A053806 -id:A053806 - cp's OEIS frontend

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

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}]

A378040 Union of A377783(n) = least nonsquarefree number > prime(n).

Original entry on oeis.org

4, 8, 12, 16, 18, 20, 24, 32, 40, 44, 48, 54, 60, 63, 68, 72, 75, 80, 84, 90, 98, 104, 108, 112, 116, 128, 132, 140, 150, 152, 160, 164, 168, 175, 180, 184, 192, 196, 198, 200, 212, 224, 228, 232, 234, 240, 242, 252, 260, 264, 270, 272, 279, 284, 294, 308, 312

Offset: 1

Gus Wiseman, Nov 20 2024

nonn

Numbers k such that, if p is the greatest prime < k, all numbers from p to k (exclusive) are squarefree.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

For squarefree we have A112926 (diffs A378037), opposite A112925 (diffs A378038).
For prime-power instead of nonsquarefree we have A345531, differences A377703.
Union of A377783 (diffs A377784), restriction of A120327 (diffs A378039).
Nonsquarefree numbers not appearing are A378084, see also A378082, A378083.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
A070321 gives the greatest squarefree number up to n.
A071403(n) = A013928(prime(n)) counts squarefree numbers up to prime(n).
A378086(n) = A057627(prime(n)) counts nonsquarefree numbers up to prime(n).
Cf. A378034 (differences of A378032), restriction of A378036 (differences A378033).
Cf. A000015, A053797, A053806, A072284, A224363, A337030, A377047, A377049, A377430, A377431.

Union[Table[NestWhile[#+1&,Prime[n],SquareFreeQ],{n,100}]]
lns[p_]:=Module[{k=p+1},While[SquareFreeQ[k],k++];k]; Table[lns[p],{p,Prime[Range[70]]}]//Union (* Harvey P. Dale, Jun 12 2025 *)

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}]

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]

A377050 Position of first appearance of zero in the n-th differences of the nonsquarefree numbers, or 0 if it does not appear.

Original entry on oeis.org

0, 0, 5, 11, 4, 129, 10, 89, 16, 161, 72, 77325, 71, 4870, 70, 253, 75, 737923, 166, 1648316, 165, 8753803, 164, 208366710, 163, 99489971, 162, 49493333, 161

Offset: 0

Gus Wiseman, Oct 19 2024

If a(29) is not 0, then it is > 10^12. - Lucas A. Brown, Oct 25 2024

			The fourth differences of A013929 begin: -6, -2, 5, 0, -7, 9, -6, 6, -7, ... so a(4) = 4.

Lucas A. Brown, Python program.

The version for primes is A376678, noncomposites A376855, composites A377037.
For squarefree instead of nonsquarefree numbers we have A377042.
For antidiagonal-sums we have A377047, absolute A377048.
For leading column we have A377049.
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. A000961, A007674, A053797, A053806, A061398, A072284, A084758, A112925, A120992, A376311, A376591, A377051.

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(17)-a(28) from Lucas A. Brown, Oct 25 2024

A051681 Smallest term of first run of exactly n consecutive integers which are not squarefree.

Original entry on oeis.org

4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com)

			a(5) = 844: 844 = 2^2*211, 845 = 5*13^2, 846 = 2*3^2*47, 847 = 7*11^2, 848 = 2^4*53.

a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al.

L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. See also the author page.
"sikefield3", double-square (2019).
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A045882 (another version), A013929, A053806.

Module[{tb=Table[If[SquareFreeQ[n],0,1],{n,11*10^5}]},Table[ SequencePosition[ tb,PadRight[{},k,1],1][[All,1]],{k,8}]]//Flatten (* The program generates the first 8 terms of the sequence. To generate more, increase the constants for n and k but the program may take a long time to run. *) (* Harvey P. Dale, Mar 24 2022 *)

iscons(x, n)=if (issquarefree(x-1) && issquarefree(x+n), for (k = 0, n-1, if (issquarefree(x+k), return (0));); return (1);); return (0);
a(n) = {my(x = 1); while (! iscons(x, n), x++); x;} \\ Michel Marcus, Jan 13 2014

a(16) reported by Louis Marmet (louis(AT)marmet.org), Jul 24 2000
a(17) was obtained as a result of a team effort by E. Wong et al.
a(18) = 125781000834058568 was obtained as a result of a team effort by L. Marmet et al.

