A120992 -id:A120992 - cp's OEIS frontend

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.

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

A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

Original entry on oeis.org

4, 4, -3, 5, -6, 4, 3, -15, 25, -10, -84, 369, -1067, 2610, -5824, 12246, -24622, 47577, -88233, 155962, -259086, 393455, -512281, 456609, 191219, -2396571, 8213890, -21761143, 50923029, -110269263, 225991429, -444168664, 844390152, -1561482492, 2817844569

Offset: 0

Gus Wiseman, Oct 19 2024

sign

Harvey P. Dale, Table of n, a(n) for n = 0..400

The version for primes is A007442, noncomposites A030016, composites A377036.
For squarefree instead of nonsquarefree numbers we have A377041.
For antidiagonal-sums we have A377047, absolute A377048.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A084758, A112925, A120992, A251092, A376311, A376591.

nn=20;
Table[First[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k]],{k,0,nn}]
With[{nsf=Select[Range[1000],!SquareFreeQ[#]&]},Table[Differences[nsf,n],{n,0,40}]][[;;,1]] (* Harvey P. Dale, Nov 28 2024 *)

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

A377431 Numbers k such that there is at least one squarefree number between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86

Offset: 1

Gus Wiseman, Oct 29 2024

nonn

			Primes 4 and 5 are 7 and 11, and the interval (8,9,10) contains 10, which is squarefree, so 4 is in the sequence.

These are the positive positions in A061398, or terms >= 2 in A373198.
The complement (no squarefree numbers) is A068360.
For prime-power instead of squarefree we have A377057, strict version A377287.
For exactly one squarefree number we have A377430.
A000040 lists the primes, differences A001223, seconds A036263.
A002808 lists the composites, complement A008578.
A005117 lists the squarefree numbers, complement A013929.
A377038 gives k-differences of squarefree numbers.
Cf. A007674, A013603, A029707, A036378, A046933, A072284, A076259, A077641, A077643, A075526, A078147, A112925, A112926, A120992, A293697.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],SquareFreeQ]]>=1&]

A373199 Least k such that the k-th maximal run of nonsquarefree numbers has length n. Position of first appearance of n in A053797.

Original entry on oeis.org

1, 2, 13, 68, 241, 6278, 61921, 311759, 2530539

Offset: 1

Gus Wiseman, Jun 08 2024

A run of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by one. The a(n)-th run of nonsquarefree numbers begins with A045882 = A051681, subset of A053806.

			The maximal runs 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
The a(n)-th rows are:
     4
     8     9
    48    49    50
   242   243   244   245
   844   845   846   847   848
For example, (48, 49, 50) is the first maximal run of 3 nonsquarefree numbers, so a(3) = 13.

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

For composite instead of nonsquarefree we have A073051.
The version for squarefree runs is A373128.
For prime instead of nonsquarefree we have A373400.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A007674, A020754, A045882, A120992, A061398, A061399, A068781, A101836, A251092, A294242, A373410, A373412.

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

A376305 Run-compression of the sequence of first differences of squarefree numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3

Offset: 1

Gus Wiseman, Sep 20 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, run-compression removes 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, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
The run-compression is A376305 (this sequence).

This is the run-compression of first differences of A005117.
For prime instead of squarefree numbers we have A037201, halved A373947.
Before compressing we had A076259, ones A375927.
For run-lengths instead of compression we have A376306.
For run-sums instead of compression we have A376307.
For prime-powers instead of squarefree numbers we have A376308.
For positions of first appearances instead of compression we have A376311.
The version for nonsquarefree numbers is A376312.
Positions of 1's are A376342.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed or anti-run compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A274174, A373197, A373198, A375707, A375708.

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

A377048 Antidiagonal-sums of the absolute value of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 12, 13, 22, 28, 31, 39, 64, 85, 132, 395, 1103, 2650, 5868, 12297, 24694, 47740, 88731, 157744, 265744, 418463, 605929, 805692, 1104513, 2396645, 8213998, 21761334, 50923517, 110270883, 225997492, 444193562, 844498084, 1561942458, 2819780451, 4973173841

Offset: 1

Gus Wiseman, Oct 19 2024

nonn

These are the row-sums of the absolute value triangle version of A377046.

			The third antidiagonal of A377046 is (9, 1, -3), so a(3) = 13.

The version for primes is A376681, noncomposites A376684, composites A377035.
For squarefree instead of nonsquarefree numbers we have A377040.
The non-absolute version is A377047.
For leading column we have A377049.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A376591, A376592.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}];
Total/@Table[Abs[t[[j,i-j+1]]],{i,nn},{j,i}]

A377047 Antidiagonal-sums of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

Original entry on oeis.org

4, 12, 7, 22, 14, 17, 39, 0, 37, 112, -337, 1103, -2570, 5868, -12201, 24670, -47528, 88283, -155910, 259140, -393399, 512341, -456546, -191155, 2396639, -8213818, 21761218, -50922953, 110269343, -225991348, 444168748, -844390064, 1561482582, -2817844477

Offset: 1

Gus Wiseman, Oct 19 2024

sign

These are the row-sums of the triangle-version of A377046.

