A373198 -id:A373198 - cp's OEIS frontend

A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).

1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, 131964, 502651, 664312, 4387185, 5392318, 20613826

Offset: 1

Gus Wiseman, Oct 07 2024

Warning: Do not confuse with A246655 (prime-powers exclusive).

			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, ...
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, ...
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, ...
with sorted first appearances at (A376655):
  1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, ...

For first differences we had A376311 (first appearances in A076259).
These are the sorted positions of first appearances in A376590.
For prime-powers instead of squarefree numbers we have A376653/A376654.
For primes instead of squarefree numbers we have A376656.
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 second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree: A376591 (inflections and undulations), A376592 (nonzero curvature).
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

q=Differences[Select[Range[1000],SquareFreeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

a(14)-a(19) from Chai Wah Wu, Oct 07 2024

A373200 Numbers k such that the k-th maximal antirun of squarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373127.

Original entry on oeis.org

1, 3, 8, 10, 19, 162, 1633, 1853, 2052, 26661, 46782, 1080330, 3138650

Offset: 1

Gus Wiseman, Jun 10 2024

The unsorted version is A373128.

An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of squarefree numbers begin:
   1
   2
   3   5
   6
   7  10
  11  13
  14
  15  17  19  21
  22
  23  26  29
  30
  31  33
  34
  35  37
The a(n)-th rows are:
    1
    3    5
   15   17   19   21
   23   26   29
   47   51   53   55   57
  483  485  487  489  491  493

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.
The unsorted version is A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For composite runs we have A373400, unsorted A073051.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

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

A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 27 2024

nonn

			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), ...
with lengths (A376267):
  1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...

Robert Israel, Table of n, a(n) for n = 1..10000

For prime instead of nonsquarefree numbers we have A333254.
For run-sums instead of run-lengths we have A376264.
For squarefree instead of nonsquarefree we have A376306.
For prime-powers instead of nonsquarefree numbers we have A376309.
For compression instead of run-lengths we have A376312.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
Cf. A007674, A053797, A053806, A072284, A112925, A120992, A373198, A375707, A376305, A376307, A376311, A380595.

nsf:= remove(numtheory:-issqrfree, [$4..1000]):
S:= nsf[2..-1]-nsf[1..-2]:
R:= NULL: x:= 4: t:= 1:
for i from 2 to nops(S) do
  if S[i] = x then t:= t+1
  else R:= R,t; x:= S[i]; t:= 1
  fi
od:
R; # Robert Israel, Jan 27 2025

Length/@Split[Differences[Select[Range[1000], !SquareFreeQ[#]&]]]//Most

A376265 Position of first appearance of n in A376264 (run-sums of first differences of nonsquarefree numbers), or 0 if there are none.

Original entry on oeis.org

2, 8, 3, 1, 6222, 14, 308540, 18

Offset: 1

Gus Wiseman, Sep 27 2024

			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), ...
with sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, ...
with first appearances at (A376265):
  2, 8, 3, 1, 6222, 14, 308540, 18, ...

This is the position of first appearance of n in A376264.
The sorted version is A376266.
For run-lengths instead of firsts of run-sums we have A376267.
For compression instead of firsts of run-sums we have A376312.
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.
A333254 lists run-lengths of differences between consecutive primes.
A376305 gives run-compression of first differences of squarefree numbers.
A376307 gives run-sums of first differences of squarefree numbers.
Cf. A037201, A053797, A053806, A120992, A373198, A373413, A373822, A375707, A376306, A376310, A376311.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Total/@Split[Differences[Select[Range[10000],!SquareFreeQ[#]&]]]//Most;
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A376264(a(n)) = n.

A376266 Sorted positions of first appearances in A376264 (run-sums of first differences of nonsquarefree numbers).

Original entry on oeis.org

1, 2, 3, 8, 10, 14, 18, 53, 1437, 6222, 40874

Offset: 1

Gus Wiseman, Sep 27 2024

			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), ...
with sums (A376264):
  4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, ...
with first appearances at (A376266):
  1, 2, 3, 8, 10, 14, 18, 53, 1437, 6222, 40874, ...

These are the positions of first appearances in A376264.
The unsorted version is A376265.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A333254 lists run-lengths of differences between consecutive primes.
A376267 gives run-lengths of first differences of nonsquarefree numbers.
A376312 gives run-compression of first differences of nonsquarefree numbers.
A376305 gives run-compression of differences of squarefree numbers, ones A376342.
Cf. A037201, A053797, A053806, A120992, A373197, A373198, A373413, A373822, A375707, A376306, A376307, A376521.

q=Total/@Split[Differences[Select[Range[10000], !SquareFreeQ[#]&]]]//Most;
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A378615 Number of non prime powers <= prime(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 14, 18, 21, 22, 25, 29, 34, 35, 39, 42, 43, 48, 50, 55, 62, 65, 66, 69, 70, 73, 84, 86, 91, 92, 101, 102, 107, 112, 115, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 182, 186, 191, 196, 197, 202, 205

