A033833 -id:A033833 - cp's OEIS frontend

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

0, 1, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset: 1

Amiram Eldar, May 02 2019

nonn

The corresponding record values are 0, 1, 2, 3, 4, 8, 20, 26, 48, 76, 112, 132, 208, ... (see the link for more values).
Deléglise et al. (2008) proved that the number of powerful (A001694) terms in this sequence is finite. They ask whether a(391) = 485432135516160000 (the 112th powerful term) is the largest. - Amiram Eldar, Aug 20 2019
Is abs(omega(a(n)) - omega(a(n+1))) <= 1? (Cf. A001221.) - David A. Corneth, Apr 16 2020

David A. Corneth, Table of n, a(n) for n = 1..884 (terms 1..762 from Amiram Eldar, calculated by Deléglise et al.)
David A. Corneth, Table of n, a(n) for n = 1..1544 if abs(omega(a(n)) - omega(a(n + 1))) <= 1
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, Journal of Number Theory, Vol. 128, No. 6 (2008), pp. 1676-1716.
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Tables des 761 premiers champions de la fonction de Kalmar, alternative link.
Amiram Eldar, Table of n, a(n), A074206(a(n)), calculated by Deléglise et al.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A001221, A001694, A002093, A033833, A074206, A163272, A330686 (after primorial deflation).

a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; s = {}; am=-1; Do[a1 = a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 0, 10000}]; s

For n >= 1, a(1+n) = A108951(A330686(n)). - Antti Karttunen, Dec 31 2019

A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

Original entry on oeis.org

1, 2, 3, 4, 6, 15, 44

Offset: 1

Gus Wiseman, Jan 07 2020

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. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
  ()  (2)  (4)    (6)    (12)     (72)
           (2*2)  (2*3)  (2*6)    (8*9)
                         (3*4)    (2*36)
                         (2*2*3)  (3*24)
                                  (4*18)
                                  (6*12)
                                  (2*4*9)
                                  (2*6*6)
                                  (3*3*8)
                                  (3*4*6)
                                  (2*2*18)
                                  (2*3*12)
                                  (2*2*2*9)
                                  (2*2*3*6)
                                  (2*3*3*4)
                                  (2*2*2*3*3)

The same for prime numbers (instead of powers of 2) is A330993,
Factorizations are A001055, with image A045782.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly 2^n factorizations is A330989.
Cf. A033833, A045778, A045783, A181821, A305936, A318283, A318284, A330972, A330973, A330976, A330998, A331022.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],IntegerQ[Log[2,Length[facs[Times@@Prime/@nrmptn[#]]]]]&]

A001055(A181821(a(n))) = 2^k for some k >= 0.

A331200 Least number with each record number of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 6, 12, 24, 48, 60, 96, 120, 180, 240, 360, 480, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 8640, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 120960, 151200, 181440, 221760

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A330997 in lacking 64.

			Strict factorizations of the initial terms:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

A subset of A330997.
All terms belong to A025487.
This is the strict version of highly factorable numbers A033833.
The corresponding records are A331232(n) = A045778(a(n)).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n)
Cf. A045783, A325238, A330972, A330973, A331023/A331024, A331201.

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]],{n,nn}];
Table[Position[qv,i][[1,1]],{i,Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]}]

a(37) and beyond from Giovanni Resta, Jan 17 2020

A331232 Record numbers of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 16, 18, 25, 34, 38, 57, 59, 67, 70, 91, 100, 117, 141, 161, 193, 253, 296, 306, 426, 552, 685, 692, 960, 1060, 1067, 1216, 1220, 1589, 1591, 1912, 2029, 2157, 2524, 2886, 3249, 3616, 3875, 4953, 5147, 5285, 5810, 6023, 6112, 6623, 8129

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

			Representatives for the initial records and their strict factorizations:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

The non-strict version is A272691.
The first appearance of a(n) in A045778 is at index A331200(n).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n).
Cf. A033833, A045783, A325238, A330972, A330973, A330997, A331023/A331024, A331201.

