cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 24 results. Next

A053289 First differences of consecutive perfect powers (A001597).

Original entry on oeis.org

3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100
Offset: 1

Views

Author

Labos Elemer, Mar 03 2000

Keywords

Comments

Michel Waldschmidt writes: Conjecture 1.3 (Pillai). Let k be a positive integer. The equation x^p - y^q = k where the unknowns x, y, p and q take integer values, all >= 2, has only finitely many solutions (x,y,p,q). This means that in the increasing sequence of perfect powers [A001597] the difference between two consecutive terms [the present sequence] tends to infinity. It is not even known whether for, say, k=2, Pillai's equation has only finitely many solutions. A related open question is whether the number 6 occurs as a difference between two perfect powers. See Sierpiński [1970], problem 238a, p. 116. - Jonathan Vos Post, Feb 18 2008
Are there are any adjacent equal terms? - Gus Wiseman, Oct 08 2024

Examples

			Consecutive perfect powers are A001597(14) = 121, A001597(13) = 100, so a(13) = 121 - 100 = 21.
		

References

  • Wacław Sierpiński, 250 problems in elementary number theory, Modern Analytic and Computational Methods in Science and Mathematics, No. 26, American Elsevier, Warsaw, 1970, pp. 21, 115-116.
  • S. S. Pillai, On the equation 2^x - 3^y = 2^X - 3^Y, Bull, Calcutta Math. Soc. 37 (1945) 15-20.

Crossrefs

For non-perfect-powers (A007916) we have A375706.
The union is A023055.
For prime-powers (A000961 or A246655) we have A057820.
Sorted positions of first appearances are A376268, complement A376519.
For second differences we have A376559.
Ascending and descending points are A376560 and A376561.
A001597 lists perfect-powers.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.

Programs

  • Mathematica
    Differences@ Select[Range@ 3200, # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* Michael De Vlieger, Jun 30 2016, after Ant King at A001597 *)
  • Python
    from sympy import mobius, integer_nthroot
    def A053289(n):
        if n==1: return 3
        def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
        kmin, kmax = 1,2
        while f(kmax)+1 >= kmax:
            kmax <<= 1
        rmin, rmax = 1, kmax
        while True:
            kmid = kmax+kmin>>1
            if f(kmid)+1 < kmid:
                kmax = kmid
            else:
                kmin = kmid
            if kmax-kmin <= 1:
                break
        while True:
            rmid = rmax+rmin>>1
            if f(rmid) < rmid:
                rmax = rmid
            else:
                rmin = rmid
            if rmax-rmin <= 1:
                break
        return kmax-rmax # Chai Wah Wu, Aug 13 2024

Formula

a(n) = A001597(n+1) - A001597(n). - Jonathan Vos Post, Feb 18 2008
From Amiram Eldar, Jun 30 2023: (Start)
Formulas from Jakimczuk (2016):
Lim sup_{n->oo} a(n)/(2*n) = 1.
Lim inf_{n->oo} a(n)/(2*n)^(2/3 + eps) = 0. (End)
Can be obtained by inserting 0 between 3 and 6 in A375702 and then adding 1 to all terms. In particular, for n > 2, a(n+1) - 1 = A375702(n). - Gus Wiseman, Sep 14 2024

A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

Original entry on oeis.org

1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Oct 01 2024

Keywords

Comments

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

Examples

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
		

Crossrefs

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros are A376588, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376588 (inflections and undulations), A376589 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
    Differences[Select[Range[100],radQ],2]
  • Python
    from itertools import count
    from sympy import mobius, integer_nthroot, perfect_power
    def A376562(n):
        def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        r = m+((k:=next(i for i in count(1) if not perfect_power(m+i)))<<1)
        return next(i for i in count(1-k) if not perfect_power(r+i)) # Chai Wah Wu, Oct 02 2024

A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

Original entry on oeis.org

1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, -16, -1, 21, 2, 2, -15, -11, 30, 2, 2, 2, 2, 2, 2, 2, -22, -15, 41, 2, 2, 2, -36, 3, 37, 2, 2, 2, -34, -11, 49, 2, 2, -66, 45, 3, -61, 2, 83, 2, 2, 2, 2, -63, 25, 42, 2, -9, -89
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2024

Keywords

Comments

Perfect-powers A007916 are numbers with a proper integer root.
Does this sequence contain zero?