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

			The non prime powers counted under each term:
  n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9  n=10
  -------------------------------------------------
   1    1    1    6   10   12   15   18   22   28
                  1    6   10   14   15   21   26
                       1    6   12   14   20   24
                            1   10   12   18   22
                                 6   10   15   21
                                 1    6   14   20
                                      1   12   18
                                          10   15
                                           6   14
                                           1   12
                                               10
                                                6
                                                1

Restriction of A356068 (first-differences A143731).
First-differences are A368748.
Maxima are A378616.
Other classes of numbers (instead of non prime powers):
- prime: A000027 (diffs A000012), restriction of A000720 (diffs A010051)
- squarefree: A071403 (diffs A373198), restriction of A013928 (diffs A008966)
- nonsquarefree: A378086 (diffs A061399), restriction of A057627 (diffs A107078)
- prime power: A027883 (diffs A366833), restriction of A025528 (diffs A010055)
- composite: A065890 (diffs A046933), restriction of A065855 (diffs  A005171)
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Cf. A027883, A053607, A304521, A343249, A345531, A377057, A377281, A377286, A377289, A377703, A377781, A378032.

Table[Length[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

from sympy import prime, primepi, integer_nthroot
def A378615(n): return int((p:=prime(n))-n-sum(primepi(integer_nthroot(p,k)[0]) for k in range(2,p.bit_length()))) # Chai Wah Wu, Dec 07 2024

a(n) = prime(n) - A027883(n). - Chai Wah Wu, Dec 08 2024

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

A376657 Number of integer factorizations of n into nonsquarefree factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 07 2024

nonn

			The a(n) factorizations for n = 16, 64, 72, 144, 192, 256, 288:
  (16)   (64)     (72)    (144)    (192)     (256)      (288)
  (4*4)  (8*8)    (8*9)   (4*36)   (4*48)    (4*64)     (4*72)
         (4*16)   (4*18)  (8*18)   (8*24)    (8*32)     (8*36)
         (4*4*4)          (9*16)   (12*16)   (16*16)    (9*32)
                          (12*12)  (4*4*12)  (4*8*8)    (12*24)
                          (4*4*9)            (4*4*16)   (16*18)
                                             (4*4*4*4)  (4*8*9)
                                                        (4*4*18)

For prime-powers we have A000688.
Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050320, strict A050326.
For nonprime numbers we have A050370.
The version for partitions is A114374.
For perfect-powers we have A294068.
For non-perfect-powers we have A303707.
For non-prime-powers we have A322452.
The strict case is A376679.
Nonsquarefree numbers:
- A078147 (first differences)
- A376593 (second differences)
- A376594 (inflections and undulations)
- A376595 (nonzero curvature)
A000040 lists the prime numbers, differences A001223.
A001055 counts integer factorizations, strict A045778.
A005117 lists squarefree numbers, differences A076259.
A317829 counts factorizations of superprimorials, strict A337069.
Cf. A008480, A053797, A053806, A061398, A089259, A120992, A124010, A182853, A373198, A375707, A376312.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],NoneTrue[SquareFreeQ]]],{n,100}]

A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

Original entry on oeis.org

0, 1, 11, 14, 39, 53, 109, 179, 222, 240, 251, 319, 337, 481, 505, 508, 578, 664, 674, 738, 818, 835, 877, 905, 933, 1041, 1069, 1098, 1325, 1352, 1392, 1535, 1539, 1567, 1652, 1663, 1732, 1817, 1849, 1960, 2134, 2148, 2158, 2220, 2387, 2428, 2457, 2622, 2625

Offset: 1

Gus Wiseman, Feb 06 2025

nonn

A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061399 counts nonsquarefree integers between primes, see A068361, A061398, A068360, A377783, A378086.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers < prime(n), see A373198, A337030.
A112925 gives the greatest squarefree number between primes, least A112926.
Cf. A057627, A065890, A378032 (differences A378034), A378033 (differences A378036).
Cf. A072284, A120327, A224363, A378040, A378082, A378084.

y=Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}];
Select[Most[Union[y]],Count[y,#]==2&]

a(n) = A378086(A068361(n)) = A378086(A068361(n)+1).

cp's OEIS Frontend

A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).

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A373200 Numbers k such that the k-th maximal antirun of squarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373127.

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A376267 Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

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A376265 Position of first appearance of n in A376264 (run-sums of first differences of nonsquarefree numbers), or 0 if there are none.

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A376266 Sorted positions of first appearances in A376264 (run-sums of first differences of nonsquarefree numbers).

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A378615 Number of non prime powers <= prime(n).

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A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

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A376657 Number of integer factorizations of n into nonsquarefree factors > 1.

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A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

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