A067535 -id:A067535 - cp's OEIS frontend

A378036 First differences of A378033 (greatest positive integer < n that is 1 or nonsquarefree).

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

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65539

Positions of 0 are A005117 - 1, complement A013929 - 1.
Sums for squarefree numbers are A070321 (restriction A112925).
The restricted opposite is A377784, differences of A377783 (union A378040).
First-differences of A378033.
The restriction is A378034, differences of A378032.
The restricted opposite for squarefree is A378037, differences of A112926.
The opposite is A378039, differences of A120327 (union A162966).
For squarefree numbers we have A378085, restriction A378038.
The opposite for squarefree is A378087, differences of A067535.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259, seconds A376590.
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.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A065514, A377047, A377048, A377049, A377703, A378082, A378083, A378084.

Differences[Table[NestWhile[#-1&,n,#>1&&SquareFreeQ[#]&],{n,100}]]

A378033(n) = if(n<=3, 1, forstep(k=n, 0, -1, if(!issquarefree(k), return(k))));
A378036(n) = (A378033(1+n)-A378033(n)); \\ Antti Karttunen, Jan 28 2025

a(prime(n)) = A378034(n).

Data section extended to a(107) by Antti Karttunen, Jan 28 2025

A378037 First differences of A112926 (smallest squarefree integer > prime(n)).

Original entry on oeis.org

2, 1, 4, 3, 1, 5, 2, 5, 4, 3, 5, 4, 4, 5, 4, 6, 1, 7, 4, 1, 8, 3, 6, 10, 1, 3, 4, 1, 4, 15, 4, 5, 3, 10, 3, 4, 7, 5, 4, 7, 1, 11, 1, 5, 2, 12, 13, 3, 1, 5, 6, 5, 7, 5, 7, 6, 2, 5, 4, 3, 10, 14, 4, 1, 4, 16, 5, 10, 4, 1, 8, 8, 4, 7, 4, 5, 8, 4, 8, 11, 1, 11, 1

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

First differences of A112926, restriction of A067535, differences A378087.
For prime powers we have A377703.
The nonsquarefree version is A377784 (differences of A377783), restriction of A378039.
The nonsquarefree opposite is A378034, first differences of A378032.
The opposite is A378038, differences of A112925.
The unrestricted opposite is A378085 except first term, differences of A070321.
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.
Cf. A007674, A013928, A045882, A053797, A053806, A072284, A073247, A120327, A280892, A378036, A378040, A378084.

Differences[Table[NestWhile[#+1&,Prime[n]+1,!SquareFreeQ[#]&],{n,100}]]

A378038 First differences of A112925 = greatest squarefree number < prime(n).

Original entry on oeis.org

1, 1, 3, 4, 1, 4, 2, 5, 4, 4, 5, 4, 3, 4, 5, 7, 1, 7, 4, 1, 7, 4, 5, 8, 2, 5, 4, 1, 4, 12, 7, 4, 4, 8, 3, 6, 6, 5, 4, 8, 1, 11, 1, 4, 2, 13, 12, 4, 1, 4, 7, 1, 10, 6, 7, 5, 2, 5, 4, 4, 9, 14, 5, 1, 3, 16, 5, 11, 1, 2, 9, 8, 5, 6, 5, 4, 9, 4, 8, 11, 1, 11, 1, 7

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

First differences of A112925, restriction of A070321, differences A378085.
For prime powers we have A377781, opposite A377703.
The nonsquarefree opposite is A377784 (differences of A377783), restriction of A378039.
The nonsquarefree version is A378034, first differences of A378032.
The opposite is A378037, differences of A112926.
The unrestricted opposite is A378087, differences of A067535.
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.
Cf. A007674, A013928, A045882, A053797, A053806, A072284, A073247, A120327, A280892, A378036, A378040, A378084.

Differences[Table[NestWhile[#-1&,Prime[n]-1,!SquareFreeQ[#]&],{n,100}]]

A372683 Least squarefree number >= 2^n.

Original entry on oeis.org

1, 2, 5, 10, 17, 33, 65, 129, 257, 514, 1027, 2049, 4097, 8193, 16385, 32770, 65537, 131073, 262145, 524289, 1048577, 2097154, 4194305, 8388609, 16777217, 33554433, 67108865, 134217730, 268435457, 536870913, 1073741826, 2147483649, 4294967297, 8589934594

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The terms together with their binary expansions and binary indices begin:
       1:                    1 ~ {1}
       2:                   10 ~ {2}
       5:                  101 ~ {1,3}
      10:                 1010 ~ {2,4}
      17:                10001 ~ {1,5}
      33:               100001 ~ {1,6}
      65:              1000001 ~ {1,7}
     129:             10000001 ~ {1,8}
     257:            100000001 ~ {1,9}
     514:           1000000010 ~ {2,10}
    1027:          10000000011 ~ {1,2,11}
    2049:         100000000001 ~ {1,12}
    4097:        1000000000001 ~ {1,13}
    8193:       10000000000001 ~ {1,14}
   16385:      100000000000001 ~ {1,15}
   32770:     1000000000000010 ~ {2,16}
   65537:    10000000000000001 ~ {1,17}
  131073:   100000000000000001 ~ {1,18}
  262145:  1000000000000000001 ~ {1,19}
  524289: 10000000000000000001 ~ {1,20}

For primes instead of powers of two we have A112926, opposite A112925, sum A373197, length A373198.
Counting zeros instead of all bits gives A372473, firsts of A372472.
These are squarefree numbers at indices A372540, firsts of A372475.
Counting ones instead of all bits gives A372541, firsts of A372433.
The opposite (greatest squarefree number <= 2^n) is A372889.
The difference from 2^n is A373125.
For prime instead of squarefree we have:
- bits A372684, firsts of A035100
- zeros A372474, firsts of A035103
- ones A372517, firsts of A014499
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A005117 lists squarefree numbers.
A030190 gives binary expansion, reversed A030308, length A070939 or A029837.
A061398 counts squarefree numbers between primes (exclusive).
A077643 counts squarefree terms between powers of 2, run-lengths of A372475.
A143658 counts squarefree numbers up to 2^n.
Cf. A029931, A048793, A049095, A049096, A059015, A069010, A076259, A077641, A211997, A230877.

Table[NestWhile[#+1&,2^n,!SquareFreeQ[#]&],{n,0,10}]

a(n) = my(k=2^n); while (!issquarefree(k), k++); k; \\ Michel Marcus, May 29 2024

from itertools import count
from sympy import factorint
def A372683(n): return next(i for i in count(1<Chai Wah Wu, Aug 26 2024

a(n) = A005117(A372540(n)).

a(n) = A067535(2^n). - R. J. Mathar, May 31 2024

A081221 Number of consecutive numbers >= n having at least one square divisor > 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 10 2003

nonn

The first time terms 0..7 occur is at n = 1, 4, 8, 48, 242, 844, 22020, 217070. - Antti Karttunen, Sep 22 2017

			For n = 3, 3 is a squarefree number, thus a(3) = 0.
For n = 48, neither 48 = 2^4 * 3 nor 49 = 7^2, nor 50 = 2^2 * 5 are squarefree, but 51 = 3*17 is, thus a(48) = 3. - _Antti Karttunen_, Sep 22 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005117, A008683, A045882, A051681, A068781, A067535, A080733.

Flatten@ Map[If[First@ # == 0, #, Reverse@ Range@ Length@ #] &, SplitBy[Table[DivisorSum[n, 1 &, And[# > 1, IntegerQ@ Sqrt@ #] &], {n, 120}], # > 0 &]] (* Michael De Vlieger, Sep 22 2017 *)

A081221(n) = { my(k=0); while(!issquarefree(n+k),k++); k; }; \\ Antti Karttunen, Sep 22 2017

from itertools import count
from sympy import factorint
def A081221(n): return next(m for m in count(0) if max(factorint(n+m).values(),default=0)<=1) # Chai Wah Wu, Dec 04 2024

mu(k) = 0 for n <= k < n+a(n) and mu(n+a(n)) <> 0, where mu = A008683 (Moebius function).
a(n)*mu(n) = 0.
a(A068781(n)) > 0.
a(n) = A067535(n) - n. - Amiram Eldar, Oct 10 2023

A378085 First differences of A070321 (greatest squarefree number <= n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 04 2024

nonn

			The greatest squarefree number <= 50 is 47, and the greatest squarefree number <= 51 is 51, so a(51) = 4.

Ones are A007674.
Zeros are A013929 - 1.
Twos are A280892.
Positions of first appearances are A020755 - 1 (except first term).
First-differences of A070321.
The nonsquarefree restriction is A378034, differences of A378032.
For nonsquarefree numbers we have A378036, differences of A378033.
The opposite restriction to primes is A378037, differences of A112926.
The restriction to primes is A378038, differences of A112925.
The nonsquarefree opposite is A378039, restriction A377784.
The opposite version is A378087.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
Cf. A013928, A020754, A045882, A053797, A053806, A067535, A068781, A072284, A073247, A073248, A120327.

Differences[Table[NestWhile[#-1&,n,#>1&&!SquareFreeQ[#]&],{n,100}]]

A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

Original entry on oeis.org

1, 2, 1, 4, 2, 5, 1, 2, 8, 2, 3, 5, 4, 2, 6, 4, 6, 5, 3, 4, 2, 8, 2, 6, 8, 4, 2, 4, 2, 16, 3, 3, 6, 2, 10, 2, 6, 6, 6, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 4, 13, 1, 6, 6, 2, 6, 4, 8, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 8, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Nov 14 2024

nonn

Note 1 is a power of a prime but not a prime-power.

