A130091 -id:A130091 - cp's OEIS frontend

A342029 Starts of runs of 3 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

1, 2, 3, 7, 11, 16, 17, 18, 23, 27, 43, 47, 48, 52, 71, 79, 96, 97, 107, 135, 147, 151, 162, 171, 191, 241, 242, 243, 331, 351, 359, 367, 387, 423, 431, 486, 507, 539, 547, 567, 575, 576, 599, 603, 639, 907, 927, 1051, 1107, 1123, 1151, 1215, 1249, 1250, 1323

Offset: 1

Amiram Eldar, Feb 25 2021

nonn

			2 is a term since 2, 3 and 4 = 2^2 all have a single exponent in their prime factorization.
4 is not a term since in the run {4, 5, 6} the third member 6 = 2*3 has two equal exponents (1) in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
Bernardo Recamán Santos, Consecutive numbers with mutually distinct exponents in their canonical prime factorization, MathOverflow, Mar 30 2015.

Subsequence of A130091 and A342028.

Subsequences: A342030, A342031.

q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; v = q /@ Range[3]; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 3]]; v = Join[Rest[v], {q[k]}], {k, 4, 1500}]; seq

A336942 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 5, 95, 8823, 4952323, 20285515801, 714092378624317

Offset: 0

Gus Wiseman, Aug 14 2020

			The a(0) = 1 through a(2) = 5 chains:
  {1}  {2,1}  {12,1}
              {12,2,1}
              {12,3,1}
              {12,4,1}
              {12,4,2,1}

A076954 can be used instead of A006939 (cf. A307895, A325337).
A336423 and A336571 are not restricted to A006939.
A336941 is the version not restricted by A130091.
A337075 is the version for factorials.
A074206 counts 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.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
Cf. A000005, A001055, A002033, A032741, A067824, A124010, A167865, A336419, A336420, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[chern[n]],{n,0,3}]

a(n) = A336423(A006939(n)) = A336571(A006939(n)).

A342030 Starts of runs of 4 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

Original entry on oeis.org

1, 2, 16, 17, 47, 96, 241, 242, 575, 1249, 2644, 2645, 4049, 4372, 4373, 4799, 9124, 12248, 33749, 72250, 120049, 130436, 281249, 303748, 1431124, 1431125, 1531250, 2101247, 3693761, 4085656, 4910975, 12502348, 12502349, 14268481, 22997761, 25486324, 26693549

Offset: 1

Amiram Eldar, Feb 25 2021

nonn

			2 is a term since 2, 3, 4 = 2^2, and 5 all have a single exponent in their prime factorization.
3 is not a term since in the run {3, 4, 5, 6} the fourth member 6 = 2*3 has two equal exponents (1) in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..196 (terms below 10^11)
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
Bernardo Recamán Santos, Consecutive numbers with mutually distinct exponents in their canonical prime factorization, MathOverflow, Mar 30 2015.

Subsequence of A130091, A342028 and A342029.

A342031 is a subsequence.

q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; v = q /@ Range[4]; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 4]]; v = Join[Rest[v], {q[k]}], {k, 5, 10^5}]; seq

A342031 Starts of runs of 5 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

Original entry on oeis.org

1, 16, 241, 2644, 4372, 1431124, 12502348, 112753348, 750031648, 2844282247, 5882272324, 6741230497, 8004453748, 87346072024, 130489991521, 218551872247, 245127093748, 460925878624, 804065433748, 1176638279524, 2210511903748, 2404792968748, 2483167488748, 3121595927521

Offset: 1

Amiram Eldar, Feb 25 2021

nonn

Bernardo Recamán Santos (2015) showed that there is no run of more than 23 consecutive numbers, since numbers of the form 36*k - 6 and 36*k + 6 do not have distinct exponents. Pace Nielsen and Adam P. Goucher showed that there can be only finitely many runs of 23 consecutive numbers (see MathOverflow link).

Aktaş and Ram Murty (2017) gave an explicit upper bound to such a run of 23 numbers. They found the first 5 terms of this sequence (and stated that there are a few more known up to 7*10^8), and said that we may conjecture (based on numerical evidence) that there are no 6 consecutive numbers.

			16 is a term since 16 = 2^4, 17, 18 = 2*3^2, 19 and 20 = 2^2*5 all have distinct exponents in their prime factorization.

Martin Ehrenstein, Table of n, a(n) for n = 1..43
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
Bernardo Recamán Santos, Consecutive numbers with mutually distinct exponents in their canonical prime factorization, MathOverflow, Mar 30 2015.

