A375710 -id:A375710 - cp's OEIS frontend

A375707 First differences minus 1 of nonsquarefree numbers.

3, 0, 2, 3, 1, 1, 3, 0, 1, 0, 3, 3, 3, 3, 0, 2, 0, 0, 1, 1, 1, 3, 2, 0, 3, 3, 2, 0, 3, 0, 2, 3, 1, 1, 3, 1, 0, 0, 3, 3, 3, 3, 0, 2, 0, 2, 0, 0, 1, 3, 2, 0, 3, 3, 2, 0, 1, 1, 0, 2, 3, 1, 1, 3, 0, 1, 0, 2, 0, 3, 3, 3, 0, 2, 3, 1, 1, 3, 2, 0, 3, 3, 3, 3, 0, 2, 3

Offset: 1

Gus Wiseman, Sep 16 2024

Also the number of squarefree numbers between the nonsquarefree numbers A013929(n) and A013929(n+1).
Delete all 0's to get A120992.
The image is {0,1,2,3}.
Add 1 to all terms for A078147.

			The runs of squarefree numbers begin:
  (5,6,7)
  ()
  (10,11)
  (13,14,15)
  (17)
  (19)
  (21,22,23)
  ()
  (26)
  ()
  (29,30,31)
  (33,34,35)

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

Positions of 0, 1, 2, 3 are A375709, A375710, A375711, A375712. This is a set partition of the positive integers into four blocks.
For runs of squarefree numbers:
- length: A120992, anti A373127
- min: A072284, anti A373408
- max: A373415, anti A007674
- sum: A373413, anti A373411
For runs of nonsquarefree numbers:
- length: A053797, anti A373409
- min: A053806, anti A373410
- max: A376164, anti A068781
- sum: A373414, anti A373412
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A046933 counts composite numbers between consecutive primes.
A073784 counts primes between consecutive composite numbers.
A093555 counts non-prime-powers between consecutive prime-powers.
Cf. A061398, A061399, A077641, A077643, A143658, A173143, A251092, A294242, A373125, A373126, A373573.

Differences[Select[Range[100],!SquareFreeQ[#]&]]-1

lista(nmax) = {my(prev = 4); for (n = 5, nmax, if(!issquarefree(n), print1(n - prev - 1, ", "); prev = n));} \\ Amiram Eldar, Sep 17 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 6/(Pi^2-6) = 1.550546... . - Amiram Eldar, Sep 17 2024

A375709 Numbers k such that A013929(k+1) = A013929(k) + 1. In other words, the k-th nonsquarefree number is 1 less than the next nonsquarefree number.

Original entry on oeis.org

2, 8, 10, 15, 17, 18, 24, 28, 30, 37, 38, 43, 45, 47, 48, 52, 56, 59, 65, 67, 69, 73, 80, 85, 92, 93, 94, 100, 106, 108, 111, 115, 122, 125, 128, 133, 134, 137, 138, 141, 143, 145, 148, 153, 158, 165, 166, 171, 178, 183, 184, 192, 196, 198, 203, 205, 207, 210

Offset: 1

Gus Wiseman, Sep 01 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) (this)
- 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 one after the 2nd and 8th terms.

Positions of 1's in A078147.
For prime-powers (A246655) we have A375734.
First differences are A373409.
For prime numbers we have A375926.
For squarefree instead of nonsquarefree we have A375927.
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, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373410, A373573.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],1]

Complement of A375710 U A375711 U A375712.

A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

Original entry on oeis.org

3, 16, 23, 27, 31, 44, 46, 51, 55, 60, 68, 74, 79, 86, 95, 101, 105, 107, 112, 116, 121, 126, 129, 146, 147, 152, 159, 164, 167, 172, 177, 182, 185, 191, 195, 199, 204, 209, 220, 223, 229, 234, 237, 242, 244, 257, 262, 270, 275, 285, 286, 291, 299, 305, 312

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 3 after the third term.

Positions of 3'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, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A375927.

Join@@Position[Differences[Select[Range[1000],!SquareFreeQ[#]&]],3]

Complement of A375709 U A375710 U A375712.

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A375707 First differences minus 1 of nonsquarefree numbers.

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A375709 Numbers k such that A013929(k+1) = A013929(k) + 1. In other words, the k-th nonsquarefree number is 1 less than the next nonsquarefree number.

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A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

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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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