A071625 -id:A071625 - cp's OEIS frontend

A367685 Numbers divisible by their multiset multiplicity kernel.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107

Offset: 1

Gus Wiseman, Nov 30 2023

nonn

First differs from A344586 in lacking 120.
We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.
First differs from A212165 at n=73: A212165(73)=120 is not a term of this. - Amiram Eldar, Dec 04 2023

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}

Includes all prime-powers A000961.
The only squarefree terms are the primes A008578.
Partitions of this type are counted by A367684.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A367579 lists MMK, ranks A367580, sum A367581, max A367583.
Cf. A005117, A039956, A051904, A052409, A072774, A130091, A367582, A367584, A367586, A367682.

mmk[n_Integer]:= Product[Min[#]^Length[#]&[First/@Select[FactorInteger[n], Last[#]==k&]], {k,Union[Last/@FactorInteger[n]]}];
Select[Range[100], Divisible[#,mmk[#]]&]

A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 9, 7, 2, 3, 25, 11, 6, 13, 49, 25, 2, 17, 6, 19, 10, 49, 121, 23, 6, 5, 169, 3, 14, 29, 125, 31, 2, 121, 289, 49, 9, 37, 361, 169, 10, 41, 343, 43, 22, 15, 529, 47, 6, 7, 10, 289, 26, 53, 6, 121, 14, 361, 841, 59, 50, 61, 961, 21, 2, 169, 1331

Offset: 1

Gus Wiseman, Dec 03 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define the multiset multiplicity cokernel MMC(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}. As an operation on multisets MMC is represented by A367858, and as an operation on their ranks it is represented by A367859.

			90 has prime factorization 2^1*3^2*5^1, so for k = 1 we have 5^2, and for k = 2 we have 3^1, so a(90) = 75.

Positions of 2's are A000079 without 1.
Positions of 3's are A000244 without 1.
Positions of primes (including 1) are A000961.
Depends only on rootless base A052410, see A007916.
Positions of prime powers are A072774.
Positions of squarefree numbers are A130091.
For kernel instead of cokernel we have A367580, ranks of A367579.
Rows of A367858 have this rank, sum A367860, max A061395, min A367587.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
Cf. A005117, A020639, A052409, A175781, A181819, A353742, A367584, A367585.

mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Times@@mmc[Join@@ConstantArray@@@FactorInteger[n]], {n,30}]

a(n^k) = a(n) for all positive integers n and k.
If n is squarefree, a(n) = A006530(n)^A001222(n).
A055396(a(n)) = A367587(n).
A056239(a(n)) = A367860(n).
A061395(a(n)) = A061395(n).
A001222(a(n)) = A001221(n).
A001221(a(n)) = A071625(n).
A071625(a(n)) = A323022(n).

A136567 a(n) is the number of exponents occurring only once each in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Jan 07 2008

nonn

Records are in A006939: 1, 2, 12, 360, 75600, ..., . - Robert G. Wilson v, Jan 20 2008

			4200 = 2^3 * 3^1 * 5^2 * 7^1. The exponents of the prime factorization are therefore 3,1,2,1. The exponents occurring exactly once are 2 and 3. So a(4200) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A056169, A071625, A133924, A136566, A181819.

For a(n)=0 see A130092 plus the term 1; for a(n)=1 see A000961.

f[n_] := Block[{fi = Sort[Last /@ FactorInteger@n]}, Count[ Count[fi, # ] & /@ Union@fi, 1]]; f[1] = 0; Array[f, 105] (* Robert G. Wilson v, Jan 20 2008 *)
Table[Boole[n != 1] Count[Split@ Sort[FactorInteger[n][[All, -1]]], ?(Length@ # == 1 &)], {n, 105}] (* _Michael De Vlieger, Jul 24 2017 *)

A136567(n) = { my(exps=(factor(n)[, 2]), m=prod(i=1, length(exps), prime(exps[i])), f=factor(m)[, 2]); sum(i=1, #f, f[i]==1); }; \\ Antti Karttunen, Jul 24 2017

(define (A136567 n) (A056169 (A181819 n))) ;; Antti Karttunen, Jul 24 2017

a(n) = A056169(A181819(n)). - Antti Karttunen, Jul 24 2017

More terms from Robert G. Wilson v, Jan 20 2008

A136568 Irregular triangle read by rows: row n contains the distinct values (in numerical order) making up the nonzero exponents in the prime factorization of n. (Row 1 = (0).).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 3, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 6, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Leroy Quet, Jan 07 2008

Row n, for n>=2, contains A071625(n) terms.

			4200 = 2^3 * 3^1 * 5^2 * 7^1. The exponents of the prime factorization are therefore 3,1,2,1. Therefore row 4200 is (1,2,3).
Table begins:
  0;
  1;
  1;
  2;
  1;
  1;
  1;
  3;
  2;
  1;
  1;
  1, 2;

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index to sequences related to prime signature.