Subsequence of A130091, A342028, A342029 and A342030.

q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; v = q /@ Range[5]; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 5]]; v = Join[Rest[v], {q[k]}], {k, 6, 1.3*10^6}]; seq

a(15) and beyond from Martin Ehrenstein, Mar 08 2021

A337104 Number of strict chains of divisors from n! to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 0, 14, 0, 384, 0, 0, 0, 21077680, 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, 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, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

The support appears to be {0, 1, 2, 4, 6, 10}.

			The a(4) = 14 chains:
  24/1
  24/2/1
  24/3/1
  24/4/1
  24/8/1
  24/12/1
  24/4/2/1
  24/8/2/1
  24/8/4/1
  24/12/2/1
  24/12/3/1
  24/12/4/1
  24/8/4/2/1
  24/12/4/2/1

A336867 appears to be the positions of zeros.
A336868 is the characteristic function (image under A057427).
A336942 is the version for superprimorials (n > 1).
A337105 does not require distinct prime multiplicities.
A337074 does not require chains to end with 1.
A337075 is the version for chains not containing n!.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts 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.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336425 counts divisible pairs of divisors of n!, both in A130091.
A336571 counts chains of divisors 1 < d < n using A130091.
A337071 counts chains of divisors starting with n!.
Cf. A002033, A022559, A071626, A098859, A124010, A167865, A327523, A336424, A336941.

strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[DeleteCases[Divisors[n],n],UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n!],{n,0,8}]

a(n) = A337075(n) whenever A337075(n) != 0.
a(n) = A337074(n)/2 for n > 1.
a(n) = A336423(n!).

A343012 Lexicographically earliest sequence of distinct numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 8, 9, 10, 7, 12, 15, 14, 11, 16, 18, 20, 21, 22, 13, 24, 25, 27, 28, 30, 32, 33, 26, 17, 35, 36, 40, 42, 44, 39, 34, 19, 45, 48, 49, 50, 54, 55, 52, 51, 38, 23, 56, 60, 63, 64, 66, 65, 68, 57, 46, 29, 70, 72, 75, 77, 78, 80, 81, 84, 85, 76, 69

Offset: 1

Amiram Eldar, Apr 02 2021

Is this sequence a permutation of the positive integers?

			The first partial products are:
1
1 * 2 = 2 = 2^1
1 * 2 * 4 = 8 = 2^3
1 * 2 * 4 * 3 = 24 = 2^3 * 3^1
1 * 2 * 4 * 3 * 6 = 144 = 2^4 * 3^2

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

Cf. A130091, A343013.

q[n_] := UnsameQ @@ FactorInteger[n][[;; , 2]]; seq = {1}; prod = 1; Do[k = 1; While[MemberQ[seq, k] || ! q[k*prod], k++]; AppendTo[seq, k]; prod *= k, {100}]; seq

A343013 Lexicographically earliest strictly increasing sequence of numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 12, 15, 16, 17, 18, 20, 24, 25, 27, 30, 32, 34, 35, 36, 40, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 70, 72, 75, 78, 79, 80, 81, 84, 85, 90, 91, 96, 98, 100, 102, 104, 105, 108, 112, 119, 120, 121, 125, 126, 128, 130, 132, 135, 136, 140, 143

Offset: 1

Amiram Eldar, Apr 02 2021

nonn

The numbers of terms not exceeding 10^k, for k=1,2,..., are 6, 46, 293, 1939, 13534, 97379, .... Apparently, this sequence has an asymptotic density 0.

Are there infinitely many terms of each prime signature? In particular, the prime terms seem to be sparse: 2, 5, 17, 79, 491, 2011, 8191 and no other below 10^6. Are there infinitely many prime terms in this sequence?

			The first partial products are:
1
1 * 2 = 2 = 2^1
1 * 2 * 4 = 8 = 2^3
1 * 2 * 4 * 5 = 40 = 2^3 * 5^1
1 * 2 * 4 * 5 * 8 = 320 = 2^6 * 5^1

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

Cf. A130091, A343012.

q[n_] := UnsameQ @@ FactorInteger[n][[;; , 2]]; seq = {1}; n = 1; prod = 1; Do[k = n + 1; While[!q[k*prod], k++]; AppendTo[seq, k]; prod *= k; n = k, {100}]; seq

A353693 a(n) is the least multiplier k such that the exponents in the prime factorization of k*n are mutually distinct (A130091).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 3, 2, 5, 2, 1, 2, 3, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 2, 1, 6, 1, 2, 1, 1, 5, 12, 1, 1, 3, 20, 1, 1, 1, 2, 1, 1, 7, 12, 1, 1, 1, 2, 1, 6, 5, 2

