A051904 -id:A051904 - cp's OEIS frontend

A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

1, 6, 30, 210, 900, 7776, 27000, 279936, 810000, 9261000, 24300000, 362797056, 729000000, 13060694016, 21870000000, 408410100000, 656100000000, 16926659444736, 19683000000000, 609359740010496, 590490000000000, 18010885410000000, 17714700000000000

Offset: 0

Gus Wiseman, Feb 18 2025

nonn

Is this sequence strictly increasing?
From David Consiglio, Jr., Feb 20 2025: (Start)
The answer to the question above is: no, a(21) < a(20). And all subsequent odd indexed terms are lower than their even predecessors.
All terms must be a product of x primes (with multiplicity) to the y power where x-y = n and x mod y = 0. There are very few combinations of numbers that meet these criteria, so checking all of them to find the minimum outcome is quite fast.
Example --> n=5
6 primes to the 1 power --> 6 distinct primes
  2*3*5*7*11*13 = 30030
7 primes to the 2 power -- disallowed (5 mod 2 = 1)
8 primes to the 3 power -- disallowed (4 mod 3 = 1)
9 primes to the 4 power -- disallowed (9 mod 4 = 1)
10 primes to the 5 power --> 2 distinct primes
  2*2*2*2*2*3*3*3*3*3 = 7776
The minimum value is 7776 and thus a(5) = 7776. (End)

			The terms together with their prime indices begin:
        1: {}
        6: {1,2}
       30: {1,2,3}
      210: {1,2,3,4}
      900: {1,1,2,2,3,3}
     7776: {1,1,1,1,1,2,2,2,2,2}
    27000: {1,1,1,2,2,2,3,3,3}
   279936: {1,1,1,1,1,1,1,2,2,2,2,2,2,2}
   810000: {1,1,1,1,2,2,2,2,3,3,3,3}
  9261000: {1,1,1,2,2,2,3,3,3,4,4,4}

David Consiglio, Jr., Table of n, a(n) for n = 0..100
David Consiglio, Jr., Python program

Position of first appearance of n in A001222 - A136565.
For factors instead of exponents we have A280286 (sorted A381075), firsts of A280292.
For indices instead of exponents we have A380956 (sorted A380957), firsts of A380955.
A000040 lists the primes, differences A001223.
A005361 gives product of prime exponents.
A055396 gives least prime index, greatest A061395.
A056239 (reverse A296150) adds up prime indices, row sums of A112798.
A124010 lists prime exponents (signature); A001221, A051903, A051904.
Cf. A046660, A071625, A075254, A075255, A076694, A130091, A130092, A290106, A380986.

prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
q=Table[Total[prisig[n]]-Total[Union[prisig[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

a(10)-a(11) from Michel Marcus, Feb 20 2025

a(12) and beyond from David Consiglio, Jr., Feb 20 2025

A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121

Offset: 1

Gus Wiseman, Mar 02 2025

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.
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

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

In A048767:
- fixed points are A048768, A217605
- conjugate is A381431, fixed points A000961, A000005
- all numbers present are A351294, conjugate A381432
- numbers missing are A351295, conjugate A381433
- numbers appearing only once are A381540 (this), conjugate A381434
- numbers appearing more than once are A381541, conjugate A381435
A000040 lists the primes.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381440 lists Look-and-Say partition of prime indices, conjugate A381436.
Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]==1&]

A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 1, 1, 1, 1, 4

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) divides A003557 but is not equal to it.

a(n) is least d such that the prime power exponents of n/d are all equal; see also A066636. - David James Sycamore, Jun 13 2024

			n=1800: the squarefree kernel is 2*3*5 = 30 and 900 = 30^2 divides n, a(1800) = 2, the quotient of 1800/900.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001694, A003557, A007947, A051904, A003557, A052485, A062759.
Cf. A059404 (n such that a(n)>1), A072774 (n such that a(n)=1).
Cf. A066636.

f:= proc(n) local F,m,t;
  F:= ifactors(n)[2];
  m:= min(seq(t[2],t=F));
  mul(t[1]^(t[2]-m),t=F)
end proc:
map(f, [$1..200]); # Robert Israel, Nov 03 2017

{1}~Join~Table[n/#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 104}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] -= e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A051904(n) = if(1==n,0,vecmin(factor(n)[, 2])); \\ After Charles R Greathouse IV's code
A062760(n) = n/(A007947(n)^A051904(n)); \\ Antti Karttunen, Sep 23 2017

a(n) = n/(A007947(n)^A051904(n)).

a(n) = n/A062759(n). - Amiram Eldar, Feb 12 2023

A082949 Numbers of the form p^q * q^p, with distinct primes p and q.

