A013929 -id:A013929 - cp's OEIS frontend

A120327 Smallest nonsquarefree number >= n.

4, 4, 4, 4, 8, 8, 8, 8, 9, 12, 12, 12, 16, 16, 16, 16, 18, 18, 20, 20, 24, 24, 24, 24, 25, 27, 27, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 44, 44, 44, 45, 48, 48, 48, 49, 50, 52, 52, 54, 54, 56, 56, 60, 60, 60, 60, 63, 63, 63, 64, 68, 68, 68, 68, 72, 72, 72, 72

Offset: 1

Zak Seidov, Aug 16 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

For squarefree instead of nonsquarefree we have A067535, differences A378087.
The opposite for squarefree is A070321, differences A378085.
The run-lengths are A078147 if we prepend 4, differences A376593.
The restriction to primes is A377783 (union A378040), differences A377784.
The opposite is A378033 (differences A378036), for prime powers A031218.
First differences are A378039 if we assume that a(1) = 1.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers.
A061398 counts squarefree numbers between primes, zeros A068360.
A061399 counts nonsquarefree numbers between primes, zeros A068361.
Restriction to primes: A112925, A112926, A337030, A378032, A378034, A378037, A378038, A378082, A378084.
Cf. A000015, A007675, A013928, A053797, A053806, A068781, A072284, A073247, A120992, A224363.

Table[NestWhile[ #+1&,n,SquareFreeQ],{n,100}] (* simplified by Harvey P. Dale, Apr 08 2014 *)

A057627 Number of nonsquarefree numbers not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29

Offset: 1

Labos Elemer, Oct 10 2000

Number of integers k in A013929 in the range 1 <= k <= n.
This sequence is different from A013940, albeit the first 35 terms are identical.
Asymptotic to k*n where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Daniel Forgues, Jan 28 2011
This sequence is the sequence of partial sums of A107078 (not of A056170). - Jason Kimberley, Feb 01 2017
Number of partitions of 2n into two parts with the smallest part nonsquarefree. - Wesley Ivan Hurt, Oct 25 2017

			a(36)=13 because 13 nonsquarefree numbers exist which do not exceed 36:{4,8,9,12,16,18,20,24,25,27,28,32,36}.

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

Cf. A005117, A008683, A013929, A013940, A013928, A002321, A028442, A107078, A229099, A294242.

N:= 1000: # to get terms up to a(N)
B:= Array(1..N, numtheory:-issqrfree):
C:= map(`if`,B,0,1):
A:= map(round,Statistics:-CumulativeSum(C)):
seq(A[n],n=1..N); # Robert Israel, Jun 03 2014

Accumulate[Table[If[SquareFreeQ[n],0,1],{n,80}]] (* Harvey P. Dale, Jun 04 2014 *)

a(n) = my(s=0); forsquarefree(k=1, sqrtint(n), s += (-1)^(#k[2]~) * (n\k[1]^2)); n - s; \\ Charles R Greathouse IV, May 18 2015; corrected by Daniel Suteu, May 11 2023

from math import isqrt
from sympy import mobius
def A057627(n): return n-sum(mobius(k)*(n//k**2) for k in range(1,isqrt(n)+1)) # Chai Wah Wu, May 10 2024

(define (A057627 n) (- n (A013928 (+ n 1))))

a(n) = n - A013928(n+1) = n - Sum_{k=1..n} mu(k)^2.

G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x). - Ilya Gutkovskiy, Apr 17 2017

Offset and formula corrected by Antti Karttunen, Jun 03 2014

A347437 Number of factorizations of n with integer alternating product.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 6, 1, 2, 2, 6, 1, 1, 1, 2, 1, 1, 1, 7

Offset: 1

Gus Wiseman, Sep 06 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The factorizations for n = 4, 16, 36, 48, 54, 64, 108:
  (4)   (16)      (36)      (48)        (54)    (64)          (108)
  (2*2) (4*4)     (6*6)     (2*4*6)     (2*3*9) (8*8)         (2*6*9)
        (2*2*4)   (2*2*9)   (3*4*4)     (3*3*6) (2*4*8)       (3*6*6)
        (2*2*2*2) (2*3*6)   (2*2*12)            (4*4*4)       (2*2*27)
                  (3*3*4)   (2*2*2*2*3)         (2*2*16)      (2*3*18)
                  (2*2*3*3)                     (2*2*4*4)     (3*3*12)
                                                (2*2*2*2*4)   (2*2*3*3*3)
                                                (2*2*2*2*2*2)