Offset: 1

Amiram Eldar, May 04 2022

nonn

First differs from A327499 at n = 30.
If n = Product_{i=1..k} p_i is squarefree (A005117), and p_1 < p_2 < ... < p_k are its k ordered prime divisors, then a(n) = Product_{i} p_i^(k-i).
If n is powerful (A001694) then a(n) = a(n/rad(n)), where rad(n) is the squarefree kernel of n (A007947). In general, if k = A051904(n) is the minimal exponent in the prime factorization of n, then a(n) = a(n/(rad(n)^(k-1))).

			a(2) = 1 since 2 = 2^1 has only one exponent (1) in its prime factorization.
a(6) = 2 since 6 = 2*3 has two equal exponents (1) in its prime factorization, and 2*6 = 12 = 2^2*3 has two distinct exponents (1 and 2).

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

Cf. A001694, A005117, A007947, A130091, A130092, A327498, A327499, A353694.

a[n_] := Module[{k = 1}, While[!UnsameQ @@ FactorInteger[k*n][[;; , 2]], k++]; k]; Array[a, 100]

a(n) = my(k=1, f=factor(n)[,2]); while(#Set(f) != #f, k++; f=factor(k*n)[,2]); k; \\ Michel Marcus, May 05 2022

a(n) = 1 if and only if n is in A130091.
a(A130092(n)) > 1.
rad(a(n)) | rad(n).
a(n) = A353694(n)/n.

A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

Original entry on oeis.org

1, 1, 1, 3, 14, 48, 384, 1308, 40288, 933848, 21077680, 75690016, 5471262080, 7964665440, 54595767744, 17948164982144, 3454946386353664, 5010658671663616, 723456523262697984, 950502767770273280, 165679731871366906880, 8443707247468681128448

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(1) = 1 through a(4) = 14 chains (with n! prepended):
  1  2/1  6/1    24/1
          6/2/1  24/2/1
          6/3/1  24/3/1
                 24/4/1
                 24/8/1
                 24/12/1
                 24/4/2/1
                 24/8/2/1
                 24/8/4/1
                 24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/8/4/2/1
                 24/12/4/2/1

A336571 is the generalization to not just factorial numbers.
A337104 is the version for chains containing n!.
A000005 counts divisors.
A001055 counts factorizations.
A032741 counts proper divisors.
A071625 counts distinct prime multiplicities.
A074206 counts 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.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336424 counts factorizations using A130091.
A336425 counts divisible pairs of divisors of n!, both in A130091.
Cf. A002033, A098859, A124010, A294068, A327499, A327500, A327523, A337071.

chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[n!],{n,0,5}]

a(n) = A337104(n) whenever A337104(n) != 0.

a(n) = A336571(n!).

A353694 a(n) is the least multiple of n with mutually distinct exponents in its prime factorization (A130091).

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 12, 13, 28, 45, 16, 17, 18, 19, 20, 63, 44, 23, 24, 25, 52, 27, 28, 29, 360, 31, 32, 99, 68, 175, 72, 37, 76, 117, 40, 41, 504, 43, 44, 45, 92, 47, 48, 49, 50, 153, 52, 53, 54, 275, 56, 171, 116, 59, 360, 61, 124, 63, 64, 325

Offset: 1

Amiram Eldar, May 04 2022

nonn

			a(2) = 2 since 2 = 2^1 has only one exponent (1) in its prime factorization.
a(6) = 12 since 6 = 2*3 has two equal exponents (1) in its prime factorization, and 2*6 = 12 = 2^2*3 has two distinct exponents (1 and 2).

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

Cf. A130091, A130092, A353693.

a[n_] := Module[{k = n}, While[!UnsameQ @@ FactorInteger[k][[;; , 2]], k += n]; k]; Array[a, 100]

a(n) = n if and only if n is in A130091.
a(A130092(n)) > n.
a(n) = n * A353693(n).

cp's OEIS Frontend

A342029 Starts of runs of 3 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A336942 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with the superprimorial A006939(n) and ending with 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A342030 Starts of runs of 4 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A342031 Starts of runs of 5 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A337104 Number of strict chains of divisors from n! to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A343012 Lexicographically earliest sequence of distinct numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A343013 Lexicographically earliest strictly increasing sequence of numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A353693 a(n) is the least multiplier k such that the exponents in the prime factorization of k*n are mutually distinct (A130091).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI