A036966 -id:A036966 - cp's OEIS frontend

A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

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

Offset: 1

Amiram Eldar, Feb 11 2023

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

Cf. A004709, A036966, A057521, A360539.

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

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

a(n) = 1 if and only if n is a cubefree number (A004709).
a(n) = n if and only if n is a cubefull number (A036966).
a(n) <= A057521(n) with equality if and only if n is in A337050.
a(n) = n/A360539(n).
Multiplicative with a(p^e) = p^e if e >= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - p^(1-s) + p^(-s) - p^(1-3*s) - p^(1-2*s) + p^(-2*s) + p^(3-3*s)).

A258600 a(n) is the index m such that A036966(m) = prime(n)^3.

Original entry on oeis.org

2, 4, 8, 13, 23, 29, 39, 45, 57, 75, 81, 99, 110, 117, 130, 149, 169, 176, 197, 209, 212, 236, 250, 270, 295, 309, 317, 328, 337, 354, 399, 414, 436, 445, 477, 483, 506, 529, 541, 563, 585, 591, 631, 635, 654, 657, 697, 747, 758, 765, 781, 803, 809, 845, 864

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

			.   n |  p |  a(n) | A036966(a(n)) = A030078(n) = p^3
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             8
.   2 |  3 |     4 |            27
.   3 |  5 |     8 |           125
.   4 |  7 |    13 |           343
.   5 | 11 |    23 |          1331
.   6 | 13 |    29 |          2197
.   7 | 17 |    39 |          4913
.   8 | 19 |    45 |          6859
.   9 | 23 |    57 |         12167
.  10 | 29 |    75 |         24389
.  11 | 31 |    81 |         29791
.  12 | 37 |    99 |         50653
.  13 | 41 |   110 |         68921
.  14 | 43 |   117 |         79507
.  15 | 47 |   130 |        103823
.  16 | 53 |   149 |        148877
.  17 | 59 |   169 |        205379
.  18 | 61 |   176 |        226981
.  19 | 67 |   197 |        300763
.  20 | 71 |   209 |        357911
.  21 | 73 |   212 |        389017
.  22 | 79 |   236 |        493039
.  23 | 83 |   250 |        571787
.  24 | 89 |   270 |        704969
.  25 | 97 |   295 |        912673  .

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A258568, A000040, A030078, A036966, A258599, A258601, A258602, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258600 = (+ 1) . fromJust . (`elemIndex` a258568_list) . a000040

With[{m = 60}, c = Select[Range[Prime[m]^3], Min[FactorInteger[#][[;; , 2]]] > 2 &]; 1 + Flatten[FirstPosition[c, #] & /@ (Prime[Range[m]]^3)]] (* Amiram Eldar, Feb 07 2023 *)

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258600(n):
    c, m = 0, prime(n)**3
    for w in range(1,integer_nthroot(m,5)[0]+1):
        if all(d<=1 for d in factorint(w).values()):
            for y in range(1,integer_nthroot(z:=m//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 # Chai Wah Wu, Sep 10 2024

A036966(a(n)) = A030078(n) = prime(n)^3.
A036966(m) mod prime(n) > 0 for m < a(n).
Also smallest number m such that A258568(m) = prime(n):
A258568(a(n)) = A000040(n) and A258568(m) != A000040(n) for m < a(n).

a(11)-a(55) and example corrected by Amiram Eldar, Feb 07 2023

A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 16, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356192 at n=16.

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

Cf. A002117, A036966, A360541.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356194.

f[p_, e_] := p^Max[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^max(f[i,2],3))};

Multiplicative with a(p^e) = p^max(e,3).
a(n) = n iff n is in A036966.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p-2)/(p^3*(p-1))) = 1.76434793373691907811... . - Amiram Eldar, Jul 29 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(3)/4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.1559111567... . - Amiram Eldar, Nov 13 2022
a(n) = n * A360541(n). - Amiram Eldar, Sep 01 2023

A337736 The number of cubefull numbers (A036966) between the consecutive cubes n^3 and (n+1)^3.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 17 2020

nonn

For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1991).

			a(2) = 1 since there is one cubefull number, 16 = 2^4, between 2^3 = 8 and 3^3 = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295. See section 3, p. 291.

Cf. A000578, A036966, A119241, A337737, A362974.

cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; a[n_] := Count[Range[n^3 + 1, (n + 1)^3 - 1], _?cubQ]; Array[a, 100]

from math import gcd
from sympy import integer_nthroot, factorint
def A337736(n):
    def f(x):
        c = 0
        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 f((n+1)**3-1)-f(n**3) # Chai Wah Wu, Sep 10 2024

Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = A362974 - 1 = 3.659266... . - Amiram Eldar, May 11 2023

A362986 a(n) = A000203(A036966(n)), the sum of divisors of the n-th cubefull number A036966(n).

Original entry on oeis.org

1, 15, 31, 40, 63, 127, 121, 156, 255, 600, 364, 511, 400, 1240, 1023, 781, 1815, 1093, 2520, 2340, 2047, 3751, 1464, 5080, 5460, 4836, 4095, 3280, 2380, 2801, 7623, 6000, 3906, 6240, 10200, 11284, 9828, 8191, 5220, 11715, 15367, 12400, 16395, 9841, 7240, 20440

Offset: 1

Amiram Eldar, May 12 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (7).

Cf. A000203, A036966, A180114, A362974, A362985.

DivisorSigma[1, Select[Range[10^4], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(sigma(k), ", ")));

from itertools import count, islice
from math import prod
from sympy import factorint
def A362986_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e>2 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A362986_list = list(islice(A362986_gen(),20)) # Chai Wah Wu, May 21 2023

Sum_{A036966(k) < x} a(k) = c * x^(4/3) + O(x^(113/96 + eps)), where c = A362985 * A362974 / 4 = 2.8912833599... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

Sum_{k=1..n} a(k) ~ c * n^4, where c = A362985 / (4 * A362974^3) = 0.006135085083... .

A379718 The second Jordan totient function applied to the cubefull numbers: a(n) = A007434(A036966(n)).

Original entry on oeis.org

1, 48, 192, 648, 768, 3072, 5832, 15000, 12288, 31104, 52488, 49152, 115248, 124416, 196608, 375000, 279936, 472392, 497664, 720000, 786432, 1119744, 1756920, 1990656, 2519424, 2880000, 3145728, 4251528, 4798248, 5647152, 4478976, 5531904, 9375000, 9720000, 7962624

Offset: 1

Amiram Eldar, Dec 31 2024

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

Cf. A007434, A013661, A036966, A371412 (analogous with J_1 = phi), A379715, A379716, A379717.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; Join[{1}, j2 /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]

j2(f) = prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));
list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmin(f[, 2]) > 2, print1(j2(f), ", "))); }

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6) = 1.02964361441212748276... .
In general, Sum_{m cubefull} 1/J_k(m) = zeta(k)^2 * Product_{p prime} (1 - 2/p^k + 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.
In general, Sum_{m k-full} 1/J_2(m) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^(2*k)), for k >= 2.

A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 33, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A368248(n), and the largest of them is A360540(n).

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

The unitary analog of A385005.
Cf. A034448, A036966, A046100, A360540, A368248.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), this sequence (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[e <= 2, 1, p^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] < 3, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = p^e + 1 if e >= 3.
a(n) = A034448(n) / A371242(n).
a(n) <= A034448(n), with equality if and only if n is cubefull (A036966).
a(n) <= A385005(n), with equality if and only if n is biquadratefree (A046100).
Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - 1/p^(s-1) + 1/p^(3*s-3) - 1/p^(4*s-3)).

A385136 The sum of divisors d of n such that n/d is a cubefull number (A036966).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 19, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 39, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 57, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 79, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A013661, A036966, A385005.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), this sequence (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := (p^(e+1) - p^e + p^(e-2) - 1)/(p-1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p) = p and a(p^e) = (p^(e+1) - p^e + p^(e-2) - 1)/(p-1) for e >= 2.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.022486596136980366... .

A360840 3-full numbers (A036966) sandwiched between twin primes.

Original entry on oeis.org

432, 2592, 139968, 444528, 472392, 995328, 3456000, 5174928, 6561000, 10125000, 15552000, 15804072, 17496000, 25299648, 28449792, 37340352, 54675000, 63700992, 85957848, 88723728, 99574272, 120891312, 144027072, 169869312, 177147000, 197413632, 253125000, 259308000

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			432 = 2^4 * 3^3 is a term since it is 3-full and 431 and 433 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Intersection of A014574 and A036966.

Subsequence of A113839.

Select[6*Range[10^6], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 2 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 2;

A363195 Number of divisors of the n-th cubefull number A036966(n).

Original entry on oeis.org

1, 4, 5, 4, 6, 7, 5, 4, 8, 16, 6, 9, 4, 20, 10, 5, 20, 7, 24, 16, 11, 25, 4, 28, 24, 20, 12, 8, 4, 5, 30, 16, 6, 16, 32, 30, 24, 13, 4, 20, 35, 20, 28, 9, 4, 36, 36, 28, 14, 16, 25, 20, 40, 16, 24, 35, 4, 40, 5, 42, 7, 32, 15, 6, 20, 32, 16, 20, 10, 30, 45, 20

Offset: 1

Amiram Eldar, May 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).

Cf. A000005, A036966, A362986.

Similar sequences: A072048, A076400, A363194.

DivisorSigma[0, Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(numdiv(k), ", ")));

from itertools import count, islice
from math import prod
from sympy import factorint
def A363195_gen(): # generator of terms
    for n in count(1):
        f = factorint(n).values()
        if all(e>2 for e in f):
            yield prod(e+1 for e in f)
A363195_list = list(islice(A363195_gen(),20)) # Chai Wah Wu, May 21 2023

a(n) = A000005(A036966(n)).

Sum_{A036966(k) < x} a(k) = c_1 * x^(1/3) * log(x)^3 + c_2 * x^(1/3) * log(x)^2 + c_3 * x^(1/3) * log(x) + c_4 * x^(1/3) + O(x^(7/24 + eps)), where c_1, c_2, c_3 and c_4 are constants. c_1 = Product_{p prime} ((1-1/p)^4 * (1 + 1/((p^(1/3) - 1)^2*p^(1/3)) + 3/(p-p^(2/3))))/162 = 0.1346652397135839416... . [corrected Sep 21 2024]

cp's OEIS Frontend

A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

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A258600 a(n) is the index m such that A036966(m) = prime(n)^3.

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A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

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A337736 The number of cubefull numbers (A036966) between the consecutive cubes n^3 and (n+1)^3.

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A362986 a(n) = A000203(A036966(n)), the sum of divisors of the n-th cubefull number A036966(n).

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A379718 The second Jordan totient function applied to the cubefull numbers: a(n) = A007434(A036966(n)).

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A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

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A385136 The sum of divisors d of n such that n/d is a cubefull number (A036966).