Original entry on oeis.org

72, 800, 6272, 30375, 247808, 750141, 1384448, 37879808, 189267968, 235782657, 1313046875, 3502727631, 4437573632, 451508436992, 634465620819, 2063731785728, 7863818359375, 7971951402153, 188153927303168, 453238525390625, 1145440056788109

Offset: 1

Reinhard Zumkeller, May 26 2003

nonn

A001221(a(n)) = 2;
A001222(a(n)) = A001414(a(n)) = A020639(a(n)) + A006530(a(n)) = A051904(a(n)) + A051903(a(n));
A020639(a(n)) = A051904(a(n));
A006530(a(n)) = A051903(a(n)).

			2^7 * 7^2 = 128*49 = 6272, therefore 6272 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A053810, A006881, A051674.
Cf. A098096, numbers of the form 2^p * p^2.
Cf. A001221, A001222, A001414, A020639, A006530, A051903, A051904.
Cf. A151800.

import Data.Set (singleton, deleteFindMin, insert)
a082949 n = a082949_list !! (n-1)
a082949_list = f $ singleton (2 ^ 3 * 3 ^ 2, 2, 3) where
   f s = y : f (if p' < q then insert (p' ^ q * q ^ p', p', q) s'' else s'')
         where s'' = insert (p ^ q' * q' ^ p, p, q') s'
               p' = a151800 p; q' = a151800 q
               ((y, p, q), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 07 2015

Take[Union[Select[Flatten[Table[If[p != q, Prime[p]^Prime[q]*Prime[q]^Prime[p]], {p, 100}, {q, 100}]], IntegerQ]], 30] (* Alonso del Arte, Oct 28 2005 *)
Select[Range[10! ],Length[FactorInteger[ # ]]==2&&FactorInteger[ # ][[1,1]]==FactorInteger[ # ][[2,2]]&&FactorInteger[ # ][[1,2]]==FactorInteger[ # ][[2,1]]&] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2010 *)
With[{nn=30},Take[Union[First[#]^Last[#] Last[#]^First[#]&/@ Subsets[ Prime[Range[nn]],{2}]],nn]] (* Harvey P. Dale, Aug 19 2012 *)

term(p,q)=p^q*q^p;
l=listcreate(465); for(m=1,30, for(n=m+1,31, listput(l,term(prime(m), prime(n))))); listsort(l) \\ Rick L. Shepherd, Sep 07 2003

Corrected and extended by Rick L. Shepherd, Sep 07 2003

A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2025

nonn

First differs from A382525 at a(216) = 1, A382525(216) = 2.

			The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For ordered set partitions we have A382525.
Positions of first appearances are A382775.
The separable case is A386575.
The inseparable case is A386582, see A386632.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A239455 counts Look-and-Say partitions, complement A351293.
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[disjointFamilies[prix[n]]],{n,100}]

A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 25, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 49, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69, 70, 71, 36, 73

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) is a first power if and only if n is not a powerful number (A001694, A052485).

			n = 1800: squarefree kernel is 2*3*5 = 30 and a(1800) = 900 = 30^2 divides n, exponent of 30 is the smallest prime exponent of 1800 = 2*2*2*3*3*5*5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001694, A003557, A007947, A051904, A052485, A062759, A065463.

a062759 n = a007947 n ^ a051904 n  -- Reinhard Zumkeller, Jul 15 2012

{1}~Join~Table[#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 73}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] = e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmin(f[,2])); prod(i = 1, #f~, f[i,1]^e));} \\ Amiram Eldar, Feb 12 2023

a(n) = A007947(n)^A051904(n).
From Amiram Eldar, Feb 12 2023: (Start)
a(n) = n/A062759(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 / 2 = 0.352221... . (End)

A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 0, 0, 0, 0, 2, 0

Offset: 1

Henry Bottomley, Jul 24 2001

			a(24) = 2 since 24 = 2^3*3^1 and max(3,1) - min(3,1) = 3 - 1 = 2;
a(25) = 0 since 25 = 5^2 and max(2) - min(2) = 2 - 2 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 4000 terms from Harry J. Smith)
Index entries for sequences computed from exponents in factorization of n.

