A246549 -id:A246549 - cp's OEIS frontend

A384785 The number of unordered factorizations of the n-th cubefull number into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 6, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 9, 1, 1, 2, 1, 2, 3, 1, 3, 1, 2, 10, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 2, 2, 2, 13, 1, 1, 2, 1, 1, 4, 1, 3, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 1, 2, 3, 17, 2

Offset: 1

Amiram Eldar, Jun 10 2025

The positive values of the multiplicative function f(n) with f(p^e) = A008483(e). Or, equivalently, a(n) is the value of this function at A036966(n).

			a(6) = 2 since the 6th cubefull number, A036966(6) = 64, has 2 factorizations: 2^3 * 2^3 and 2^6.
a(12) = 3 since the 12th cubefull number, A036966(12) = 256, has 3 factorizations: 2^3 * 2^5, 2^4 * 2^4, and 2^8.

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

Cf. A008483, A036966, A246549, A384783 (powerful analog), A384786.

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := Module[{cub = Union[Flatten[Table[i^3*j^4*k^5, {k, 1, Surd[lim, 5]}, {j, 1, Surd[lim/k^5, 4]}, {i, 1, Surd[lim/(j^4*k^5), 3]}]]]}, Select[s /@ cub, # > 0 &]]; seq[10^5]

s(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1)-numbpart(x-2)+numbpart(x-3), factor(n)[, 2]));
cubs(lim) = {my(c = List()); for(k = 1, sqrtnint(lim, 5), for(j = 1, sqrtnint(lim \ k^5, 4), for(i = 1, sqrtnint(lim \ (j^4*k^5), 3), listput(c, i^3*j^4*k^5)))); Set(c); }
list(lim) = {my(c = cubs(lim), v = List(), s1); for(k = 1, #c, s1 = s(c[k]); if(s1 > 0, listput(v, s1))); Vec(v);}

A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

Original entry on oeis.org

1, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

The least term that is not a power of 2 is a(85) = 2^61 * 3^18.
Indices of records of the multiplicative function f(n) with f(p^e) = A008483(e).
All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
The corresponding record values are 1, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..567
Amiram Eldar, Table of n, a(n), and record values for n = 1..567
Index entries for sequences related to powerful numbers.

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A008483, A046055, A246549, A384784 (powerful analog), A384785.

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378

Offset: 1

Amiram Eldar, Feb 19 2025

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

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

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));

A246551 Prime powers p^e where p is a prime and e is odd.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313

Offset: 1

Joerg Arndt, Aug 29 2014

nonn

These are the integers with only one prime factor whose cototient is square, so this sequence is a subsequence of A063752. Indeed, cototient(p^(2k+1)) = (p^k)^2 and cototient(p) = 1 = 1^2. - Bernard Schott, Jan 08 2019

With 1 prepended, this sequence is the lexicographically earliest sequence of distinct numbers whose partial products are all numbers whose exponents in their prime power factorization are squares (A197680). - Amiram Eldar, Sep 24 2024

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000961, A246547, A246549, A168363, A197680, subsequence of A171561.
Cf. also A056798 (prime powers with even exponents >= 0).
Subsequence of A063752.

[n:n in [2..1000]| #PrimeDivisors(n) eq 1 and IsSquare(n-EulerPhi(n))]; // Marius A. Burtea, May 15 2019

Take[Union[Flatten[Table[Prime[n]^(k + 1), {n, 100}, {k, 0, 14, 2}]]], 100] (* Vincenzo Librandi, Jan 10 2019 *)

for(n=1, 10^4, my(e=isprimepower(n)); if(e%2==1, print1(n, ", ")))

from sympy import primepi, integer_nthroot
def A246551(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0])for k in range(1,x.bit_length(),2)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 13 2024

A304326 Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 0, 1, 3, 1, 3, 1, 3, 3, 0, 1, 3, 1, 3, 3, 3, 1, 2, 1, 3, 0, 3, 1, 7, 1, 0, 3, 3, 3, 3, 1, 3, 3, 2, 1, 7, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 7, 1, 3, 3, 0, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 2, 0, 3, 1, 7, 3, 3, 3, 2, 1

Offset: 1

Gus Wiseman, May 10 2018

nonn

			The a(180) = 7 ways are (6*30), (12*15), (18*10), (30*6), (60*3), (90*2), (180*1).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Positions of zeros are A246549. Range appears to be A075427.

Cf. A000961, A001055, A001597, A001694, A005117, A007916, A034444, A091050, A183096, A303386, A303707, A304327, A304328.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
Table[Length[Select[Divisors[n],radQ[#]&&SquareFreeQ[n/#]&]],{n,100}]

a(n)={sumdiv(n, d, d<>1 && !ispower(d) && issquarefree(n/d))} \\ Andrew Howroyd, Aug 26 2018

A304327 Number of ways to write n as a product of a perfect power and a squarefree number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 10 2018

nonn

First term greater than 2 is a(746496) = 3.

			The a(746496) = 3 ways are 12^5*3, 72^3*2, 864^2*1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000961, A001055, A001597, A005117, A007916, A034444, A091050, A183096, A203025, A246549, A303386, A304326, A304328.

Table[Length[Select[Divisors[n],(#===1||GCD@@FactorInteger[#][[All,2]]>1)&&SquareFreeQ[n/#]&]],{n,100}]

A304327(n) = sumdiv(n,d,issquarefree(n/d)*((1==d)||ispower(d))); \\ Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

A152441 Decimal expansion of Sum_{primes p} 1/(p^2*(p-1)).

Original entry on oeis.org

3, 2, 0, 9, 0, 9, 2, 4, 9, 0, 0, 8, 7, 2, 9, 6, 2, 9, 3, 5, 7, 8, 2, 4, 0, 9, 5, 0, 2, 3, 6, 9, 4, 4, 6, 1, 4, 4, 5, 5, 0, 9, 9, 9, 2, 8, 4, 3, 2, 9, 3, 6, 2, 6, 5, 7, 4, 5, 8, 7, 1, 3, 7, 0, 0, 5, 5, 4, 4, 0, 0, 1, 1, 2, 5, 3, 2, 2, 5, 2, 3, 3, 8, 4, 8, 4, 1, 2, 1, 4, 4, 6, 8, 4, 1, 3, 9, 6, 0, 1, 0, 6, 1, 3

Offset: 0

R. J. Mathar, Dec 04 2008

Generally, sum_p 1/(p^s*(p-1)) equals A136141 minus the sum over all prime zeta functions with index 2 to s (see A085964 to A085969).

			0.320909249008729629357824095023694461445509992843293626574587137005544001125... = 1/(4*1) + 1/(9*2) + 1/(25*4) + 1/(49*6) + ...

Index to constants which are prime zeta sums {2,1,0}

Cf. A085548, A136141, A246549.

digits = 104; sp = NSum[PrimeZetaP[n], {n, 3, Infinity},  WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)

sumeulerrat(1/(p^2*(p-1))) \\ Amiram Eldar, Mar 18 2021

Equals A136141 minus A085548 .

Equals Sum_{n>=1} 1/A246549(n). - Amiram Eldar, Oct 27 2020

More digits from Jean-François Alcover, Sep 11 2015

A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.

Original entry on oeis.org

8, 16, 27, 32, 64, 81, 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 1536, 2048, 2187, 2197, 2304, 2401, 2560, 3072, 3125, 3456, 3584, 4096, 4608, 4913, 5120, 5184, 5632, 6144, 6400, 6561, 6656, 6859, 6912, 7168, 8192, 8704, 9216, 9728, 10240, 11264

Offset: 1

Gus Wiseman, Aug 04 2025

nonn

First cubefull number (A246549) not in this sequence is 216.
The first term that is not a prime power is 1536.
A set partition is inseparable iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.

			The prime indices of 2304 are {1,1,1,1,1,1,1,1,2,2}, and we have disjoint inseparable choice {{4,3,1},{2}}, so 2304 is in the sequence.
The terms together with their prime indices begin:
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   343: {4,4,4}
   512: {1,1,1,1,1,1,1,1,1}
   625: {3,3,3,3}
   729: {2,2,2,2,2,2}

This is the inseparable case of A351294, positives in A386575, counted by A239455.
Also positions of positive terms in A386582.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A025065/A386638 counts inseparable type partitions, ranks A335126, sums of A386586.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts separable type partitions, ranks A335127, sums of A386585.
A386633 counts separable type set partitions, row sums of A386635.
A386634 counts inseparable type set partitions, row sums of A386636.
Cf. A001221, A001222, A051903, A051904, A056239, A130091, A279790, A351293, A373957, A382525, A386587.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dsj[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
insepQ[y_]:=2*Max[y]>Total[y]+1;
Join@@Position[Sign[Table[Length[Select[dsj[prix[n]],insepQ[Length/@#]&]],{n,1000}]],1]

A246550 Prime powers p^e where p is a prime and e >= 4.

Original entry on oeis.org

16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 14641, 15625, 16384, 16807, 19683, 28561, 32768, 59049, 65536, 78125, 83521, 117649, 130321, 131072, 161051, 177147, 262144, 279841, 371293, 390625, 524288, 531441, 707281, 823543, 923521, 1048576, 1419857, 1594323, 1771561

Offset: 1

Joerg Arndt, Aug 29 2014

nonn

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000961, A246547, A246549, A168363.

N:= 10^7: # to get all terms <= N
{seq(seq(p^m, m=4..floor(log[p](N))), p = select(isprime,[2,seq(2*i+1,i=1..floor(N^(1/4)))]))}; # Robert Israel, Aug 29 2014

With[{max = 10^6}, Sort @ Flatten @ Table[p^Range[4, Floor[Log[p, max]]], {p, Select[Range[Surd[max, 4]], PrimeQ]}]] (* Amiram Eldar, Oct 24 2020 *)

m=10^7; v=[]; forprime(p=2, m^(1/4), e=4; while(p^e<=m, v=concat(v, p^e); e++)); v=vecsort(v) \\ Jens Kruse Andersen, Aug 29 2014

from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A246550(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, k)[0]) for k in range(4, x.bit_length())))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^3*(p-1)) = 0.1461466097... - Amiram Eldar, Oct 24 2020

A375145 Numbers whose prime factorization has exactly one exponent that is larger than 2.

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 343, 344

Offset: 1

Amiram Eldar, Aug 01 2024

Subsequence of A046099 and first differs from it at n = 35: A046099(35) = 216 = 2^3 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p^3-1) = A286229 / A002117 = 0.16148833663564192901... .

			8 = 2^3 is a term since its prime factorization has exactly one exponent, 3, that is larger than 2.

Subsequence of A046099, A356862.

Cf. A002117, A057521, A190641, A246549, A286229, A375146.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 2 &)] == 1; Select[Range[350], q]

is(k) = #select(x -> x > 2, factor(k)[, 2]) == 1;

cp's OEIS Frontend

A384785 The number of unordered factorizations of the n-th cubefull number into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

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A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

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A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

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A246551 Prime powers p^e where p is a prime and e is odd.

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A304326 Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.

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A304327 Number of ways to write n as a product of a perfect power and a squarefree number.

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A152441 Decimal expansion of Sum_{primes p} 1/(p^2*(p-1)).

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A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.

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