A376306 Run-lengths of the sequence of first differences of squarefree numbers.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 21 2024

nonn

			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, ...
with runs:
  (1,1),(2),(1,1),(3),(1),(2),(1,1),(2,2,2),(1,1),(3,3),(1,1),(2),(1,1), ...
with lengths A376306 (this sequence).

Run-lengths of first differences of A005117.
Before taking run-lengths we had A076259, ones A375927.
For prime instead of squarefree numbers we have A333254.
For compression instead of run-lengths we have A376305.
For run-sums instead of run-lengths we have A376307.
For prime-powers instead of squarefree numbers we have A376309.
For positions of first appearances instead of run-lengths we have A376311.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first 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, A373198, A375707, A376312.

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

A376312 Run-compression of first differences (A078147) of nonsquarefree numbers (A013929).

Original entry on oeis.org

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

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 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), ...
and run-compression (A376312):
  4, 1, 3, 4, 2, 4, 1, 2, 1, 4, 1, 3, 1, 2, 4, 3, 1, 4, 3, 1, 4, 1, 3, 4, ...

For nonprime instead of squarefree numbers we have A037201, halved A373947.
Before compressing we had A078147.
For run-sums instead of compression we have A376264.
For squarefree instead of nonsquarefree we have A376305, ones A376342.
For prime-powers instead of nonsquarefree numbers we have A376308.
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.
Cf. A007674, A053797, A053806, A072284, A112925, A120992, A274174, A373198, A375707, A376306, A376307, A376311.

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

A378084 Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).

Original entry on oeis.org

9, 25, 27, 28, 36, 45, 49, 50, 52, 56, 64, 76, 81, 88, 92, 96, 99, 100, 117, 120, 121, 124, 125, 126, 135, 136, 144, 147, 148, 153, 156, 162, 169, 171, 172, 176, 188, 189, 204, 207, 208, 216, 220, 225, 236, 243, 244, 245, 248, 250, 256, 261, 268, 275, 276, 280

Offset: 1

Gus Wiseman, Nov 23 2024

nonn

Warning: do not confuse with A377784.

			The terms together with their prime indices begin:
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   36: {1,1,2,2}
   45: {2,2,3}
   49: {4,4}
   50: {1,3,3}
   52: {1,1,6}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   76: {1,1,8}
   81: {2,2,2,2}
   88: {1,1,1,5}
   92: {1,1,9}
   96: {1,1,1,1,1,2}

Disjoint from A377783 (union A378040), first-differences A377784.
Appearing once: A378082.
Appearing twice: A378083.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A070321 gives the greatest squarefree number up to n.
A112925 gives least squarefree number > prime(n), differences A378038.
A112926 gives greatest squarefree number < prime(n), differences A378037.
A120327 (union A162966) gives least nonsquarefree number >= n, differences A378039.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A224363, A377430, A377431, A378032, A378033.

nn=100;
y=Table[NestWhile[#+1&,Prime[n],SquareFreeQ[#]&],{n,nn}];
Complement[Select[Range[Prime[nn]],!SquareFreeQ[#]&],y]

Complement of A378040 in A013929.

A375703 Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

Original entry on oeis.org

2, 5, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729

Offset: 1

Gus Wiseman, Aug 28 2024

nonn

Non-perfect-powers A007916 are numbers without a proper integer root.

			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 A375702, first a(n), last A375704, sum A375705.

For prime numbers we have A045344.
For nonsquarefree numbers we have A053806, anti-runs A373410.
For nonprime numbers we have A055670, anti-runs A005381.
For squarefree numbers we have A072284, anti-runs A373408.
The anti-run version is A216765 (same as A375703 with 2 exceptions).
For non-prime-powers we have A373673, anti-runs A120430.
For prime-powers we have A373676, anti-runs A373575.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1.
- first: A375703 (this)
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
Cf. A007674, A045542, A375708, A375713, A375714.

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

Numbers k > 0 such that k-1 is a perfect power (A001597) but k is not.

cp's OEIS Frontend

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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A378040 Union of A377783(n) = least nonsquarefree number > prime(n).

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

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A377050 Position of first appearance of zero in the n-th differences of the nonsquarefree numbers, or 0 if it does not appear.

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A051681 Smallest term of first run of exactly n consecutive integers which are not squarefree.

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A376306 Run-lengths of the sequence of first differences of squarefree numbers.

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A376312 Run-compression of first differences (A078147) of nonsquarefree numbers (A013929).

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A378084 Nonsquarefree numbers not appearing in A377783 (least nonsquarefree number > prime(n)).

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