A376595 -id:A376595 - cp's OEIS frontend

A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

-3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, 0, -1, -2, 3, -3, 2, 1, -2, 0, 2, -2, -1, 0, 3, 0, 0, 0, -3, 2, -2, 2, -2, 0, 1, 2, -1, -2, 3, 0, -1, -2, 1, 0, -1, 2, 1, -2, 0, 2, -3, 1, -1, 2, -2, 3, 0, 0, -3, 2, 1, -2, 0, 2

Offset: 1

Gus Wiseman, Oct 01 2024

sign

The range is {-3, -2, -1, 0, 1, 2, 3}.

			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, ...
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, ...
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, ...

The version for A000002 is A376604, first differences of A054354.
The first differences were A078147.
Zeros are A376594, complement A376595.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A064113 lists positions of adjacent equal prime gaps.
A114374 counts partitions into nonsquarefree numbers.
A246655 lists prime-powers exclusive, inclusive A000961.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376594 (inflections and undulations), A376595 (nonzero curvature).
Cf. A061398, A053797, A053806, A120992, A182853, A251092, A373198, A375707, A376305, A376306, A376311, A376312, A376655.

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

from math import isqrt
from sympy import mobius, factorint
def A376593(n):
    def f(x): return n+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)
    k = next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values()))
    return next(i for i in range(1-k,5-k) if any(d>1 for d in factorint(m+(k<<1)+i).values())) # Chai Wah Wu, Oct 02 2024

A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are nonzero.

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with nonzero terms at (A376603):
  2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of nonzero curvature in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
These are the positions of nonzero terms in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For prime instead of composite we have A333214.
The complement is A376602.
For upward concavity (instead of nonzero) we have A376651, downward A376652.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).
For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
Cf. A000961, A064113, A246655, A251092, A258025, A333254.

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

A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

Original entry on oeis.org

4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with nonzeros at (A376598):
  4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...

Gus Wiseman, Points of nonzero curvature in the prime-powers inclusive.

The first differences were A057820, see also A376340.
First differences are A376309.
These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376597.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
`A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]

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]

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]

A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, 38, 40, 42, 46, 49, 53, 54, 60, 61, 63, 65, 67, 69, 71, 72, 78, 79, 81, 82, 83, 84, 85, 89, 91, 99, 101, 105, 106, 110, 111, 113, 115, 117, 118, 122, 124, 132, 134, 136, 138, 148

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376599) are nonzero.

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, subtract 1 and shift left.

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  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, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with nonzero terms (A376601) at:
  1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, ...

Gus Wiseman, Points of nonzero curvature in the non-prime-powers inclusive.

For first differences we had A375735, ones A375713(n) - 1.
These are the nonzeros of A376599.
The complement is A376600.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A024619/A361102 list non-prime-powers inclusive.
A057820 gives first differences of prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376600 (inflections and undulations).
For nonzero curvature: A333214 (prime), A376603 (composite), A376588 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Join@@Position[Sign[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2]],1|-1]

A376679 Number of strict 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, 1, 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, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 08 2024

nonn

			The a(3456) = 28 factorizations are:
  (4*8*9*12)  (4*9*96)    (36*96)   (3456)
              (8*9*48)    (4*864)
              (4*12*72)   (48*72)
              (4*16*54)   (54*64)
              (4*18*48)   (8*432)
              (4*24*36)   (9*384)
              (4*27*32)   (12*288)
              (4*8*108)   (16*216)
              (8*12*36)   (18*192)
              (8*16*27)   (24*144)
              (8*18*24)   (27*128)
              (9*12*32)   (32*108)
              (9*16*24)
              (12*16*18)

Dominic McCarty, Table of n, a(n) for n = 1..10000

Positions of zeros are A005117 (squarefree numbers), complement A013929.
For squarefree instead of nonsquarefree we have A050326, non-strict A050320.
For prime-powers we have A050361, non-strict A000688.
For nonprime numbers we have A050372, non-strict A050370.
The version for partitions is A256012, non-strict A114374.
For perfect-powers we have A323090, non-strict A294068.
The non-strict version is A376657.
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, A303707, A322452, A375707, A376312.

function nextNonSquareFree(val){val+=1;for(let i=2;i*i<=val;i+=1){if(val%i==0&&val%(i*i)==0){return val}}return nextNonSquareFree(val)}function strictFactorCount(val,maxFactor){if(val==1){return 1}let sum=0;while(maxFactorDominic McCarty, Oct 19 2024

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],UnsameQ@@#&&NoneTrue[#,SquareFreeQ]&]],{n,100}] (* corrected by Gus Wiseman, Jun 27 2025 *)

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}]

cp's OEIS Frontend

A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

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A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

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A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

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

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

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A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

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A376679 Number of strict integer factorizations of n into nonsquarefree factors > 1.

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

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