Cf. A071625, A136565.

Join[{0},Flatten[Rest[Table[Union[Transpose[FactorInteger[n]][[2]]],{n,100}]]]] (* Harvey P. Dale, Mar 19 2013 *)

row(n) = if(n == 1, [0], Set(factor(n)[,2])); \\ Amiram Eldar, Aug 04 2024

More terms from Harvey P. Dale, Mar 19 2013

A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

Original entry on oeis.org

174636000, 206388000, 244490400, 261954000, 269892000, 274428000, 288943200, 291060000, 301644000, 309582000, 343980000, 349272000, 365148000, 366735600, 377848800, 383292000, 404838000, 411642000, 412776000, 422301600, 433414800, 449820000, 452466000, 457380000

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(5) = 174636000.
Positions of 5's in A071625.
Numbers k such that A001221(A181819(k)) = 5.

			174636000 = 2^5 * 3^4 * 5^3 * 7^2 * 11^1 has five distinct exponents so belongs to the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

One distinct exponent: A062770 or A072774.
Two distinct exponents: A323055.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A006939, A051270, A059404, A071625, A118914, A181819, A182855, A323014, A323022, A323024.

Select[Range[300000000],Length[Union[Last/@FactorInteger[#]]]==5&]

is(n) = #Set(factor(n)[, 2]) == 5 \\ David A. Corneth, Jan 12 2019

a(13)-a(24) from Daniel Suteu, Jan 12 2019

A325264 Numbers whose omega-sequence sums to 7.

Original entry on oeis.org

30, 36, 42, 64, 66, 70, 78, 100, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 196, 222, 225, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The sequence of terms together with their prime indices and omega-sequences begins:
   30: {1,2,3} (3,3,1)
   36: {1,1,2,2} (4,2,1)
   42: {1,2,4} (3,3,1)
   64: {1,1,1,1,1,1} (6,1)
   66: {1,2,5} (3,3,1)
   70: {1,3,4} (3,3,1)
   78: {1,2,6} (3,3,1)
  100: {1,1,3,3} (4,2,1)
  102: {1,2,7} (3,3,1)
  105: {2,3,4} (3,3,1)
  110: {1,3,5} (3,3,1)
  114: {1,2,8} (3,3,1)
  130: {1,3,6} (3,3,1)
  138: {1,2,9} (3,3,1)
  154: {1,4,5} (3,3,1)
  165: {2,3,5} (3,3,1)
  170: {1,3,7} (3,3,1)
  174: {1,2,10} (3,3,1)
  182: {1,4,6} (3,3,1)
  186: {1,2,11} (3,3,1)
  190: {1,3,8} (3,3,1)
  195: {2,3,6} (3,3,1)
  196: {1,1,4,4} (4,2,1)

Positions of 7's in A325249.
Numbers with omega-sequence summing to m: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A060687, A062770, A118914, A130091, A303555, A323023, A325238, A325265, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],Total[omseq[#]]==7&]

A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 50, 51, 52, 55, 57, 58, 60, 62, 63, 65, 68, 69, 74, 75, 76, 77, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 99, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 126, 129, 132

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

Also numbers whose adjusted frequency depth is one plus their number of prime factors counted with multiplicity. The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose adjusted frequency depth is equal to their length plus 1. The enumeration of these partitions by sum is given by A127002.

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
   6:     {1,2} (2,2,1)
  10:     {1,3} (2,2,1)
  12:   {1,1,2} (3,2,2,1)
  14:     {1,4} (2,2,1)
  15:     {2,3} (2,2,1)
  18:   {1,2,2} (3,2,2,1)
  20:   {1,1,3} (3,2,2,1)
  21:     {2,4} (2,2,1)
  22:     {1,5} (2,2,1)
  26:     {1,6} (2,2,1)
  28:   {1,1,4} (3,2,2,1)
  33:     {2,5} (2,2,1)
  34:     {1,7} (2,2,1)
  35:     {3,4} (2,2,1)
  38:     {1,8} (2,2,1)
  39:     {2,6} (2,2,1)
  44:   {1,1,5} (3,2,2,1)
  45:   {2,2,3} (3,2,2,1)
  46:     {1,9} (2,2,1)
  50:   {1,3,3} (3,2,2,1)
  51:     {2,7} (2,2,1)
  52:   {1,1,6} (3,2,2,1)
  55:     {3,5} (2,2,1)
  57:     {2,8} (2,2,1)
  58:    {1,10} (2,2,1)
  60: {1,1,2,3} (4,3,2,2,1)

Cf. A056239, A112798, A118914, A127002, A181819, A323023, A325246, A325259, A325266, A325270, A325277, A325282.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

fdadj[n_Integer]:=If[n==1,0,Length[NestWhileList[Times@@Prime/@Last/@FactorInteger[#]&,n,!PrimeQ[#]&]]];
Select[Range[100],fdadj[#]==PrimeOmega[#]+1&]

A336618 Maximum divisor of n! with equal prime multiplicities.