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, ...
		

Crossrefs

The version for A000002 is A376604, first differences of A054354.
For first differences we have A053289, union A023055, firsts A376268, A376519.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
For perfect-powers: A053289 (first differences), A376560 (positive curvature), A376561 (negative curvature).
For second differences: A036263 (prime), A073445 (composite), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
    Differences[Select[Range[1000],perpowQ],2]
  • PARI
    lista(nn) = my(v = concat (1, select(ispower, [1..nn])), w = vector(#v-1, i, v[i+1] - v[i])); vector(#w-1, i, w[i+1] - w[i]); \\ Michel Marcus, Oct 02 2024
  • Python
    from sympy import mobius, integer_nthroot
    def A376559(n):
        def bisection(f,kmin=0,kmax=1):
            while f(kmax) > kmax: kmax <<= 1
            while kmax-kmin > 1:
                kmid = kmax+kmin>>1
                if f(kmid) <= kmid:
                    kmax = kmid
                else:
                    kmin = kmid
            return kmax
        def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
        a = bisection(f,n,n)
        b = bisection(lambda x:f(x)+1,a,a)
        return a+bisection(lambda x:f(x)+2,b,b)-(b<<1) # Chai Wah Wu, Oct 02 2024
    

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

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, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Aug 03 2020

Keywords

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

Examples

			The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
  2  4    8      5*12     16       4*9      32         48
     2*2  2*4    3*20     4*4      3*12     4*8        4*12
          2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
                 2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
                          2*2*2*2  2*2*9    2*2*8      2*24
                                   2*2*3*3  2*2*2*4    2*3*8
                                            2*2*2*2*2  2*2*12
                                                       2*2*3*4
                                                       2*2*2*2*3
		

Crossrefs

A327523 is the case when n is restricted to belong to A130091 also.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty chains of divisors.
A281116 counts factorizations with no common divisor.
A302696 lists numbers whose prime indices are pairwise coprime.
A305149 counts stable factorizations.
A320439 counts factorizations using A289509.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A336568 = not a product of two numbers with distinct prime multiplicities.
A336569 counts maximal chains of elements of A130091.
A337256 counts chains of divisors.

Programs

  • Mathematica
    facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
    Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]

A336571 Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 5, 1, 5, 3, 3, 1, 14, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 3, 17, 1, 3, 3, 14, 1, 4, 1, 5, 5, 3, 1, 36, 2, 5, 3, 5, 1, 14, 3, 14, 3, 3, 1, 16, 1, 3, 5, 32, 3, 4, 1, 5, 3, 4, 1, 35, 1, 3, 5, 5, 3, 4, 1, 36, 8, 3, 1
Offset: 1

Views

Author

Gus Wiseman, Jul 29 2020

Keywords

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

Examples

			The a(n) sets for n = 4, 6, 12, 16, 24, 84, 36:
  {}   {}   {}     {}       {}        {}        {}
  {2}  {2}  {2}    {2}      {2}       {2}       {2}
       {3}  {3}    {4}      {3}       {3}       {3}
            {4}    {8}      {4}       {4}       {4}
            {2,4}  {2,4}    {8}       {7}       {9}
                   {2,8}    {12}      {12}      {12}
                   {4,8}    {2,4}     {28}      {18}
                   {2,4,8}  {2,8}     {2,4}     {2,4}
                            {4,8}     {2,12}    {3,9}
                            {2,12}    {2,28}    {2,12}
                            {3,12}    {3,12}    {2,18}
                            {4,12}    {4,12}    {3,12}
                            {2,4,8}   {4,28}    {3,18}
                            {2,4,12}  {7,28}    {4,12}
                                      {2,4,12}  {9,18}
                                      {2,4,28}  {2,4,12}
                                                {3,9,18}
		

Crossrefs

A336423 is the version for chains containing n.
A336570 is the maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.

Programs

  • Mathematica
    strchns[n_]:=If[n==1,1,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
    Table[strchns[n],{n,100}]

A322452 Number of factorizations of n into factors > 1 not including any prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2018

Keywords

Comments

Also the number of multiset partitions of the multiset of prime indices of n with no constant parts.

Examples

			The a(840) = 11 factorizations are (6*10*14), (6*140), (10*84), (12*70), (14*60), (15*56), (20*42), (21*40), (24*35), (28*30), (840).
		

Crossrefs

Positions of 0's are the prime powers A000961.

Programs

  • Mathematica
    acfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[acfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimePowerQ[#]&]}]];
    Table[Length[acfacs[n]],{n,100}]
  • PARI
    A322452(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(1A322452(n/d, d))); (s)); \\ Antti Karttunen, Jan 03 2019
    
  • PARI
    first(n) = my(res=vector(n)); for(i=1, n, f=factor(i); v=vecsort(f[,2] , , 4); f[, 2] = v; fb = factorback(f); if(fb==i, res[i] = A322452(i), res[i] = res[fb])); res \\ A322452 the function above \\ David A. Corneth, Jan 03 2019

Extensions

More terms from Antti Karttunen, Jan 03 2019

A336417 Number of perfect-power divisors of superprimorials A006939.

Original entry on oeis.org

1, 1, 2, 5, 15, 44, 169, 652, 3106, 15286, 89933, 532476, 3698650, 25749335, 204947216, 1636097441, 14693641859, 132055603656, 1319433514898, 13186485900967, 144978145009105, 1594375302986404, 19128405558986057, 229508085926717076, 2983342885319348522
Offset: 0

Views

Author

Gus Wiseman, Jul 24 2020

Keywords

Comments

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.
The n-th superprimorial number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Examples

			The a(0) = 1 through a(4) = 15 divisors:
  1  2  12  360  75600
-------------------------
  1  1   1    1      1
         4    4      4
              8      8
              9      9
             36     16
                    25
                    27
                    36
                   100
                   144
                   216
                   225
                   400
                   900
                  3600
		

Crossrefs

A000325 is the uniform version.
A076954 can be used instead of A006939.
A336416 gives the same for factorials instead of superprimorials.
A000217 counts prime power divisors of superprimorials.
A000961 gives prime powers.
A001597 gives perfect powers, with complement A007916.
A006939 gives superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A091050 counts perfect power divisors.
A181818 gives products of superprimorials.
A294068 counts factorizations using perfect powers.
A317829 counts factorizations of superprimorials.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
    perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
    Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]
  • PARI
    a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ Andrew Howroyd, Aug 30 2020

Formula

a(n) = A091050(A006939(n)).
a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - Andrew Howroyd, Aug 30 2020

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 30 2020

A336426 Numbers that cannot be written as a product of superprimorials {2, 12, 360, 75600, ...}.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76
Offset: 1

Views

Author

Gus Wiseman, Jul 26 2020

Keywords

Comments

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Examples

			We have 288 = 2*12*12 so 288 is not in the sequence.
		

Crossrefs

A181818 is the complement.
A336497 is the version for superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.

Programs

  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
    facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&],{d,Select[s,Divisible[n,#]&]}]];
    Select[Range[100],facsusing[Array[chern,30],#]=={}&]

A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 5, 0, 3, 0, 3, 1, 1, 0, 7, 1, 1, 2, 3, 0, 4, 0, 7, 1, 1, 1, 15, 0, 1, 1, 7, 0, 4, 0, 3, 3, 1, 0, 16, 1, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 18, 0, 1, 3, 16, 1, 4, 0, 3, 1, 4, 0, 32, 0, 1, 3, 3, 1, 4, 0, 16, 5, 1, 0, 18, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Examples

			The a(36) = 15 pairs of factorizations:
  (2*2*3*3)|(4*9)
  (2*2*3*3)|(6*6)
  (2*2*3*3)|(36)
    (2*2*9)|(6*6)
    (2*2*9)|(36)
    (2*3*6)|(4*9)
    (2*3*6)|(36)
     (2*18)|(36)
    (3*3*4)|(6*6)
    (3*3*4)|(36)
     (3*12)|(36)
      (4*9)|(6*6)
      (4*9)|(36)
      (6*6)|(4*9)
      (6*6)|(36)
		

Crossrefs

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
    Table[Length[Select[Tuples[facs[n],2],!Or@@Divisible@@@Tuples[#]&]],{n,100}]

A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Gus Wiseman, Dec 08 2018

Keywords

Comments

First differs from A322438 at a(144) = 3, A322438(144) = 4.
From Antti Karttunen, Dec 11 2020: (Start)
Zeros occur on numbers that are either of the form p^k, or q * p^k, or p*q*r, for some primes p, q, r, and exponent k >= 0. [Note also that in all these cases, when x > 1, A307408(x) = 2+A307409(x) = 2 + (A001222(x) - 1)*A001221(x) = A000005(x)].
Proof:
It is easy to see that for such numbers it is not possible to obtain two such distinct factorizations, that no factor of the other would not divide some factor of the other.
Conversely, the complement set of above is formed of such composites n that have at least one unitary divisor that is either of the form
(1) p^x * q^y, with x, y >= 2,
or
(2) p^x * q^y * r^z, with x >= 2, and y, z >= 1,
or
(3) p^x * q^y * r^z * s^w, with x, y, z, w >= 1,
where p, q, r, s are distinct primes. Let's indicate with C the remaining portion of k coprime to p, q, r and s (which could be also 1). Then in case (1) we can construct two factorizations, the first having factors (p*q*C) and (p^(x-1) * q^(y-1)), and the second having factors (p^x * C) and (q^y) that are guaranteed to satisfy the condition that no factor in the other factorization divides any of the factors of the other factorization. For case (2) pairs like {(p * q^y * C), (p^(x-1) * r^z)} and {(p^x * C), (q^y * r^z)}, and for case (3) pairs like {(p^x * q^y * C), (r^z * s^w)} and {(p^x * r^z * C), (q^y * s^w)} offer similar examples, therefore a(n) > 0 for all such cases.
(End)

Examples

			The a(120) = 2 pairs of such factorizations:
   (6*20)|(8*15)
   (8*15)|(10*12)
The a(144) = 3 pairs of factorizations:
   (6*24)|(9,16)
   (8*18)|(12*12)
   (9*16)|(12*12)
The a(210) = 3 pairs of factorizations:
   (6*35)|(10*21)
   (6*35)|(14*15)
  (10*21)|(14*15)
[Note that 210 is the first squarefree number obtaining nonzero value]
The a(240) = 4 pairs of factorizations:
   (6*40)|(15*16)
   (8*30)|(12*20)
  (10*24)|(15*16)
  (12*20)|(15*16)
The a(1728) = 14 pairs of factorizations:
    (6*6*48)|(27*64)
   (6*12*24)|(27*64)
     (6*288)|(27*64)
    (8*8*27)|(12*12*12)
  (12*12*12)|(27*64)
  (12*12*12)|(32*54)
    (12*144)|(27*64)
    (12*144)|(32*54)
    (16*108)|(24*72)
     (18*96)|(27*64)
     (24*72)|(27*64)
     (24*72)|(32*54)
     (27*64)|(36*48)
     (32*54)|(36*48)
		

Crossrefs

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
    Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]],{n,100}]
  • PARI
    factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
    is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
    number_of_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_ndf_pair(z[i],z[j])));
    A322437(n) = number_of_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

Formula

For n > 0, a(A002110(n)) = A322441(n)/2 = A339626(n). - Antti Karttunen, Dec 10 2020

Extensions

Data section extended up to a(120) and more examples added by Antti Karttunen, Dec 10 2020
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