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

Positions of 1's are A005117, complement A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The restriction to powers of 2 is A344607.
The even-length case is A347438, also the case of alternating product 1.
The reciprocal version is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441.
The reverse version is A347442.
The additive version is A347446, ranked by A347457.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347458, reciprocal A347459.
The ordered version is A347463.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors of n (reverse: A071322).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A062312, A088218, A119620, A316523, A330972, A332269, A347445, A347447, A347451, A347454.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],IntegerQ@*altprod]],{n,100}]

A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A344607(n).

a(n^2) = A347458(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

Original entry on oeis.org

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054

Offset: 1

Benoit Cloitre, Feb 28 2002

nonn

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

A subset of A029707 (lesser index of twin primes).
Prime index of each (prime) term of A061351.
Positions of zeros in A061399.
For perfect power instead of squarefree we have A377436, zeros of A377432.
Positions of zeros in A377784.
The rest of the twin primes are at A378620, indices of A061368.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A061398 counts squarefree numbers between primes, zeros A068360.
A120327 gives the least nonsquarefree number >= n.
Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)

isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

n such that A061398(n) = prime(n+1)-prime(n)-1.

prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024

A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 9, 9, 9, 12, 12, 12, 12, 16, 16, 18, 18, 20, 20, 20, 20, 24, 25, 25, 27, 28, 28, 28, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 45, 45, 45, 48, 49, 50, 50, 52, 52, 54, 54, 56, 56, 56, 56, 60, 60, 60, 63, 64, 64, 64, 64, 68

Offset: 1

Gus Wiseman, Nov 18 2024

nonn

			The nonsquarefree numbers <= 10 are {4, 8, 9}, so a(10) = 9.

For prime-powers we have A031218, differences A377782.
Greatest of the nonsquarefree numbers counted by A057627.
The opposite for squarefree is A067535, differences A378087.
For squarefree we have A070321, differences A378085.
The opposite is A120327 (union A162966), differences A378039.
The restriction to the primes is A378032, opposite A377783 (union A378040).
First-differences are A378036, restriction A378034.
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.
A112925 gives the greatest squarefree number < prime(n), differences A378038.
A112926 gives the least squarefree number > prime(n), differences A378037.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A377047, A377048, A377049, A377784, A378082, A378083, A378084.

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

a(n) = my(k=n); while (issquarefree(k), k--); if(!k, 1, k); \\ Michel Marcus, Jul 26 2025

a(prime(n)) = A378032(n).

a(n) = A013929(A057627(n)), for n > 3. - Ridouane Oudra, Jul 26 2025

A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

Offset: 0

Alois P. Heinz, Sep 05 2014

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

			The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3;
   5,  6;
   7, 10;
  11, 14, 15;
  13, 21, 22, 30;
  17, 26, 33, 35, 42;
  19, 34, 39, 55, 66,  70;
  23, 38, 51, 65, 77,  78, 105, 110;
  29, 46, 57, 85, 91, 102, 130, 154, 165, 210;
  ...
Corresponding triangle of strict integer partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (42) (51) (321)
        (7) (61) (52) (43) (421)
     (8) (71) (62) (53) (521) (431)
(9) (81) (72) (63) (54) (621) (432) (531). - _Gus Wiseman_, Feb 23 2018

Alois P. Heinz, Rows n = 0..42, flattened

Column k=1 gives: A008578(n+1).
Last elements of rows give: A246868.
Row sums give A147655.
Row lengths are: A000009.
Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(
      map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..14);

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

A038838 Numbers that are divisible by the square of an odd prime.

