A054744 -id:A054744 - cp's OEIS frontend

A368333 The largest term of A054744 that divide n.

1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Dec 21 2023

The largest divisor d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

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

Cf. A034444, A048103, A054744, A327939, A368329, A368332, A368334, A368335.

f[p_, e_] := If[e < p, 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = p^e if e >= p.
A034444(a(n)) = A368334(n).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= n, with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^(p*s) + 1/p^(p*(s-1)) + 1/p^((p+1)*s-1) - 1/p^((p+1)*(s-1)+1)).

A368335 The number of divisors of the largest term of A054744 that divides of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

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

Cf. A000005, A048103, A054744, A368331, A368332, A368333, A368334, A368336.

f[p_, e_] := If[e < p, 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2]+1));}

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e+1 if e >= p.
a(n) = A000005(A368333(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A000005(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s-1) + 1/p^((p+1)*s) - 1/p^((p+1)*s-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1 + (p-1)*p)/((p-1)*p^p)) = 1.98019019497523582894... .

A368332 The number of terms of A054744 that divide n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 21 2023

The number of divisors d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).

The largest of these divisors is A368333(n).

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

Cf. A054744, A365632, A368328, A368333, A368334, A368335.

f[p_, e_] := If[e < p, 1, e - p + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2] - f[i,1] + 2));}

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e - p + 2 if e >= p.
a(n) >= 1, with equality if and only if n is in A048103.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .

A368334 The number of terms of A054744 that are unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 21 2023

First differs from A081117 at n = 28.

Also, the number of terms of A072873 that are unitary divisors of n.

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

Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.

Cf. A081117.

f[p_, e_] := If[e < p, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, 2));}

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = 2 if e >= p.
a(n) = A034444(A368333(n)).
a(n) = A034444(A327939(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A034444(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

N. J. A. Sloane

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

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

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020

A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024

from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023

;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

More terms from James Sellers, Apr 22 2000

A100716 Numbers k such that p^p divides k for some prime p.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208, 212

Offset: 1

Leroy Quet, Dec 10 2004

nonn

Complement of A048103; A129251(a(n)) > 0; A051674 is a subsequence; A129254 = (terms a(k) such that a(k+1)=a(k)+1). - Reinhard Zumkeller, Apr 07 2007

A027748(a(n),k) <= A124010(a(n),k) for some k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012

			54 is included because 3^3 divides 54.

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

Complement: A048103.
Positions of nonzeros in A129251.
Cf. A100717, A129150, A129152.
Cf. A054744.
Cf. A051674 (a subsequence).
Subsequence of A276079 from which it differs for the first time at n=175, where a(175) = 628, while A276079(175) = 625, a value missing from here.

a100716 n = a100716_list !! (n-1)
a100716_list = filter (\x -> or $
   zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012
(Scheme, with Antti Karttunen's IntSeq-library)
(define A100716 (NONZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

fQ[n_] := Union[ Table[ #[[1]] <= #[[2]]] & /@ FactorInteger[n]][[ -1]] == True; Select[ Range[2, 215], fQ[ # ] &] (* Robert G. Wilson v, Dec 14 2004 *)
f[n_] := Module[{aux=FactorInteger[n]}, Last@Union@Table[aux[[i,1]] <=  aux[[i,2]], {i,Length[aux]}] == True]; Select[Range[2,215], f] (* José María Grau Ribas, Jan 25 2012 *)
Rest@ Select[Range@ 216, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] == 0 &] (* Michael De Vlieger, Aug 19 2016 *)

is(n)=forprime(p=2,default(primelimit),if(n%p^p==0,return(1));if(p^p>n,return(0))) \\ Charles R Greathouse IV, Jan 24 2012

from itertools import count, islice
from sympy import factorint
def A100716_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(map(lambda d:d[1]>=d[0],factorint(n).items())),count(max(startvalue,1)))
A100716_list = list(islice(A100716_gen(),30)) # Chai Wah Wu, Jan 05 2023

a(n) ~ k*n with k = 1/(1 - Product(1 - p^-p)) = 3.5969959469... where the product is over all primes p. - Charles R Greathouse IV, Jan 24 2012

More terms from Robert G. Wilson v, Dec 14 2004

A054743 If n = Product p_i^e_i then p_i < e_i (where e_i > 0) for all i.