Cf. A072774 (positions of zeros), A059404 (of nonzeros).

Cf. A001222, A033150, A051903, A051904, A071178, A108951, A325226.

dlsp[n_]:=Module[{xp=FactorInteger[n][[All,2]]},Max[xp]-Min[xp]]; Join[ {0},Array[ dlsp,120]] (* Harvey P. Dale, Jan 28 2021 *)

{ for (n=1, 4000, if (n<2, M=m=0, f=factor(n)~; M=m=f[2, 1]; for (i=2, length(f), M=max(M, f[2, i]); m=min(m, f[2, i]))); write("b062977.txt", n, " ", M - m) ) } \\ Harry J. Smith, Aug 14 2009

A062977(n) = if((1==n),0,n=(factor(n)[, 2]); vecmax(n)-vecmin(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A051903(n) - A051904(n).
a(A108951(n)) = A325226(n) = A001222(n) - A071178(n). - Antti Karttunen, Nov 17 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150 - 1 = 0.705211... . - Amiram Eldar, Jan 05 2024

A078315 Minimum exponent in prime factorization of n*rad(n)+1, where rad = A007947 (the radical or squarefree kernel).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

2 = a(4) = a(45) = a(48) = a(140) = a(529) = a(682) = a(3264) = a(3564) = a(4680) = a(4756) = a(166320) = a(194873) = a(330096) = a(364905) = a(2100332) = a(4160200) with all terms in between equal to 1. Is there an n with a(n) > 2? - Charles R Greathouse IV, May 20 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A051904, A078310, A078314, A078316.

a078315 = a051904 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Min[FactorInteger[1 + n * Times @@ FactorInteger[n][[;;, 1]]][[;;, 2]]]; Array[a, 100] (* Amiram Eldar, Sep 08 2024 *)

a(n)=my(f=factor(n));f[,2]=apply(n->n+1,f[,2]);vecmin(factor(factorback(f)+1)[,2]) \\ Charles R Greathouse IV, May 20 2013

a(n) = A051904(A078310(n)).

A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 180, 182, 186, 190, 195, 222, 230, 231, 238, 246, 252, 255, 258, 266, 273, 282, 285, 286, 290, 300, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 396, 399, 402, 406, 410, 418, 426, 429

Offset: 1

Reinhard Zumkeller, May 03 2013

nonn

Union of A007304, A179643 and A179695; subsequence of A033992;
A001221(a(n)) = 3 and A051903(a(n)) <= A051904(a(n)) + 1 and A001222(a(n)) = 3 or 5 or 7;
A050326(a(n)) = 5.

			A007304(1) = 2*3*5 = 30, A206778(30,1..8)=[1,2,3,5,6,10,15,30]:
A050326(30) = #{30, 15*2, 10*3, 6*5, 5*3*2} = 5;
A179643(1) = 2^2*3^2*5 = 180, A206778(180,1..8)=[1,2,3,5,6,10,15,30]:
A050326(180) = #{30*6, 30*3*2, 15*6*2, 10*6*3, 6*5*3*2} = 5;
A179695(1) = 2^3*3^2*5^2 = 1800, A206778(1800,1..8)=[1,2,3,5,6,10,15,30]:
A050326(1800) = #{30*10*6, 30*6*5*2, 30*10*3*2, 15*10*6*2, 10*6*5*3*2} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A124010.

a225228 n = a225228_list !! (n-1)
a225228_list = filter f [1..] where
   f x = length es == 3 && sum es `elem` [3,5,7] &&
                           maximum es - minimum es <= 1
         where es = a124010_row x

is(n)=my(f=vecsort(factor(n)[,2]~)); f==[1,1,1] || f==[1,2,2] || f==[2,2,3] \\ Charles R Greathouse IV, Jul 28 2016

a(n) ~ 2n log n / (log log n)^2. - Charles R Greathouse IV, Jul 28 2016

A325240 Numbers whose minimum prime exponent is 2.

Original entry on oeis.org

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

Or barely powerful numbers, a subset of powerful numbers A001694.
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 minimum multiplicity is 2 (counted by A244515).
Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

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

Positions of 2's in A051904.
Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Cf. A065483, A082695.

Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023

from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    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):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

cp's OEIS Frontend

A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

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A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

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A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

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A082949 Numbers of the form p^q * q^p, with distinct primes p and q.

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A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

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A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

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A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

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