Original entry on oeis.org

9, 18, 25, 27, 36, 45, 49, 50, 54, 63, 72, 75, 81, 90, 98, 99, 100, 108, 117, 121, 125, 126, 135, 144, 147, 150, 153, 162, 169, 171, 175, 180, 189, 196, 198, 200, 207, 216, 225, 234, 242, 243, 245, 250, 252, 261, 270, 275, 279, 288, 289, 294, 297, 300, 306

Offset: 1

David W. Wilson

nonn

Condition 1 of Theorem 7.5 (Robinson, 1979) includes: "k is a multiple of a square of an odd prime." - Jonathan Vos Post, Aug 06 2007
If m is a term, every k*m with k > 1 is another term and the primitive terms are the square of odd primes. The subsequence of odd terms is A053850 while the even terms 18, 36, 50, 54, 72, 90, 98, ... are exactly twice the terms of this sequence. - Bernard Schott, Nov 20 2020
The asymptotic density of this sequence is 1 - 8/Pi^2 = 0.189430... - Amiram Eldar, Nov 21 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM ITEM 45.
Raphael M. Robinson, Multiple tiling of n-dimensional space by unit cubes, Math. Z., Vol. 166 (1979), pp. 225-264.
Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., Vol. 42, No. 2 (2005), pp. 181-211; Robinson theorem cited on p. 204.

Cf. A000040, A065091, A122132 (complement).
Cf. A013929 (supersequence of nonsquarefrees).
Subsequences: A001248 \ {2} (primitives), A053850 (odds), A036785 (divisible by the squares of two distinct primes).
Subsequence of A167662. - Reinhard Zumkeller, Nov 08 2009

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 22 2006 */

list(lim)=my(v=List(),n,e,t); forfactored(k=9,lim\1, e=k[2][,2]; t=#e; n=k[1]; if(if(n%2 && t, vecmax(e)>1, t>1, vecmax(e[2..t])>1, 0), listput(v, k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018

{a(n)} = {j such that for some k>1 A001248(k)|j} = {j such that for some k>0 (A065091(k)^2)|j}. - Jonathan Vos Post, Aug 06 2007

A008966(A000265(a(n))) = 0. - Reinhard Zumkeller, Nov 08 2009

A347439 Number of factorizations of n with integer reciprocal alternating product.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 6, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 4, 0, 0, 0, 1, 0, 0, 0, 5

Offset: 1

Gus Wiseman, Sep 07 2021

nonn

All of these factorizations have an even number of factors, so their reverse-alternating product is also an integer.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
The value of a(n) does not depend solely on the prime signature of n. See the example comparing a(144) and a(400). - Antti Karttunen, Jul 28 2024

			The a(n) factorizations for
n    = 16,       36,       64,           72,       128,          144:
a(n) = 3,        4,        6,            5,        7,            11
--------------------------------------------------------------------------------
       2*8       6*6       8*8           2*36      2*64          2*72
       4*4       2*18      2*32          3*24      4*32          3*48
       2*2*2*2   3*12      4*16          6*12      8*16          4*36
                 2*2*3*3   2*2*2*8       2*2*3*6   2*2*4*8       6*24
                           2*2*4*4       2*3*3*4   2*4*4*4       12*12
                           2*2*2*2*2*2             2*2*2*16      2*2*6*6
                                                   2*2*2*2*2*4   2*3*3*8
                                                                 3*3*4*4
                                                                 2*2*2*18
                                                                 2*2*3*12
                                                                 2*2*2*2*3*3
From _Antti Karttunen_, Jul 28 2024 (Start)
For n=400, there are 12 such factorizations:
  2*200
  4*100
  5*80
  10*40
  20*20
  2*2*2*50
  2*2*5*20
  2*2*10*10
  2*4*5*10
  2*5*5*8
  4*4*5*5
  2*2*2*2*5*5.
Note that 400 = 2^4 * 5^2 has the same prime signature as 144 = 2^4 * 3^2. 400 = 2*4*5*10 is the factorization for which there is no analogous factorization of 144, as 2*3*4*6 doesn't satisfy the condition of having an integer reciprocal alternating product.
(End)

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

Positions of 0's are A005117 \ {1}.
Positions of non-0's are 1 and A013929.
The restriction to powers of 2 is A027187, reverse A035363.
Positions of 1's are 1 and A082293.
The additive version is A119620, ranked by A347451 and A028982.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reciprocal version is A347437.
The reverse version is A347438.
Allowing any alternating product < 1 gives A347440.
The non-reciprocal reverse version is A347442.
Allowing any alternating product >= 1 gives A347456.
The restriction to perfect squares is A347459, non-reciprocal A347458.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A236913, A316523, A330972, A332269, A344606, A344607, A347445, A347446, A347454, A347457, A347463.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i,{i,Length[q]}];
Table[Length[Select[facs[n],IntegerQ[recaltprod[#]]&]],{n,100}]

A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

A347439(n, m=0, ap=1, e=1) = if(1==n, 1==denominator(ap), sumdiv(n, d, if(d>1 && d>=m, A347439(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(2^n) = A027187(n).

a(n^2) = A347459(n).

Data section extended up to a(108) by Antti Karttunen, Jul 28 2024

A347442 Number of factorizations of n with integer reverse-alternating product.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 8, 2, 3, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 8, 5, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 9, 1, 3, 3, 8, 1, 1, 1, 3, 1, 1, 1, 12

Offset: 1

Gus Wiseman, Sep 08 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(n) factorizations for n = 4, 8, 16, 32, 36, 54, 64:
  (4)    (8)      (16)       (32)         (36)       (54)     (64)
  (2*2)  (2*4)    (2*8)      (4*8)        (6*6)      (3*18)   (8*8)
         (2*2*2)  (4*4)      (2*16)       (2*18)     (2*3*9)  (2*32)
                  (2*2*4)    (2*2*8)      (3*12)     (3*3*6)  (4*16)
                  (2*2*2*2)  (2*4*4)      (2*2*9)             (2*4*8)
                             (2*2*2*4)    (2*3*6)             (4*4*4)
                             (2*2*2*2*2)  (3*3*4)             (2*2*16)
                                          (2*2*3*3)           (2*2*2*8)
                                                              (2*2*4*4)
                                                              (2*2*2*2*4)
                                                              (2*2*2*2*2*2)

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

The restriction to powers of 2 is A000041, reverse A344607.
Positions of 2's are A001248.
Positions of 1's are A005117.
Positions of non-1's are A013929.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
The non-reverse version is A347437.
The reciprocal version is A347438.
The even-length case is A347439.
Allowing any alternating product < 1 gives A347440.
The odd-length case is A347441, ranked by A347453.
The additive version is A347445, ranked by A347457.
The non-reverse additive version is A347446, ranked by A347454.
Allowing any alternating product >= 1 gives A347456.
The ordered version is A347463.
A038548 counts possible reverse-alternating products of factorizations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
Cf. A000290, A025047, A330972, A347443, A347449, A347451, A347458, A347459, A347460, A347462.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],IntegerQ@*revaltprod]],{n,100}]

A347442(n, m=n, ap=1, e=0) = if(1==n, 1==denominator(ap), sumdiv(n, d, if((d>1)&&(d<=m), A347442(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A000041(n).

Data section extended up to a(108) by Antti Karttunen, Oct 22 2023

A375735 First differences of non-prime-powers (inclusive).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 04 2024

nonn

Inclusive means 1 is a prime-power but not a non-prime-power.

Non-prime-powers (inclusive) are listed by A024619.

			The 5th non-prime-power (inclusive) is 15, and the 6th is 18, so a(5) = 3.

For perfect powers (A001597) we have the latter terms of A053289.
For nonprime numbers (A002808) we have the latter terms of A073783.
For squarefree numbers (A005117) we have the latter terms of A076259.
First differences of A024619.
For prime-powers (A246655) we have the latter terms of A057820.
Essentially the same as the exclusive version, A375708.
Positions of 1's are A375713(n) - 1.
For runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, first differences A001223.
A000961 lists prime-powers (inclusive).
A007916 lists non-perfect-powers, first differences A375706.
A013929 lists the nonsquarefree numbers, first differences A078147.
A246655 lists prime-powers (exclusive).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power anti-runs: A373576, min A120430, max A006549, length A373671.
Non-prime-power anti-runs: A373679, min A373575, max A255346, len A373672.
Cf. A046933, A061399, A093555, A176246, A251092, A323054, A375709, A375736.

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

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375735(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 10 2024

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A057627 Number of nonsquarefree numbers not exceeding n.

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A347437 Number of factorizations of n with integer alternating product.

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A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

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A378033 Greatest nonsquarefree number <= n, or 1 if there is none (the case n <= 3).

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A246867 Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

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A038838 Numbers that are divisible by the square of an odd prime.

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