			The third antidiagonal of A377046 is (9, 1, -3), so a(3) = 7.

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree instead of nonsquarefree numbers we have A377039.
The absolute value version is A377048.
For leading column we have A377049.
For first position of 0 in each row we have A377050.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A376311, A376591, A376592, A377051.

nn=20;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

Original entry on oeis.org

1, 3, 8, 23, 29, 33, 45, 98, 153, 188, 216, 262, 281, 366, 428, 589, 737, 1182, 1830, 1878, 2190, 2224, 3076, 3301, 3384, 3426, 3643, 3792, 4521, 4611, 7969, 8027, 8687, 12541, 14356, 14861, 15782, 17005, 19025, 23282, 30801, 31544, 33607, 34201, 34214, 38589

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A073051.

A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of composite numbers begin:
   4
   6
   8   9  10
  12
  14  15  16
  18
  20  21  22
  24  25  26  27  28
  30
  32  33  34  35  36
  38  39  40
  42
  44  45  46
  48  49  50  51  52
  54  55  56  57  58
  60
  62  63  64  65  66
  68  69  70
  72
  74  75  76  77  78
  80  81  82
  84  85  86  87  88
  90  91  92  93  94  95  96
  98  99 100
The a(n)-th rows are:
   4
   8   9  10
  24  25  26  27  28
  90  91  92  93  94  95  96
 114 115 116 117 118 119 120 121 122 123 124 125 126
 140 141 142 143 144 145 146 147 148
 200 201 202 203 204 205 206 207 208 209 210

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

The unsorted version is A073051, firsts of A176246.
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 squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite runs 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[10000],CompositeQ],#1+1==#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A375702 Length of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

2, 3, 6, 8, 1, 4, 3, 12, 14, 16, 18, 20, 3, 2, 15, 24, 26, 19, 8, 17, 12, 32, 34, 18, 17, 38, 40, 42, 27, 16, 46, 48, 50, 52, 54, 56, 58, 60, 38, 23, 64, 66, 68, 70, 34, 37, 74, 76, 78, 80, 46, 35, 84, 86, 88, 22, 67, 70, 9, 11, 94, 96, 98, 100, 102, 39, 64

Offset: 1

Gus Wiseman, Aug 27 2024

nonn

Non-perfect-powers A007916 are numbers with no proper integer roots.

			The list of all non-perfect-powers, split into runs, begins:
   2   3
   5   6   7
  10  11  12  13  14  15
  17  18  19  20  21  22  23  24
  26
  28  29  30  31
  33  34  35
  37  38  39  40  41  42  43  44  45  46  47  48
Row n has length a(n), first A375703, last A375704, sum A375705.

For nonsquarefree numbers we have A053797, anti-runs A373409.
For squarefree numbers we have A120992, anti-runs A373127.
For nonprime numbers we have A176246, anti-runs A373403.
For prime-powers we have A373675, anti-runs A373576.
For non-prime-powers we have A373678, anti-runs A373679.
The anti-run version is A375736, sum A375737.
For runs of non-perfect-powers (A007916):
- length: A375702 (this).
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
Cf. A007674, A045542, A061399, A216765, A251092, A375708, A375714.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Length/@Split[Select[Range[100],radQ],#1+1==#2&]//Most

For n > 2 we have a(n) = A053289(n+1) - 1.

A376591 Inflection and undulation points in the sequence of squarefree numbers (A005117).

Original entry on oeis.org

1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, 62, 65, 68, 71, 79, 84, 87, 96, 98, 99, 101, 103, 107, 110, 113, 118, 120, 121, 123, 128, 131, 134, 137, 142, 144, 145, 147, 152, 153, 155, 158, 163, 165, 166, 172, 175, 179, 184

Offset: 1

Gus Wiseman, Oct 04 2024

nonn

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

			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 zeros at (A376591):
 1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, ...

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

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
These are the zeros of A376590.
The complement is A376592.
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 inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
For squarefree numbers: A076259 (first differences), A376590 (second differences), A376592 (nonzero curvature).
Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A376593, A376655.

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

cp's OEIS Frontend

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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A377049 First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

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A377431 Numbers k such that there is at least one squarefree number between prime(k)+1 and prime(k+1)-1.

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A373199 Least k such that the k-th maximal run of nonsquarefree numbers has length n. Position of first appearance of n in A053797.

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A376305 Run-compression of the sequence of first differences of squarefree numbers.

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A377048 Antidiagonal-sums of the absolute value of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

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A377047 Antidiagonal-sums of the array A377046(n,k) = n-th term of k-th differences of nonsquarefree numbers (A013929).

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A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

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A375702 Length of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

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