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]],{n,nn}];
Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]

def fact(num):
    ret = []
    temp = num
    div = 2
    while temp > 1:
        while temp % div == 0:
            ret.append(div)
            temp //= div
        div += 1
    return ret
def all_partitions(lst):
    if lst:
        x = lst[0]
        for partition in all_partitions(lst[1:]):
            yield [x] + partition
            for i, _ in enumerate(partition):
                partition[i] *= x
                yield partition
                partition[i] //= x
    else:
        yield []
best = 0
terms = [0]
q = 2
while len(terms) < 100:
    total_set = set()
    factors = fact(q)
    total_set = set(tuple(sorted(x)) for x in all_partitions(factors) if len(x) == len(set(x)))
    if len(total_set) > best:
        best = len(total_set)
        terms.append(best)
        print(q,best)
    q += 2#only check evens
print(terms)
#  David Consiglio, Jr., Jan 14 2020

a(n) = A045778(A331200(n)).

a(26)-a(37) from David Consiglio, Jr., Jan 14 2020

a(38) and beyond from Giovanni Resta, Jan 17 2020

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Gus Wiseman, Jan 07 2020

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

The same for powers of 2 (instead of primes) is A330990.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Factorizations of a number's inverse prime shadow are A318284.
Numbers with a prime number of factorizations are A330991.
Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

A001055(A181821(a(n))) belongs to A000040.

A033834 Number of proper factorizations of the numbers with a record number of proper factorizations.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 11, 15, 18, 20, 28, 29, 30, 37, 46, 51, 56, 63, 76, 97, 104, 108, 117, 170, 211, 288, 381, 391, 466, 483, 661, 718, 736, 782, 842, 906, 1096, 1260, 1385, 1396, 1712, 1767, 2115, 2178, 2342, 3078, 3443, 3680, 3929, 5287, 5412, 5446, 5652

Offset: 1

Jeff Burch

nonn

Amiram Eldar, Table of n, a(n) for n = 1..235 (terms 1..118 from T. D. Noe, calculated by E. R. Canfield et al.)
E. R. Canfield, P. Erdős, C. Pomerance, On a problem of Oppenheim concerning Factorisatio Numerorum, J. Number Theory 17 (1983) 1-28, Table 1, column "number of factorizations" - 1.

Records in A001055 (see also A033833). - N. J. A. Sloane, Jun 12 2009

Cf. A028422.

0 term prepended and sequence extended by T. D. Noe, May 18 2012

A111588 Crazy Dice: number of ways to design a pair of n-sided dice with positive integers on their faces, so that the sums when they are tossed occur with the same probabilities as if a pair of standard n-sided dice were tossed.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 8, 1, 2, 2, 10, 1, 8, 1, 8, 2, 2, 1, 33, 2, 2, 4, 8, 1, 13, 1, 26, 2, 2, 2, 57, 1, 2, 2, 33, 1, 13, 1, 8, 8, 2, 1, 140, 2, 8, 2, 8, 1, 33, 2, 33, 2, 2, 1, 125, 1, 2, 8, 71, 2, 13, 1, 8, 2, 13, 1, 348, 1, 2, 8, 8, 2, 13, 1, 140, 10, 2, 1, 122, 2, 2, 2, 33, 1, 118, 2

Offset: 1

N. J. A. Sloane, Nov 17 2005

nonn

It is not required that the two dice be identical, it is not required that the entries be bounded by n and we do not ask that the entries be distinct from one another on each cube.
We pretend for the purpose of this sequence that regular n-sided dice exist for all n.
In other words, how many (unordered) pairs of polynomials B(x) = x^b_1 + x^b_2 + ... + x^b_n, C(x) = x^c_1 + x^c_2 + ... + x^c_n, are there with all exponents positive integers, such that B(x)*C(x) = (x+x^2+x^3+...+x^n)^2?
a(n) = 1 means that the only way two n-sided dice can have the same probability distribution as two normal n-sided dice (each side numbered 1 through n) is if they are normal. a(6) = 2 corresponds to normal dice and Sicherman dice (one labeled 1,2,2,3,3,4 and the other 1,3,4,5,6,8). - Charles R Greathouse IV, Jan 19 2017
Records are: 1, 2, 4, 8, 10, 33, 57, 140, 348, 583, 956, 2036, 2393, 3050, ... and they seem to occur at positions given by A033833. - Antti Karttunen, Aug 28 2017

			The first nontrivial example is for n=4: {1,2,2,3} and {1,3,3,5} together have the same sum probabilities as a pair of {1,2,3,4}. That is, (x + 2x^2 + x^3)(x + 2x^3 + x^5)=(x + x^2 + x^3 + x^4)^2.

M. Gardner, "Penrose Tiles to Trapdoor Ciphers", p. 266.

Antti Karttunen, Table of n, a(n) for n = 1..239
Donald J. Newman, A Problem Seminar, Springer; see Problem #88.

