A276079 -id:A276079 - cp's OEIS frontend

A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

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, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Aug 18 2016

nonn

Numbers not divisible by p^(1+A000720(p)) for any prime p, where A000720(p) gives the index of prime p: 1 for 2, 2 for 3, 3 for 5, and so on.
Also Heinz numbers of integer partitions where the multiplicity of i does not exceed i for any i (A052335). Differs from A048103 in lacking {625, 1250, 1875, 3750, 4375, 5625, 6875, 8125, 8750, ...}. - Gus Wiseman, Mar 09 2019
Asymptotic density is Product_{i>=1} 1-prime(i)^(-1-i) = 0.72102334... - Amiram Eldar, Oct 20 2020

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

Positions of zeros in A276077.
Complement: A276079.
Sequence A276076 sorted into ascending order.
Cf. A000040, A000720.
Subsequence of A048103 from which it differs for the first time at n=451, where a(451) = 626, while A048103(451) = 625, a value missing from here.
Cf. A052335, A056239, A062457, A087153, A109298, A112798, A115584, A117144, A118914, A124010, A197987, A276078.
Cf. A324524, A324525, A324571, A324587, A324588.

Select[Range@ 121, Or[# == 1, AllTrue[FactorInteger[#], PrimePi[#1] >= #2 & @@ # &]] &] (* Michael De Vlieger, Jun 24 2017 *)

isok(n) = my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > primepi(f[k, 1]), return(0))); return (1); \\ Michel Marcus, Jun 24 2017

is(n) = {my(t=1);forprime(p = 2, , t++; pp = p^t; if(n%pp==0, return(0)); if(pp > n, return(1)))} \\ David A. Corneth, Jun 24 2017

upto(n) = {my(v = vector(n,i,1), t=1, res=List()); forprime(p=2, , t++; pp = p^t; if(pp>n, break); for(i=1, n\pp, v[pp*i] = 0)); for(i=1, n, if(v[i]==1, listput(res, i))); res} \\ David A. Corneth, Jun 24 2017

from sympy import factorint, primepi
def ok(n):
    f = factorint(n)
    return all(f[i] <= primepi(i) for i in f)
print([n for n in range(1, 151) if ok(n)]) # Indranil Ghosh, Jun 24 2017

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

A276077 Number of distinct prime factors p of n such that p^(1+A000720(p)) is a divisor of n, where A000720(p) gives the index of prime p, 1 for 2, 2 for 3, 3 for 5, and so on.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 18 2016

			For n = 2 (= prime(1)), 2 is not divisible by 2^(1+1), thus a(2) = 0.
For n = 3 (= prime(3)), 3 is not divisible by 3^(2+1), thus a(3) = 0.
For n = 4 (= prime(1)^2), 4 is divisible by 2^(1+1), and there are no other prime factors apart from 2, thus a(4) = 1.
For n = 108 = 2^2 * 3^3, it is divisible both by 2^(1+1) and 3^(2+1), thus a(108) = 2.
For n = 625 = prime(3)^4, it is divisible by 5^(3+1), thus a(625) = 1.

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

Cf. A000720, A028234, A055396, A067029.
Cf. A276078 (positions of zeros), A276079 (nonzeros), also A276076.
Differs from A129251 for the first time at n=625, where a(625) = 1, while A129251(625) = 0.

f[p_, e_] := If[PrimePi[p] < e, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 30 2023 *)

a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i,1]) < f[i,2]);} \\ Amiram Eldar, Sep 30 2023

from sympy import primepi, isprime, primefactors, factorint
def a028234(n):
    f=factorint(n)
    return 1 if n==1 else n//(min(f)**f[min(f)])
def a067029(n):
    f=factorint(n)
    return 0 if n==1 else f[min(f)]
def a049084(n): return primepi(n)*(isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n):
    if n==1: return 0
    val = a(a028234(n))
    if a067029(n) > a055396(n):
        val += 1
    return val
print([a(n) for n in range(1, 201)]) # Indranil Ghosh, Jun 21 2017

(define (A276077 n) (if (= 1 n) 0 (+ (A276077 (A028234 n)) (if (> (A067029 n) (A055396 n)) 1 0))))

This formula uses Iverson bracket, which gives 1 as its value if the condition given inside [ ] is true and 0 otherwise:
a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) > A055396(n)].
Other identities. For all n >= 1:
a(A276076(n)) = 0.
From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = 1 if primepi(p) < e, and 0 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/prime(k)^(k+1) = 0.2886971166123417096098... . (End)

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

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

A328251 Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.

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, 216, 220, 224, 225, 228, 232, 236, 240, 243, 244, 248, 250, 252, 256, 260, 264, 268, 270, 272

Offset: 1

Antti Karttunen, Oct 11 2019

nonn

This probably is NOT an intersection of A013929 and A099309.

			We see that 225 = 3^2 * 5^2 is not squarefree, and then when starting iterating with A003415, we obtain --> 240 --> 608 --> 1552 --> ... which is a trajectory that will never reach neither a prime nor any squarefree number at all, because already 240 = 2^4 * 3 * 5 is a member of A100716, whose terms all belong into A099309, as any divisor of the form p^p of n will be always present when taking its successive arithmetic derivatives. Thus 225 is included in this sequence.

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

Row 1 of array A328250. Indices of zeros in A328248.
Cf. A003415, A099309, A327929, A327934.
Cf. A013929, A100716 (a subsequence).
Differs from A100716 and A276079 for the first time at a(63) = 225, the term which is not present in them.

A003415checked(n) = if(n<=1, 0, my(f=factor(n), s=0); for(i=1, #f~, if(f[i,2]>=f[i,1],return(0), s += f[i, 2]/f[i, 1])); (n*s));
A328248(n) = { my(k=1); while(n && !issquarefree(n), k++; n = A003415checked(n)); (!!n*k); };
isA328251(n) = (0==A328248(n));

A387112 Numbers with (strictly) choosable initial intervals of prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 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, 91, 93, 94, 95

Offset: 1

Gus Wiseman, Aug 23 2025

First differs from A371088 in having a(86) = 121.
The initial interval of a nonnegative integer x is the set {1,...,x}.
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 say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1,2,3},{1},{1,3},{2}) is not.
This sequence lists all numbers k such that if the prime indices of k are (x1,x2,...,xz), then the sequence of sets (initial intervals) ({1,...,x1},{1,...,x2},...,{1,...,xz}) is choosable.

			The prime indices of 85 are {3,7}, with initial intervals {{1,2,3},{1,2,3,4,5,6,7}}, which are choosable, so 85 is in the sequence
The prime indices of 90 are {1,2,2,3}, with initial intervals {{1},{1,2},{1,2},{1,2,3}}, which are not choosable, so 90 is not in the sequence.

Partitions of this type are counted by A238873, complement A387118.
For partitions instead of initial intervals we have A276078, complement A276079.
For prime factors instead of initial intervals we have A368100, complement A355529.
For divisors instead of initial intervals we have A368110, complement A355740.
These are all the positions of nonzero terms in A387111, complement A387134.
The complement is A387113.
For strict partitions instead of initial intervals we have A387176, complement A387137.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
A370585 counts maximal subsets with choosable prime factors.
Cf. A003963, A005703, A052335, A239312, A335433, A335448, A355739, A387110, A383706.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Range/@prix[#]],UnsameQ@@#&]!={}&]

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 2, 7, 6, 15, 0, 6, 14, 56, 6, 101, 30, 21, 0, 297, 12, 490, 14, 45, 112, 1255, 0, 42, 202, 6, 30, 4565, 42, 6842, 0, 168, 594, 105, 12, 21637, 980, 303, 0, 44583, 90, 63261, 112, 42, 2510, 124754, 0, 210, 84, 891, 202, 329931, 12, 392, 0, 1470, 9130

Offset: 1

Gus Wiseman, Aug 26 2025

			The prime factors of 9 are (3,3), and the a(9) = 6 choices are:
  ((3),(2,1))
  ((3),(1,1,1))
  ((2,1),(3))
  ((2,1),(1,1,1))
  ((1,1,1),(3))
  ((1,1,1),(2,1))

For prime factors instead of partitions we have A008966, see A355741.
Twice partitions of this type are counted by A296122.
For prime indices instead of factors we have A387110, see A387136.
For strict partitions and prime indices we have A387115.
For constant partitions and prime indices we have A387120.
Positions of zero are A387326, for indices apparently A276079 (complement A276078).
Allowing repeated partitions gives A387327, see A299200, A357977.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A063834, A261049, A299201, A335433, A335448, A355739, A383706, A387111, A387135.

Table[Length[Select[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]],UnsameQ@@#&]],{n,30}]

A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 453, 578, 740, 935, 1186, 1486, 1865, 2322, 2890, 3572, 4415, 5423, 6659, 8134, 9927, 12062, 14643, 17706, 21387, 25746, 30957, 37109, 44433, 53054, 63273, 75276, 89444, 106044

Offset: 0

Gus Wiseman, Aug 29 2025

Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.

Also the number of integer partitions of n with at least one part k satisfying that the multiplicity of k exceeds the number of integer partitions of k.

			The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (211)   (311)    (222)     (511)      (611)
               (1111)  (2111)   (411)     (2221)     (2222)
                       (11111)  (2211)    (3211)     (3311)
                                (3111)    (4111)     (4211)
                                (21111)   (22111)    (5111)
                                (111111)  (31111)    (22211)
                                          (211111)   (32111)
                                          (1111111)  (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions are ranked by A276079.
For divisors instead of partitions we have A370320, complement A239312.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of partitions we have A370593, ranks A355529.
For initial intervals instead of partitions we have A387118, complement A238873.
For just choices of strict partitions we have A387137.
The complement is counted by A387328, ranks A276078.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.

Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]]==0&]],{n,0,15}]

A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Gus Wiseman, Aug 27 2025

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 say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

The complement for all partitions appears to be A276078, counted by A052335.
For all partitions we appear to have A276079, counted by A387134.
For divisors instead of strict partitions we have A355740, counted by A370320.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873, see A387111.
For initial intervals instead of strict partitions we have A387113, counted by A387118.
These are the positions of 0 in A387115.
Partitions of this type are counted by A387137, complement A387178.
The complement is A387177.
The version for constant partitions is A387180, counted by A387329.
The complement for constant partitions is A387181, counted by A387330.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together the prime indices of n.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]

A277009 Numbers not in range of A277012: numbers such that at least one run of 1-bits in their binary expansion is longer than 1 + the total number of 0-bits anywhere right of that run.

Original entry on oeis.org

3, 7, 11, 14, 15, 19, 23, 27, 29, 30, 31, 35, 39, 43, 46, 47, 51, 55, 59, 60, 61, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 93, 94, 95, 99, 103, 107, 110, 111, 115, 119, 121, 122, 123, 124, 125, 126, 127, 131, 135, 139, 142, 143, 147, 151, 155, 157, 158, 159, 163, 167, 171, 174, 175, 179, 183, 187, 188, 189, 190, 191, 195, 199, 203

Offset: 1

Antti Karttunen, Sep 25 2016

Numbers n for which A277007(n) > 0.

Numbers n for which A276077(A005940(1+n)) > 0.

			3 ("11" in binary, A007088) is present as the length of that only run of 1's is 2, and 2 > 1+0, where 0 is the total number of 0's to the right of that run.
60 ("111100" in binary) is present as 4 > 2+1.
246 ("11110110" in binary) is present as the length of the leftmost run of 1-bits is 4, and 4 > 1+2, where 2 is the total number of 0's located anywhere to the right of that run.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Complement: A277008.
Positions of nonzeros in A277007. Numbers not present in A277012.
Cf. A007088, A005940, A276077, A276079.
Differs from its subsequence A277019 for the first time at n=20, where a(20)=60, a term not present in A277019.

cp's OEIS Frontend

A276078 Numbers n in whose prime factorization no exponent of any prime(k) exceeds k.

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

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A276077 Number of distinct prime factors p of n such that p^(1+A000720(p)) is a divisor of n, where A000720(p) gives the index of prime p, 1 for 2, 2 for 3, 3 for 5, and so on.

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A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

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A328251 Row 1 of array A328250: numbers n whose k-th arithmetic derivative is never squarefree for any k >= 0.

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A387112 Numbers with (strictly) choosable initial intervals of prime indices.

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A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

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A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

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