Original entry on oeis.org

1, 8, 16, 32, 64, 81, 128, 243, 256, 512, 648, 729, 1024, 1296, 1944, 2048, 2187, 2592, 3888, 4096, 5184, 5832, 6561, 7776, 8192, 10368, 11664, 15552, 15625, 16384, 17496, 19683, 20736, 23328, 31104, 32768, 34992, 41472, 46656

Offset: 1

James Sellers, Apr 22 2000

Closed under multiplication. Use A104126 to construct A192135 by putting A104126(n) * prime(n)^k in a list up to some chosen bound. Create this sequence by multiplying any k elements of A192135 with distinct prime factors in a list (k>1). The last list along with A192135 is this sequence when sorted. - David A. Corneth, Jun 07 2016

			8 appears in the list because 8 = 2^3 and 2<3.
Construction of elements up to 1000: 1. Put 2^3 and 3^5 in a list; {8, 81} (The terms of A104126 up to 1000.) 2. For each element, put products the last list with their distinct prime factors up to 1000. Gives: {8, 16, 32, 64, 128, 256, 512, 81, 243, 729} (Terms from A192135 up to 1000). 3. Put products of k powers of distinct primes in a new list up to 1000: {648} (k>1). Unite {648} with {8, 16, 32, 64, 128, 256, 512, 81, 243, 729}. {8, 16, 32, 64, 128, 256, 512, 81, 243, 729, 648}. Sort the list. This gives: {8, 16, 32, 64, 81, 128, 243, 256, 512, 648, 729}, which are the elements below 1000 in this sequence. - _David A. Corneth_, Jun 07 2016

Robert Israel, Table of n, a(n) for n = 1..10001

Cf. A048103, A054744, A104126, A192135.

Cf. A207481 (same construction, but with p_i>=e_i),

N:= 10^10: # to get all terms <= N
p:= 1:
S:= {1}:
do
  p:= nextprime(p);
  if p^(p+1) > N then break fi;
  pp:= [seq(p^j, j=p+1 .. ilog[p](N))];
  S:= S union select(`<=`,{seq(seq(s*q,s=S),q=pp)},N);
od:
sort(convert(S,list)); # Robert Israel, Jun 07 2016

okQ[n_] := AllTrue[FactorInteger[n], #[[1]] < #[[2]]&];
Join[{1}, Select[Range[50000], okQ]] (* Jean-François Alcover, Jun 08 2016 *)

lista(nn) = {for (n=1, nn, f = factor(n); ok = 1; for (i=1, #f~, if (f[i, 1] >= f[i, 2], ok = 0; break;);); if (ok, print1(n, ", ")););} \\ Michel Marcus, Jun 15 2013

Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/((p-1)*p^p) = 1.27325025767774256043... - Amiram Eldar, Nov 24 2020

1 prepended by Alec Jones, Jun 07 2016

A300566 Numbers z such that there is a solution to x^4 + y^5 = z^6 with x, y, z >= 1.

Original entry on oeis.org

8748, 10368, 342732

Offset: 1

M. F. Hasler, Apr 16 2018

Also in the sequence: 810000 = 2^4*3^4*5^4, 1361367 = 3^4*7^5, 3240000 = 2^6*3^4*5^4, 9335088 = 2^4*3^5*7^4, 25312500 = 2^2*3^4*5^7, 31505922 = 2*3^8*7^4, 43740000 = 2^5*3^7*5^4, 512578125 = 3^8*5^7, 1215000000 = 2^6*3^5*5^7, 1701708750 = 2*3^4*5^4*7^5, 2196150000 = 2^4*3*5^5*11^4, 2431012500 = 2^2*3^4*5^5*7^4, 4269246912 = 2^6*3^4*7^7, 4447203750 = 2*3^5*5^4*11^4, 36015000000 = 2^6*3*5^7*7^4, 48717927500 = 2^2*5^4*11^7, 75969140625 = 3^4*5^8*7^4, 91116682272 = 2^5*3^4*7^4*11^4. - Jacques Tramu, Apr 17 2018
Consider a solution (x,y,z) of x^4 + y^5 = z^6. For any m, (x*m^15, y*m^12, z*m^10) is also a solution. Reciprocally, if (x/m^15, y/m^12, z/m^10) is a triple of integers for some m, then this is also a solution. We call primitive a solution for which there is no such m > 1. - M. F. Hasler, Apr 17 2018
Observation: a(n) = A054744(n+38) = A257999(n+32), at least for 1 <= n <= 2 in both cases. - Omar E. Pol, Apr 17 2018
These relations hold only for n = 1 and 2. The next larger known term 342732 = 2^2*3*13^4 shows that in general the terms don't belong to A054744 nor A257999, although the earlier comment implies that each term gives rise to infinitely many non-primitive terms in A054744. - M. F. Hasler, Apr 19 2018
When S = a^4 + b^10/4 is a square, then z = b^5/2 + sqrt(S) is a solution, with x = a*z and y = b*z. All known solutions and further solutions 8957952, 10616832, 52200625, 216486432, ... are of this form (with rational a, b). - M. F. Hasler, Apr 19 2018

			a(1) = 8748 = 2^2*3^7 is in the sequence because 8748^6 = (2^3*3^8)^5 + (2^3*3^10)^4, using 2^3 + 1 = 3^2. Similarly, all z = 4*3^(10k-3) are in the sequence for k >= 1, with x = 8*3^(15k-5) and y = 8*3^(12k-4).
a(2) = 10368 = 2^7*3^4 is in the sequence because 10368^6 = (2^8*3^5)^5 + (2^10*3^6)^4, using 3 + 1 = 2^2. Similarly, any z = 2^7*3^(10k+4) is in the sequence for k >= 0, with x = 2^10*3^(15k+6) and y = 2^8*3^(12k+5).
z = 342732 = 2^2*3*13^4 is in the sequence because (2^2*3*13^4)^6 = (2^3*13^5)^5 + (2^3*5*13^6)^4, using 2^3*13 + 5^4 = 3^6.
z = 810000 = 2^4*3^4*5^4 is in the sequence because z^6 = x^4 + y^5 with x = 2^5*3^6*5^6 and y = 2^4*3^5*5^5 (using 1 + 3*5 = 2^4).
z = 1361367 = 3^4*7^5 is in the sequence because z^6 = x^4 + y^5 with x = 3^5*7^8 and y = 2*3^4*7^6.

Cf. A300564 (z^4 = x^2 + y^3) and A242183, A300565 (z^5 = x^3 + y^4), A302174.
Cf. A100294: numbers of the form a^5 + b^4.
See A303266 for the y-values.

is(z)=for(y=1,sqrtnint(-1+z=z^6,5),ispower(z-y^5,4)&&return(y))
/* Code below for illustration only, not guaranteed to give a complete list. Half-integral values give the additional term 31505922 for b = 63/2. Third-integral values give the additional solution z = 342732 for b = 26/3. */
S=[]; N=1e5; forstep(b=1,9,1/3, forstep(a=1,N,1/3, issquare(b^10+a^4<<2,&r)&& !frac(z=b^5/2+r/2)&& !print1(z",")&&S=setunion(S,[z])); print1([b])); S

A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

Original entry on oeis.org

1, 8, 16, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 343, 350, 360, 375, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 500, 504, 512, 525

Offset: 1

Labos Elemer, Nov 21 2003

What is the asymptotic growth of this sequence?

Answer: a(n) ~ k*n, where k = 1/A175475. That is, about 4.8% of numbers are in this sequence. - Charles R Greathouse IV, Jul 14 2014

			378 = 2 * 3^3 * 7 is a term of the sequence since 7 < 7.23... = 378^(1/3).

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

Cf. A063763, A001248, A048098, A036966, A054744, A059172, A068936, A076405, A076467, A006530, A175475.

filter:= n ->
evalb(max(seq(f[1],f=ifactors(n)[2]))^3 <= n):
select(filter, [$1..1000]); # Robert Israel, Jul 14 2014

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; Do[If[ !Greater[ma[n], gy=n^(1/3)//N]&&!PrimeQ[n], Print[n(*, {gy, ma[n]}*)]], {n, 1, 1000}]
Select[Range[1000], (FactorInteger[#][[-1,1]])^3 <= # &] (* T. D. Noe, Sep 14 2011 *)
Select[Range[1000],FactorInteger[#][[-1,1]]<=CubeRoot[#]&] (* Harvey P. Dale, Jun 30 2025 *)

is(n)=my(f=factor(n)[,1]);f[#f]^3<=n \\ Charles R Greathouse IV, Sep 14 2011

from sympy import primefactors
def ok(n):
    if n==1 or max(primefactors(n))**3<=n: return True
    else: return False
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 23 2017

Solutions to A006530(n) <= n^(1/3).

A325132 Number of integer partitions of n where the multiplicity of each part k is at least prime(k).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 6, 6, 7, 7, 10, 8, 11, 12, 12, 14, 17, 16, 20, 22, 24, 26, 31, 31, 37, 39, 43, 46, 54, 53, 63, 65, 73, 75, 87, 87, 100, 102, 115, 117, 133, 134, 151, 155, 172, 176, 197, 202, 223, 231, 254, 262, 290, 298, 327, 341, 370

Offset: 0

Gus Wiseman, Apr 01 2019

nonn

The Heinz numbers of these partitions are given by A054744.

			The first few terms count the following integer partitions:
   0: ()
   2: (11)
   3: (111)
   4: (1111)
   5: (11111)
   6: (222)
   6: (111111)
   7: (1111111)
   8: (2222)
   8: (22211)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (22222)
  10: (222211)
  10: (2221111)
  10: (1111111111)
  11: (2222111)
  11: (22211111)
  11: (11111111111)
  12: (222222)
  12: (2222211)
  12: (22221111)
  12: (222111111)
  12: (111111111111)

Cf. A052335, A054744, A062457, A087153, A117144, A324525, A324572.

Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=Prime[i],{i,Union[#]}]&]],{n,0,30}]

G.f.: Product_{k>=1} (1 + x^(prime(k)*k) / (1 - x^k)). - Ilya Gutkovskiy, Nov 28 2020

cp's OEIS Frontend

A368333 The largest term of A054744 that divide n.

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A368335 The number of divisors of the largest term of A054744 that divides of n.

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A368332 The number of terms of A054744 that divide n.

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A368334 The number of terms of A054744 that are unitary divisors of n.

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A048103 Numbers not divisible by p^p for any prime p.

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A100716 Numbers k such that p^p divides k for some prime p.

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A054743 If n = Product p_i^e_i then p_i < e_i (where e_i > 0) for all i.

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