Cf. A033833.

ok(p,e,n)=my(v=Vec(factorback(p,e))); vecmin(v)>=0 && vecsum(v)==n
a(n)=if(n<4, return(1)); my(x='x,f=factor((x^n-1)/(x-1)),p=f[,1],e=2*f[,2]~,u=vector(#e,i,[0,e[i]]),s,t); t=vecmax(e); for(i=1,#e, if(e[i]==t, u[i][2]\=2; break)); forvec(v=u, t=e-v; if(cmp(v,t)<=0 && ok(p,v,n) && ok(p,t,n), s++)); s \\ Charles R Greathouse IV, Jan 19 2017

Edited and extended by Matthew Conroy, Jan 16 2006

Correction to some terms, thanks to Adam Chalcraft. - Matthew Conroy, Apr 04 2010

A291834 First positions of records in A252665.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 432, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 14400, 15120, 20160, 25200, 30240, 40320, 50400, 60480, 75600, 80640, 90720, 100800, 120960, 151200, 181440

Offset: 1

Michael De Vlieger, Sep 03 2017

nonn

Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases.
The terms 2520, 7560, 25200, 221760, 665280, 8648640, ... are not in A033833 but are in A002182. The term 3600 is the smallest that is in neither A033833 nor A002182, but in A007416. The term 831600 is the smallest that is in none of the three aforementioned sequences.
Conjectures based on a(n) < 10^7:
1. Numbers in a(n) are products of the first several consecutive primes p.
2. Outside of a(1), the least prime factor of a(n) has multiplicity > 1. This implies no primes, primorials, or squarefree a(n) for n > 1.
3. The greatest prime factor of a(n) generally has multiplicity 1. Note, however, exceptions in a(n) for n = {1, 2, 3, 5, 7, 9, 12, 13, 15, 17, 19, 26, 29, 33, 73, ...}.
4. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.
See "Records and first positions of records in A252665" for more information. - Michael De Vlieger, Sep 03 2017

Michael De Vlieger, Table of n, a(n) for n = 1..76

Cf. A002182, A007416, A033833, A252665, A291833.

With[{s = Array[f[#, #, 5] &, 10^4]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]]

A278110 a(n) = Product_{k=1..A056811(n)} A000040(k)^A278109(n,k).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 12, 12, 12, 24, 24, 24, 24, 48, 48, 144, 144, 288, 288, 288, 288, 576, 2880, 2880, 8640, 17280, 17280, 17280, 17280, 34560, 34560, 34560, 34560, 207360, 207360, 207360, 207360, 414720, 414720, 414720, 414720, 829440, 2488320

Offset: 1

Jason Kimberley, Feb 02 2017

Jason Kimberley, Table of n, a(n) for n = 1..1000

Cf. A033833, A181817, A278108, A278109.

[&*[Integers()|p^(n div p^2):p in PrimesUpTo(Isqrt(n))]:n in[1..45]];

A291833 Records transform of A252665.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 18, 21, 28, 30, 37, 43, 51, 53, 59, 66, 92, 103, 150, 188, 189, 235, 239, 312, 351, 396, 400, 493, 593, 628, 751, 947, 954, 1283, 1433, 1632, 2063, 2074, 2088, 2446, 2629, 3143, 3582, 3952, 4008, 4121, 4602, 5803, 6046, 6323, 6899

Offset: 1

Michael De Vlieger, Sep 03 2017

nonn

See a-file "Records and first positions of records in A252665" in that sequence for more information. - Michael De Vlieger, Sep 03 2017

			A252665(n) for 1 <= n <= 24 is {1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7}; the records are {1, 2, 3, 4, 5, 7}, thus these are the first 6 terms of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..76

Cf. A033833, A252665, A291834.

f[n_, i_, t_] := b[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i],
Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Union@ FoldList[Max, Array[f[#, #, 5] &, 10^5]] (* Michael De Vlieger, Sep 03 2017, after Alois P. Heinz at A218320 *)

cp's OEIS Frontend

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

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A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

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A331200 Least number with each record number of factorizations into distinct factors > 1.

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A331232 Record numbers of factorizations into distinct factors > 1.

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A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

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A033834 Number of proper factorizations of the numbers with a record number of proper factorizations.

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A111588 Crazy Dice: number of ways to design a pair of n-sided dice with positive integers on their faces, so that the sums when they are tossed occur with the same probabilities as if a pair of standard n-sided dice were tossed.

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A291834 First positions of records in A252665.

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