Original entry on oeis.org

1, 1, 2, 6, 8, 30, 36, 210, 210, 1296, 1296, 2310, 7776, 30030, 44100, 46656, 46656, 510510, 1679616, 9699690, 9699690, 10077696, 10077696, 223092870, 223092870, 729000000, 901800900, 13060694016, 13060694016, 13060694016, 78364164096, 200560490130

Offset: 0

Gus Wiseman, Jul 30 2020

nonn

A number has equal prime multiplicities iff it is a power of a squarefree number. We call such numbers uniform, so a(n) is the maximum uniform divisor of n!.

			The sequence of terms together with their prime signatures begins:
       1: ()
       1: ()
       2: (1)
       6: (1,1)
       8: (3)
      30: (1,1,1)
      36: (2,2)
     210: (1,1,1,1)
     210: (1,1,1,1)
    1296: (4,4)
    1296: (4,4)
    2310: (1,1,1,1,1)
    7776: (5,5)
   30030: (1,1,1,1,1,1)
   44100: (2,2,2,2)

Amiram Eldar, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

A327526 is the non-factorial generalization, with quotient A327528.
A336415 counts these divisors.
A336616 is the version for distinct prime multiplicities.
A336619 is the quotient n!/a(n).
A047966 counts uniform partitions.
A071625 counts distinct prime multiplicities.
A072774 lists uniform numbers.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A319269 counts uniform factorizations.
A327524 counts factorizations of uniform numbers into uniform numbers.
A327527 counts uniform divisors.
Cf. A000005, A001222, A001597, A007916, A098859, A124010, A182853, A327498.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617, A336414, A336416.

Table[Max@@Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&],{n,0,15}]

a(n) = A327526(n!).

A342091 a(n) is the least number k such that k! has n distinct exponents in its prime factorization.

Original entry on oeis.org

1, 2, 4, 6, 10, 15, 22, 33, 44, 55, 68, 85, 102, 119, 145, 174, 203, 232, 261, 296, 333, 370, 410, 451, 492, 533, 590, 656, 708, 767, 826, 885, 944, 1005, 1072, 1143, 1207, 1278, 1422, 1455, 1562, 1652, 1778, 1917, 2032, 2134, 2235, 2328, 2425, 2540, 2682, 2831, 2929

Offset: 0

Amiram Eldar, Feb 27 2021

nonn

After n=0, first differs from A073818 at n = 27.
a(n) is the least k such that A071625(k!) = A071626(k) = n.
Is this sequence strictly increasing?

			a(1) = 2 since 2! = 2^1 is the least factorial with a single exponent (1) in its prime factorization.
a(2) = 4 since 4! = 24 = 2^3 * 3^1 is the least factorial with 2 distinct exponents (1 and 3) in its prime factorization.
a(3) = 6 since 6! = 720 = 2^4 * 3^2 * 5^1 is the least factorial with 3 distinct exponents (1, 2 and 4) in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 0..2109 (terms below 10^7)
Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer., Vol. 34 (1982), pp. 25-45.

Cf. A000142, A071625, A071626, A073818, A133924, A336616, A336619, A336867, A336868.

f[1] = 0; f[n_] := Length @ Union[FactorInteger[n!][[;; , 2]]]; seq[max_] := Module[{s = Table[0, {max}], n = 1, c = 0}, While[c < max, i = f[n] + 1; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50]

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 02 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.

The version with multiplicity is A181591.
For partitions we have A355383, with multiplicity A339006.
The version for compositions is A355384.
Positions of first appearances are A355386.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 count prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A000712, A022811, A056239, A071625, A118914, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]

cp's OEIS Frontend

A367685 Numbers divisible by their multiset multiplicity kernel.

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A367859 Multiset multiplicity cokernel (MMC) of n. Product of (greatest prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n.

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A136567 a(n) is the number of exponents occurring only once each in the prime factorization of n.

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A136568 Irregular triangle read by rows: row n contains the distinct values (in numerical order) making up the nonzero exponents in the prime factorization of n. (Row 1 = (0).).

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A323056 Numbers with exactly five distinct exponents in their prime factorization, or five distinct parts in their prime signature.

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A325264 Numbers whose omega-sequence sums to 7.

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A325281 Numbers of the form a*b, a*a*b, or a*a*b*c where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).

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A336618 Maximum divisor of n! with equal prime multiplicities.

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A325281 Numbers of the form ab, aab, or aabc where a, b, and c are distinct primes. Numbers with sorted prime signature (1,1), (1,2), or (1,1,2).