A375709 -id:A375709 - cp's OEIS frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024

			44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From _Gus Wiseman_, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
  (4,8)
  (9,12,16,18,20,24)
  (25,27)
  (28,32,36,40,44)
  (45,48)
  (49)
  (50,52,54,56,60,63)
  (64,68,72,75)
  (76,80)
  (81,84,88,90,92,96,98)
  (99)
The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
Cf. A049535, A077647, A078143, A045882.
Subsequence of A261869.
Cf. A007674, A373409, A373410, A373412, A375707, A376164.
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.
Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.

a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015

Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)

isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021

A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015

A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

Original entry on oeis.org

5, 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, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1

Offset: 1

Gus Wiseman, Aug 31 2024

nonn

Non-prime-powers (exclusive) are listed by A361102.

Warning: For this sequence, 1 is not a prime-power but is a non-prime-power.

			The 6th non-prime-power (exclusive) is 15, and the 7th is 18, so a(6) = 3.

For prime-powers (A000961, A246655) we have A057820, gaps A093555.
For perfect powers (A001597) we have A053289.
For nonprime numbers (A002808) we have A073783.
For squarefree numbers (A005117) we have A076259.
First differences of A361102, inclusive A024619.
Positions of 1's are A375713.
If 1 is considered a prime power we have A375735.
Runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
A013929 lists the nonsquarefree numbers, differences A078147.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A046933, A061399, A176246, A251092, A375709.

Differences[Select[Range[100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375708(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 09 2024

A375735 First differences of non-prime-powers (inclusive).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 04 2024

nonn

Inclusive means 1 is a prime-power but not a non-prime-power.

Non-prime-powers (inclusive) are listed by A024619.

			The 5th non-prime-power (inclusive) is 15, and the 6th is 18, so a(5) = 3.

For perfect powers (A001597) we have the latter terms of A053289.
For nonprime numbers (A002808) we have the latter terms of A073783.
For squarefree numbers (A005117) we have the latter terms of A076259.
First differences of A024619.
For prime-powers (A246655) we have the latter terms of A057820.
Essentially the same as the exclusive version, A375708.
Positions of 1's are A375713(n) - 1.
For runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, first differences A001223.
A000961 lists prime-powers (inclusive).
A007916 lists non-perfect-powers, first differences A375706.
A013929 lists the nonsquarefree numbers, first differences A078147.
A246655 lists prime-powers (exclusive).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power anti-runs: A373576, min A120430, max A006549, length A373671.
Non-prime-power anti-runs: A373679, min A373575, max A255346, len A373672.
Cf. A046933, A061399, A093555, A176246, A251092, A323054, A375709, A375736.

Differences[Select[Range[2,100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375735(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 10 2024

A375707 First differences minus 1 of nonsquarefree numbers.

Original entry on oeis.org

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

A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 10, 14, 15, 18, 19, 21, 22, 24, 25, 27, 28, 30, 35, 36, 38, 40, 41, 43, 44, 46, 48, 49, 51, 53, 54, 58, 59, 62, 63, 65, 66, 68, 69, 71, 72, 74, 76, 79, 80, 82, 84, 85, 87, 88, 90, 94, 96, 97, 101, 102, 105, 107, 108, 110, 111, 113, 114, 116

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2-1)) = 0.53071182... (A065469). - Amiram Eldar, Sep 15 2024

			The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by one after terms 1, 2, 4, 5, ...

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

Positions of 1's in A076259.
For prime-powers (A246655) we have A375734.
First differences are A373127.
For nonsquarefree instead of squarefree we have A375709.
For nonprime numbers we have A375926, differences A373403.
For composite numbers we have A375929.
The complement is A375930, differences A120992.
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, A061399, A065469, A072284, A110969, A294242, A373409, A373410, A373573.

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

lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && is2, print1(c-1, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024

A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 17, 43, 70, 1077, 6635, 12369, 43578, 105102700

Offset: 1

Gus Wiseman, Sep 04 2024

The corresponding prime-powers A246655(a(n)) are given by A006549.

From A006549, it is not known whether this sequence is infinite.

			The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.

For nonprime numbers (A002808) we have A375926, differences A373403.
Positions of 1's in A057820.
First differences are A373671.
For nonsquarefree numbers we have A375709, differences A373409.
For non-prime-powers we have A375713.
For non-perfect-powers we have A375740.
For squarefree numbers we have A375927, differences A373127.
Prime-powers:
- terms: A000961, complement A024619.
- differences: A057820.
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).
- anti-runs: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A025528 counts prime-powers up to n.
Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.

Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]

Numbers k such that A246655(k+1) - A246655(k) = 1.

The inclusive version is a(n) + 1 shifted.

a(14) from Amiram Eldar, Sep 24 2024

A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

Original entry on oeis.org

5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253

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 2 after the fifth and sixth terms.

Positions of 2's in A078147.
For prime numbers we have A029707.
For nonprime numbers we appear to have A014689.
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, A373409, A373573, A375927.

Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

Complement of A375709 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.

A375930 Numbers k such that A005117(k+1) - A005117(k) > 1. In other words, the k-th squarefree number is more than 1 less than the next.

Original entry on oeis.org

3, 6, 8, 11, 12, 13, 16, 17, 20, 23, 26, 29, 31, 32, 33, 34, 37, 39, 42, 45, 47, 50, 52, 55, 56, 57, 60, 61, 64, 67, 70, 73, 75, 77, 78, 81, 83, 86, 89, 91, 92, 93, 95, 98, 99, 100, 103, 104, 106, 109, 112, 115, 117, 120, 121, 122, 125, 127, 130, 133, 136, 139

Offset: 1

Gus Wiseman, Sep 12 2024

nonn

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2-1)) = 1 - A065469 = 0.46928817... . - Amiram Eldar, Sep 15 2024

			The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ... which first increase by more than one after positions 3, 6, 8, 11, ...

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

For nonprime numbers: A014689, complement A375926, differences A373403.
For composite numbers: A065890 shifted, complement A375929.
Positions of terms > 1 in A076259.
First differences are A120992, complement A373127.
The complement is 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.
Cf. A007674, A061399, A065469, A068781, A072284, A110969, A294242, A375709.

Join@@Position[Differences[Select[Range[100],SquareFreeQ[#]&]],_?(#>1&)]

lista(kmax) = {my(is1 = 1, is2, c = 1); for(k = 2, kmax, is2 = issquarefree(k); if(is2, c++); if(is1 && !is2, print1(c, ", ")); is1 = is2);} \\ Amiram Eldar, Sep 15 2024

cp's OEIS Frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

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A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

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A375735 First differences of non-prime-powers (inclusive).

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A375707 First differences minus 1 of nonsquarefree numbers.

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A375927 Numbers k such that A005117(k+1) - A005117(k) = 1. In other words, the k-th squarefree number is 1 less than the next.

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A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

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A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 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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