A120992 -id:A120992 - cp's OEIS frontend

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.

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.

A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Oct 04 2024

nonn

These are points at which the second differences (A376590) are nonzero.

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with nonzeros at (A376591):
  2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, ...

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

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the nonzeros of A376590.
The complement is A376591.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376591 (inflection and undulation points).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A373198, A376655.

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

A377039 Antidiagonal-sums of the array A377038(n,k) = n-th term of k-th differences of squarefree numbers (A005117).

Original entry on oeis.org

1, 3, 4, 9, 1, 18, 8, -9, 106, -237, 595, -1170, 2276, -3969, 6640, -10219, 14655, -18636, 19666, -12071, -13056, 69157, -171441, 332756, -552099, 798670, -982472, 901528, -116173, -2351795, 8715186, -23856153, 57926066, -130281007, 273804642, -535390274

Offset: 0

Gus Wiseman, Oct 18 2024

sign

These are row-sums of the triangle-version of A377038.

			The fourth antidiagonal of A377038 is (6,1,-1,-2,-3), so a(4) = 1.

The version for primes is A140119, noncomposites A376683, composites A377034.
These are the antidiagonal-sums of A377038.
The absolute version is A377040.
For nonsquarefree numbers we have A377047.
For prime-powers we have A377052.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
A377041 gives first column of A377038, for primes A007442 or A030016.
A377042 gives first position of 0 in each row of A377038.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A076259, A120992, A376311, A376590, A376591, A377046.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377042 Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.

Original entry on oeis.org

0, 0, 1, 11, 8, 57, 14, 11, 13, 1019, 44, 1250, 43, 2721, 42, 249522, 2840, 1989839, 2839, 3373774, 4933, 142715511, 42793, 435650856, 5266, 30119361, 104063, 454172978707, 100285, 434562125244, 2755089, 2409925829164, 2485612

Offset: 0

Gus Wiseman, Oct 18 2024

a(n) for n even appear to be smaller than a(n) for n odd. - Chai Wah Wu, Oct 19 2024

a(33) > 10^13, unless it is 0. - Lucas A. Brown, Nov 15 2024

			The fourth differences begin: -3, 3, 1, -6, 7, -5, 3, 0, -2, ... so a(4) = 8

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
This is the first position of 0 in each row of A377038.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
A377039 gives antidiagonal-sums of A377038, absolute version A377040.
A377041 gives first column of A377038, for primes A007442 or A030016.
Cf. A007674, A053797, A053806, A061398, A072284, A076259, A112925, A120992, A376311, A376590, A376591, A377046.

nn=10000;
u=Table[Differences[Select[Range[nn],SquareFreeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

a(15)-a(20) from Chai Wah Wu, Oct 19 2024

a(21)-a(32) from Lucas A. Brown, Nov 15 2024

A243348 Difference between the n-th squarefree number and n: a(n) = A005117(n) - n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 4, 4, 4, 5, 6, 7, 7, 7, 9, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 16, 16, 19, 20, 21, 22, 22, 22, 23, 23, 25, 25, 25, 26, 26, 26, 27, 27, 29, 29, 29, 31, 31, 32, 32, 32, 33, 34, 35, 35, 35, 36, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 42

Offset: 1

Antti Karttunen, Jun 04 2014

nonn

a(n) <= n, as A243351(n) = 2n - A005117(n) goes never negative (please see the plot A005117(n)/n given in the links section).
No runs longer than three appear, because there must be at least one gap (cf. A053806) in each range [4k+1 .. 4(k+1)] where no term(s) of A005117 appear.
See also A120992 which gives the run lengths.
Record values of first differences: a(2) - a(1) = 0, a(4) - a(3) = 1, a(7) - a(6) = 2, a(32) - a(31) = 3, a(151) - a(150) = 4, a(516) - a(515) = 5, a(13392) - a(13391) = 6, a(131965) - a(131964) = 7, a(664314) - a(664313) = 8, a(5392319) - a(5392318) = 9, and a(134453712) - a(134453711) = 11. - Charles R Greathouse IV, Nov 05 2017

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Sequence plotted together with A243351 showing how their ratio develops.
Antti Karttunen, Ratio A005117(n)/n plotted in the same way, converging to Pi^2/6.

Cf. A005117, A053797, A053806, A243351, A243289, A243347.

A120992 gives the lengths of runs.

do(x)=my(v=List([0])); forfactored(n=2,x\1, if(vecmax(n[2][,2])==1, listput(v,n[1]-#v-1))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

from math import isqrt
from sympy import mobius
def A243348(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m-n # Chai Wah Wu, Aug 12 2024

Scheme
```
(define (A243348 n) (- (A005117 n) n))
```

a(n) = A005117(n) - n.
a(n) = A243349(n) - A243289(n).
a(n) = n - A243351(n).
Limit_{n->oo} a(n)/A243351(n) = (Pi^2 - 6)/(12 - Pi^2) = 1.81637833.... - Charles R Greathouse IV, Jun 04 2014
a(n) ~ kn where k = Pi^2/6 - 1 = 0.644934.... - Charles R Greathouse IV, Nov 05 2017

A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 21, 24, 30, 35, 40, 41, 46, 50, 69, 82, 131, 140, 185, 192, 199, 210, 248, 251, 271, 277, 325, 406, 423, 458, 645, 748, 811, 815, 826, 831, 987, 1053, 1109, 1426, 1456, 1590, 1629, 1870, 1967, 2060, 2371, 2607, 2920, 2946, 3564, 3681, 4119

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A373401.