Differences of A065514, which is the restriction of A031218 (differences A377782).
The opposite is A377703 (restriction of A000015), differences of A345531.
The opposite for nonsquarefree is A377784, differences of A377783.
For nonsquarefree we have A378034, differences of A378032 (restriction of A378033).
The opposite for squarefree is A378037, differences of A112926 (restriction of A067535).
For squarefree we have A378038, differences of A112925 (restriction of A070321).
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.
A361102 lists the non-powers of primes, differences A375708.
Prime-powers between primes:
- A053607 primes
- A080101 count (exclusive)
- A304521 by bits
- A366833 count
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A001597, A007916, A008864, A053289, A120327, A343249, A375706, A377281, A377282, A377289.

Differences[Table[NestWhile[#-1&,Prime[n]-1,#>1&&!PrimePowerQ[#]&],{n,100}]]

A378371 Distance between n and the least non prime power >= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 28 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The least non prime power >= 4 is 6, so a(4) = 2.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime we have A007920 (A151800), strict A013632.
For composite we have A010051 (A113646 except initial terms).
For perfect power we have A074984 (A377468)
For squarefree we have A081221 (A067535).
For nonsquarefree we have (A120327).
For non perfect power we have A378357 (A378358).
The opposite version is A378366 (A378367).
For prime power we have A378370, strict A377282 (A000015).
This sequence is A378371 (A378372).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A053707, A065514, A276781 (A031218), A343249, A345531, A376596, A376597, A377051, A377054, A377289, A378457.

Table[NestWhile[#+1&,n,PrimePowerQ[#]&]-n,{n,100}]

a(n) = A378372(n) - n.

A378372 Least non prime power >= n, allowing 1.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 10, 10, 10, 10, 12, 12, 14, 14, 15, 18, 18, 18, 20, 20, 21, 22, 24, 24, 26, 26, 28, 28, 30, 30, 33, 33, 33, 34, 35, 36, 38, 38, 39, 40, 42, 42, 44, 44, 45, 46, 48, 48, 50, 50, 51, 52, 54, 54, 55, 56, 57, 58, 60, 60, 62, 62, 63, 65, 65, 66, 68

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The least non prime power >= 4 is 6, so a(4) = 6.

Sequences obtained by subtracting n from each term are placed in parentheses below.
For prime power we have A000015 (A378370).
For squarefree we have A067535 (A081221).
For composite we have A113646 (A010051).
For nonsquarefree we have A120327.
For prime we have A151800 (A007920), strict (A013632).
Run-lengths are 1 and A375708.
For perfect power we have A377468 (A074984).
For non-perfect power we have A378358 (A378357).
The opposite is A378367, distance A378366.
This sequence is A378372 (A378371).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007916, A031218 (A276781), A053707, A065514, A345531 (A377281), A377051, A377282, A377289, A378457.

Table[NestWhile[#+1&,n,PrimePowerQ[#]&],{n,100}]

a(n) = A378371(n) + n.

A074984 m^p-n, for smallest m^p>=n.

Original entry on oeis.org

0, 2, 1, 0, 3, 2, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 4, 3, 2, 1, 0, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8

Offset: 1

Zak Seidov, Oct 07 2002

nonn

a(n) = 0 if n = m^p that is if n is a full power (square, cube etc.).

This is the distance between n and the next perfect power. The previous perfect power is A081676, which differs from n by A069584. After a(8) = a(9) this sequence is an anti-run (no adjacent equal terms). - Gus Wiseman, Dec 02 2024

Wikipedia, Catalan's conjecture

Sequences obtained by subtracting n from each term are placed in parentheses below.
Positions of 0 are A001597.
Positions of 1 are A375704.
The version for primes is A007920 (A007918).
The opposite (greatest perfect power <= n) is A069584 (A081676).
The version for perfect powers is A074984 (this) (A377468).
The version for squarefree numbers is A081221 (A067535).
The version for non perfect powers is A378357 (A378358).
The version for nonsquarefree numbers is A378369 (A120327).
The version for prime powers is A378370 (A000015).
The version for non prime powers is A378371 (A378372).
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A131605 lists perfect powers that are not prime powers.
A377432 counts perfect powers between primes, zeros A377436.
Cf. A014210, A023055, A045542, A052410, A076412, A151800, A188951, A216765.

powerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; powerQ[1] = True; a[n_] := For[k = n, True, k++, If[powerQ[k], Return[k-n]]]; Table[a[n], {n, 1, 92}] (* Jean-François Alcover, Apr 19 2013 *)

a(n) = { if (n==1, return (0)); my(nn = n); while(! ispower(nn), nn++); return (nn - n);} \\ Michel Marcus, Apr 19 2013

a(n) = A377468(n) - n. - Gus Wiseman, Dec 02 2024

cp's OEIS Frontend

A378036 First differences of A378033 (greatest positive integer < n that is 1 or nonsquarefree).

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A378037 First differences of A112926 (smallest squarefree integer > prime(n)).

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A378038 First differences of A112925 = greatest squarefree number < prime(n).

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A372683 Least squarefree number >= 2^n.

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A081221 Number of consecutive numbers >= n having at least one square divisor > 1.

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A378085 First differences of A070321 (greatest squarefree number <= n).

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A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

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A378371 Distance between n and the least non prime power >= n, allowing 1.

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A378372 Least non prime power >= n, allowing 1.

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