 2377, 2441, 2513, 2585, 2665, 2729, 2801, 2953, 3017, 3089, 3161, 3241, 3305, 3313, 3457, 3529, 3593

Offset: 1

Franklin T. Adams-Watters, Aug 10 2011

nonn

k, k+1, k+2, k+4, k+5, and k+6 are squarefree; k+3 is divisible by 4 but no higher power of 2 and no other prime squared.
From Amiram Eldar, Nov 28 2023: (Start)
All the terms are of the form 8*k + 1.
The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 14, 140, 1384, 13774, 137784, 1378053, 13779491, 137794128, 1377940943, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0137794... . (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Subsequence of A005117, A007674, A007675 and A017077.

sfQ[n_]:=Module[{c4=FactorInteger[n[[4]]],r=Drop[n,{4}]},First[c4] == {2,2} && Max[Transpose[Rest[c4]][[2]]]==1&&And@@SquareFreeQ/@r]; Join[{1}, Transpose[ Select[Partition[Range[2,3600],7,1],sfQ]][[1]]] (* Harvey P. Dale, Nov 22 2011 *)

ap(n)={forstep(k=1,n,8,
if(issquarefree(k)&&issquarefree(k+1)&&issquarefree(k+2)&&
   issquarefree((k+3)\2)&&
   issquarefree(k+4)&&issquarefree(k+5)&&issquarefree(k+6),
  print1(k", ")))}

A229882 Squarefree oblong numbers.

Original entry on oeis.org

2, 6, 30, 42, 110, 182, 210, 462, 506, 870, 930, 1122, 1190, 1406, 1482, 1722, 1806, 2162, 3306, 3422, 3782, 4290, 4422, 4830, 4970, 5402, 6006, 6162, 6806, 7310, 7482, 8742, 8930, 10302, 10506, 11130, 11342, 11990, 12210, 12882, 13110, 14042, 15006, 16770, 17030

Offset: 1

Irina Gerasimova, Oct 02 2013

Oblong numbers that are squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002378, A005117, A007674, A061304.

Select[#*(# + 1) & /@ Range[150], SquareFreeQ] (* Amiram Eldar, Feb 22 2021 *)

for(n=1,1e4,if(issquarefree(n) && issquarefree(n+1), print1(n*(n+1)", "))) \\ Charles R Greathouse IV, Mar 18 2014

Intersection of A002378 and A005117. - Joerg Arndt, Oct 03 2013

a(n) = A007674(n)^2 + A007674(n). - Charles R Greathouse IV, Oct 04 2013

Wrong term removed and more terms added by Amiram Eldar, Feb 22 2021

A335962 Numbers k such that k^2 + 1 and k^2 + 2 are both squarefree.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 10, 11, 12, 15, 16, 17, 20, 21, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 39, 42, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 60, 61, 62, 64, 65, 66, 69, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 87, 88, 89, 90, 91, 92, 96, 97, 98, 100, 101

Offset: 1

Amiram Eldar, Jul 01 2020

nonn

Dimitrov (2020) proved that this sequence is infinite and has an asymptotic density Product_{p prime > 2} (1 - ((-1/p) + (-2/p) + 2)/p^2) = 0.67187..., where (a/p) is the Legendre symbol.

			1 is a term since 1^2 + 1 = 2 and 1^1 + 2 = 3 are both squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. I. Dimitrov, Pairs of square-free values of the type n^2+1, n^2+2, arXiv:2004.09975 [math.NT], 2020.

Subsequence of A049533.

Cf. A005117, A007674, A069987.

Select[Range[100], And @@ SquareFreeQ /@ (#^2 + {1, 2}) &]

cp's OEIS Frontend

A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

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

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A215726 Numbers k such that the k-th triangular number is squarefree.

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A369020 Numbers k such that k and k+1 have the same maximal exponent in their prime factorization.

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A069977 Numbers k such that k and k+2 are squarefree.

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Extensions

A075424 a(n) = A075423(A075423(n)).

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Maple

A172187 Numbers k such that k and k+1 are squarefree but 2*k+1 is not.

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Mathematica

A194002 Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.

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