For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.

			The maximal antiruns of prime numbers > 3 begin:
    5
    7  11
   13  17
   19  23  29
   31  37  41
   43  47  53  59
   61  67  71
   73  79  83  89  97 101
  103 107
  109 113 127 131 137
  139 149
  151 157 163 167 173 179
The a(n)-th rows begin:
    5
    7  11
   19  23  29
   43  47  53  59
   73  79  83  89  97 101
  109 113 127 131 137

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For squarefree antiruns: A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051.
The unsorted version is A373401, firsts of A027833.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[4,10000],PrimeQ],#1+2!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

4, 8, 32, 44, 104, 140, 284, 464, 572, 620, 644, 824, 860, 1232, 1292, 1304, 1484, 1700, 1724, 1880, 2084, 2132, 2240, 2312, 2384, 2660, 2732, 2804, 3392, 3464, 3560, 3920, 3932, 4004, 4220, 4244, 4424, 4640, 4724, 5012, 5444, 5480, 5504, 5660, 6092, 6200

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377783.

			The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
    32: {1,1,1,1,1}
    44: {1,1,5}
   104: {1,1,1,6}
   140: {1,1,3,4}
   284: {1,1,20}
   464: {1,1,1,1,10}
   572: {1,1,5,6}
   620: {1,1,3,11}
   644: {1,1,4,9}
   824: {1,1,1,27}
   860: {1,1,3,14}
  1232: {1,1,1,1,4,5}

Subset of A377783 (union A378040, diffs A377784), restriction of A120327 (diffs A378039).
Terms appearing once are A378082.
Terms not appearing at all are A378084.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A071403(n) = A013928(prime(n)) counts squarefree numbers < prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers < prime(n).
Cf. A112926 (diffs A378037), opposite A112925 (diffs A378038).
Cf. A378032 (diffs A378034), restriction of A378033 (diffs A378036).
Cf. A053797, A053806, A070321, A072284, A112929, A120992, A224363, A337030, A377430, A377431, A377703.

y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==2&]

A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

Original entry on oeis.org

6, 1, 18, 8, 4, 2, 10, 52, 678

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The sorted version is A373574.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			The maximal antiruns of nonsquarefree numbers begin:
   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 a(n)-th rows are:
    49
     4    8
   148  150  152
    64   68   72   75
    28   32   36   40   44
     9   12   16   18   20   24
    81   84   88   90   92   96   98
   477  480  484  486  488  490  492  495
  6345 6348 6350 6352 6354 6356 6358 6360 6363

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

For composite runs we have A073051, firsts of A176246, sorted A373400.
For squarefree runs we have the triple (5,3,1), firsts of A120992.
For prime runs we have the triple (1,3,2), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127, sorted A373200.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373401, firsts of A027833, sorted A373402.
For composite antiruns we have the triple (2,7,1), firsts of A373403.
Positions of first appearances in A373409.
The sorted version is A373574.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A025157, A049094, A061399, A068781, A072284, A110969, A251092, A294242, A373410, A373412.

t=Length/@Split[Select[Range[10000],!SquareFreeQ[#]&],#1+1!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A376342 Positions of 1's in the run-compression (A376305) of the first differences (A076259) of the squarefree numbers (A005117).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 51, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 124, 126, 128, 130

Offset: 1

Gus Wiseman, Sep 24 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with run-compression (A376305):
  1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, ...
with ones at (A376342):
  1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, ...

Before compressing we had A076259.
Positions of 1's in A376305.
The version for nonsquarefree numbers gives positions of ones in A376312.
For prime instead of squarefree numbers we have A376343.
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.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A037201, A053797, A053806, A061398, A072284, A120992, A373198, A376306, A376307, A376308, A376311.

Join@@Position[First /@ Split[Differences[Select[Range[100],SquareFreeQ]]],1]

A376594 Inflection and undulation points in the sequence of nonsquarefree numbers (A013929).

Original entry on oeis.org

5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, 81, 82, 83, 88, 92, 93, 96, 98, 103, 109, 113, 118, 123, 130, 131, 133, 137, 139, 146, 149, 154, 155, 156, 161, 165, 168, 169, 174, 179, 180, 183, 187, 188, 189, 193, 201, 211, 213

Offset: 1

Gus Wiseman, Oct 04 2024

nonn

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

			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 zeros (A376594) at:
  5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, ...

Gus Wiseman, Inflection and undulation points in the nonsquarefree numbers.

The first differences were A078147.
These are the zeros of A376593.
The complement is A376595.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A064113 lists positions of adjacent equal prime gaps.
A114374 counts partitions into nonsquarefree numbers.
For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376597 (prime-power), A376600 (non-prime-power).
For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376593 (second differences), A376595 (nonzero curvature).
Cf. A007674, A053797, A053806, A061398, A112926, A120992, A251092, A375707, A376312, A376590, A376593.

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&],2],0]

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A377039 Antidiagonal-sums of the array A377038(n,k) = n-th term of k-th differences of squarefree numbers (A005117).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A377042 Position of first zero in the n-th differences of the squarefree numbers (A005117), or 0 if it does not appear.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A243348 Difference between the n-th squarefree number and n: a(n) = A005117(n) - n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A373402 Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A378083 Nonsquarefree numbers appearing exactly twice in A377783 (least nonsquarefree number > prime(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs