A258600 -id:A258600 - cp's OEIS frontend

A030078 Cubes of primes.

8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091, 2571353, 2685619, 3307949

Offset: 1

Patrick De Geest

Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001
Intersection of A014612 and A000578. Intersection of A014612 and A030513. - Wesley Ivan Hurt, Sep 10 2013
Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020
There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021

			a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.

Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen über Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Eric Weisstein's World of Mathematics, Prime Power.
Wikipedia, p-group, Classification.
Index to sequences related to prime signature

Other sequences that are k-th powers of primes are: A000040 (k=1), A001248 (k=2), this sequence (k=3), A030514 (k=4), A050997 (k=5), A030516 (k=6), A092759 (k=7), A179645 (k=8), A179665 (k=9), A030629 (k=10), A079395 (k=11), A030631 (k=12), A138031 (k=13), A030635 (k=16), A138032 (k=17), A030637 (k=18).
Cf. A060800, A131991, A000578, subsequence of A046099.
Subsequence of A007422 and of A054397.
Cf. A036966, A085541, A088453, A157289, A258600.

a030078 = a000578 . a000040
a030078_list = map a000578 a000040_list -- Reinhard Zumkeller, May 26 2012

[p^3: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Array[Prime[ # ]^3&, 5! ] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^3 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime, primerange
def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
print(aupton(35)) # Michael S. Branicky, Aug 27 2021

[p**3 for p in prime_range(100)] # Zerinvary Lajos, May 15 2007

n such that A062799(n) = 3. - Benoit Cloitre, Apr 06 2002
a(n) = A000040(n)^3. - Omar E. Pol, Jul 27 2009
A064380(a(n)) = A000010(a(n)). - Vladimir Shevelev, Apr 19 2010
A003415(a(n)) = A079705(n). - Reinhard Zumkeller, Jun 26 2011
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
A000005(a(n)) = 4. - Wesley Ivan Hurt, Sep 10 2013
a(n) = A119959(n) * A008864(n) -1.- R. J. Mathar, Aug 13 2019
Sum_{n>=1} 1/a(n) = P(3) = 0.1747626392... (A085541). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) (A157289).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) (A088453). (End)

A036966 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.

Original entry on oeis.org

1, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 432, 512, 625, 648, 729, 864, 1000, 1024, 1296, 1331, 1728, 1944, 2000, 2048, 2187, 2197, 2401, 2592, 2744, 3125, 3375, 3456, 3888, 4000, 4096, 4913, 5000, 5184, 5488, 5832, 6561, 6859, 6912, 7776, 8000

Offset: 1

N. J. A. Sloane

Also called powerful_3 numbers.
For n > 1: A124010(a(n),k) > 2, k = 1..A001221(a(n)). - Reinhard Zumkeller, Dec 15 2013
a(m) mod prime(n) > 0 for m < A258600(n); a(A258600(n)) = A030078(n) = prime(n)^3. - Reinhard Zumkeller, Jun 06 2015

M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
P. Erdős and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
M. J. Halm, Sequences
H.-Q. Liu, The number of cubefull numbers in an interval (supplement), Funct. Approx. Comment. Math. 43 (2) 105-107, December 2010.
Index entries for sequences related to powerful numbers

Cf. A001694, A030078, A036967, A065483, A258600, A376363, A376364.

import Data.Set (singleton, deleteFindMin, fromList, union)
a036966 n = a036966_list !! (n-1)
a036966_list = 1 : f (singleton z) [1, z] zs where
   f s q3s p3s'@(p3:p3s)
     | m < p3 = m : f (union (fromList $ map (* m) ps) s') q3s p3s'
     | otherwise = f (union (fromList $ map (* p3) q3s) s) (p3:q3s) p3s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030078_list
-- Reinhard Zumkeller, Jun 03 2015, Dec 15 2013

isA036966 := proc(n)
    local p ;
    for p in ifactors(n)[2] do
        if op(2,p) < 3 then
            return false;
        end if;
    end do:
    return true ;
end proc:
A036966 := proc(n)
    option remember;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA036966(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 01 2013

Select[ Range[2, 8191], Min[ Table[ # [[2]], {1}] & /@ FactorInteger[ # ]] > 2 &]
Join[{1},Select[Range[8000],Min[Transpose[FactorInteger[#]][[2]]]>2&]] (* Harvey P. Dale, Jul 17 2013 *)

is(n)=n==1 || vecmin(factor(n)[,2])>2 \\ Charles R Greathouse IV, Sep 17 2015

list(lim)=my(v=List(),t); for(a=1,sqrtnint(lim\1,5), for(b=1,sqrtnint(lim\a^5,4), t=a^5*b^4; for(c=1,sqrtnint(lim\t,3), listput(v,t*c^3)))); Set(v) \\ Charles R Greathouse IV, Nov 20 2015

list(lim)=my(v=List(),t); forsquarefree(a=1,sqrtnint(lim\1,5), my(a5=a[1]^5); forsquarefree(b=1,sqrtnint(lim\a5,4), if(gcd(a[1],b[1])>1, next); t=a5*b[1]^4; for(c=1,sqrtnint(lim\t,3), listput(v,t*c^3)))); Set(v) \\ Charles R Greathouse IV, Jan 12 2022

from math import gcd
from sympy import integer_nthroot, factorint
def A036966(n):
    def f(x):
        c = n+x
        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
    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 10 2024

Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) (A065483). - Amiram Eldar, Jun 23 2020

Numbers of the form x^5*y^4*z^3. There is a unique representation with x,y squarefree and coprime. - Charles R Greathouse IV, Jan 12 2022

More terms from Erich Friedman

Corrected by Vladeta Jovovic, Aug 17 2002

A258599 a(n) is the index m such that A001694(m) = prime(n)^2.

Original entry on oeis.org

2, 4, 6, 10, 16, 20, 28, 31, 39, 48, 51, 65, 71, 75, 84, 94, 107, 110, 120, 129, 133, 145, 152, 163, 180, 187, 191, 199, 202, 212, 238, 246, 258, 261, 282, 286, 297, 309, 319, 330, 342, 344, 366, 372, 377, 382, 407, 431, 440, 443, 450, 463, 468, 487, 498

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

			.   n |  p |  a(n) | A001694(a(n)) = A001248(n) = p^2
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             4
.   2 |  3 |     4 |             9
.   3 |  5 |     6 |            25
.   4 |  7 |    10 |            49
.   5 | 11 |    16 |           121
.   6 | 13 |    20 |           169
.   7 | 17 |    28 |           289
.   8 | 19 |    31 |           361
.   9 | 23 |    39 |           529
.  10 | 29 |    48 |           841
.  11 | 31 |    51 |           961
.  12 | 37 |    65 |          1369
.  13 | 41 |    71 |          1681
.  14 | 43 |    75 |          1849
.  15 | 47 |    84 |          2209
.  16 | 53 |    94 |          2809
.  17 | 59 |   107 |          3481
.  18 | 61 |   110 |          3721
.  19 | 67 |   120 |          4489
.  20 | 71 |   129 |          5041
.  21 | 73 |   133 |          5329
.  22 | 79 |   145 |          6241
.  23 | 83 |   152 |          6889
.  24 | 89 |   163 |          7921
.  25 | 97 |   180 |          9409  .

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

Cf. A258567, A000040, A001248, A001694, A258600, A258601, A258602, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258599 = (+ 1) . fromJust . (`elemIndex` a258567_list) . a000040

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

from math import isqrt
from sympy import prime, integer_nthroot, factorint
def A258599(n):
    m = prime(n)**2
    return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # Chai Wah Wu, Sep 10 2024

A001694(a(n)) = A001248(n) = prime(n)^2.
A001694(m) mod prime(n) > 0 for m < a(n).
Also smallest number m such that A258567(m) = prime(n):
A258567(a(n)) = A000040(n) and A258567(m) != A000040(n) for m < a(n).

A258601 a(n) is the index m such that A036967(m) = prime(n)^4.

Original entry on oeis.org

2, 5, 10, 16, 28, 37, 55, 61, 80, 105, 113, 142, 163, 170, 190, 219, 249, 260, 286, 310, 318, 352, 374, 407, 448, 472, 482, 505, 511, 536, 614, 634, 672, 682, 740, 754, 783, 821, 842, 878, 916, 924, 984, 996, 1015, 1032, 1103, 1171, 1201, 1213, 1233, 1270, 1286, 1343, 1379

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

A036967(a(n)) = A030514(n) = prime(n)^4;
A036967(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258569(m) = prime(n):
A258569(a(n)) = A000040(n) and A258569(m) != A000040(n) for m < a(n).

			.   n |  p |  a(n) | A036967(a(n)) = A030514(n) = p^4
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            16
.   2 |  3 |     5 |            81
.   3 |  5 |    10 |           625
.   4 |  7 |    16 |          2401
.   5 | 11 |    28 |         14641
.   6 | 13 |    37 |         28561
.   7 | 17 |    55 |         83521
.   8 | 19 |    61 |        130321
.   9 | 23 |    80 |        279841
.  10 | 29 |   105 |        707281
.  11 | 31 |   113 |        923521
.  12 | 37 |   142 |       1874161
.  13 | 41 |   163 |       2825761
.  14 | 43 |   170 |       3418801
.  15 | 47 |   190 |       4879681
.  16 | 53 |   219 |       7890481
.  17 | 59 |   249 |      12117361
.  18 | 61 |   260 |      13845841
.  19 | 67 |   286 |      20151121
.  20 | 71 |   310 |      25411681
.  21 | 73 |   318 |      28398241
.  22 | 79 |   352 |      38950081
.  23 | 83 |   374 |      47458321
.  24 | 89 |   407 |      62742241
.  25 | 97 |   448 |      88529281

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A258569, A000040, A030514, A036967, A258599, A258600, A258602, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258601 = (+ 1) . fromJust . (`elemIndex` a258569_list) . a000040

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^4,4) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258601(n):
    c, m = 0, prime(n)**4
    for u in range(1,integer_nthroot(m,7)[0]+1):
        if all(d<=1 for d in factorint(u).values()):
            for w in range(1,integer_nthroot(a:=m//u**7,6)[0]+1):
                if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                    for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                        if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                            c += integer_nthroot(z//y**5,4)[0]
    return c # Chai Wah Wu, Sep 10 2024

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024

A258602 a(n) is the index m such that A069492(m) = prime(n)^5.

Original entry on oeis.org

2, 5, 12, 20, 37, 45, 68, 82, 106, 142, 154, 196, 219, 234, 260, 305, 342, 360, 407, 434, 451, 496, 528, 573, 635, 668, 681, 720, 737, 770, 885, 919, 966, 984, 1065, 1087, 1139, 1193, 1228, 1283, 1331, 1348, 1440, 1455, 1484, 1509, 1624, 1731, 1767, 1789

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

A069492(a(n)) = A050997(n) = prime(n)^5;
A069492(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258570(m) = prime(n):
A258570(a(n)) = A000040(n) and A258570(m) != A000040(n) for m < a(n).

			.   n |  p |  a(n) | A069492(a(n)) = A050997(n) = p^5
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            32
.   2 |  3 |     5 |           243
.   3 |  5 |    12 |          3125
.   4 |  7 |    20 |         16807
.   5 | 11 |    37 |        161051
.   6 | 13 |    45 |        371293
.   7 | 17 |    68 |       1419857
.   8 | 19 |    82 |       2476099
.   9 | 23 |   106 |       6436343
.  10 | 29 |   142 |      20511149
.  11 | 31 |   154 |      28629151
.  12 | 37 |   196 |      69343957
.  13 | 41 |   219 |     115856201
.  14 | 43 |   234 |     147008443
.  15 | 47 |   260 |     229345007
.  16 | 53 |   305 |     418195493
.  17 | 59 |   342 |     714924299
.  18 | 61 |   360 |     844596301
.  19 | 67 |   407 |    1350125107
.  20 | 71 |   434 |    1804229351
.  21 | 73 |   451 |    2073071593
.  22 | 79 |   496 |    3077056399
.  23 | 83 |   528 |    3939040643
.  24 | 89 |   573 |    5584059449
.  25 | 97 |   635 |    8587340257  .

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A258570, A000040, A050997, A069492, A258599, A258600, A258601, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258602 = (+ 1) . fromJust . (`elemIndex` a258570_list) . a000040

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^5,5) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258602(n):
    c, m = 0, prime(n)**5
    for t in range(1,integer_nthroot(m,9)[0]+1):
        if all(d<=1 for d in factorint(t).values()):
            for u in range(1,integer_nthroot(s:=m//t**9,8)[0]+1):
                if gcd(t,u)==1 and all(d<=1 for d in factorint(u).values()):
                    for w in range(1,integer_nthroot(a:=s//u**8,7)[0]+1):
                        if gcd(u,w)==1 and gcd(t,w)==1 and all(d<=1 for d in factorint(w).values()):
                            for y in range(1,integer_nthroot(z:=a//w**7,6)[0]+1):
                                if gcd(w,y)==1 and gcd(u,y)==1 and gcd(t,y)==1 and all(d<=1 for d in factorint(y).values()):
                                    c += integer_nthroot(z//y**6,5)[0]
    return c # Chai Wah Wu, Sep 10 2024

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024

A258603 a(n) is the index m such that A069493(m) = prime(n)^6.

Original entry on oeis.org

2, 6, 13, 22, 45, 58, 87, 102, 135, 181, 199, 252, 287, 306, 342, 401, 461, 479, 536, 583, 602, 665, 712, 776, 860, 911, 932, 975, 997, 1051, 1212, 1258, 1331, 1356, 1479, 1502, 1580, 1651, 1705, 1784, 1856, 1879, 2013, 2037, 2093, 2113, 2272, 2438, 2484, 2510

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

A069493(a(n)) = A030516(n) = prime(n)^6;
A069493(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258571(m) = prime(n):
A258571(a(n)) = A000040(n) and A258571(m) != A000040(n) for m < a(n).

			.   n |  p |  a(n) | A069493(a(n)) = A030516(n) = p^6
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            64
.   2 |  3 |     6 |           729
.   3 |  5 |    13 |         15625
.   4 |  7 |    22 |        117649
.   5 | 11 |    45 |       1771561
.   6 | 13 |    58 |       4826809
.   7 | 17 |    87 |      24137569
.   8 | 19 |   102 |      47045881
.   9 | 23 |   135 |     148035889
.  10 | 29 |   181 |     594823321
.  11 | 31 |   199 |     887503681
.  12 | 37 |   252 |    2565726409
.  13 | 41 |   287 |    4750104241
.  14 | 43 |   306 |    6321363049
.  15 | 47 |   342 |   10779215329
.  16 | 53 |   401 |   22164361129
.  17 | 59 |   461 |   42180533641
.  18 | 61 |   479 |   51520374361
.  19 | 67 |   536 |   90458382169
.  20 | 71 |   583 |  128100283921
.  21 | 73 |   602 |  151334226289
.  22 | 79 |   665 |  243087455521
.  23 | 83 |   712 |  326940373369
.  24 | 89 |   776 |  496981290961
.  25 | 97 |   860 |  832972004929  .

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A258571, A000040, A030516, A069493, A258599, A258600, A258601, A258602.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258603 = (+ 1) . fromJust . (`elemIndex` a258571_list) . a000040

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^6,6) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258603(n):
    c, m = 0, prime(n)**6
    for y1 in range(1,integer_nthroot(m,11)[0]+1):
        if all(d<=1 for d in factorint(y1).values()):
            for y2 in range(1,integer_nthroot(z2:=m//y1**11,10)[0]+1):
                if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
                    for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
                        if all(gcd(y3,x)==1 for x in (y1,y2)) and all(d<=1 for d in factorint(y3).values()):
                            for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
                                if all(gcd(y4,x)==1 for x in (y1,y2,y3)) and all(d<=1 for d in factorint(y4).values()):
                                    for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
                                        if all(gcd(y5,x)==1 for x in (y1,y2,y3,y4)) and all(d<=1 for d in factorint(y5).values()):
                                            c += integer_nthroot(z5//y5**7,6)[0]
    return c # Chai Wah Wu, Sep 10 2024

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024

A069493 6-full numbers: if p divides n then so does p^6.

Original entry on oeis.org

1, 64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 46656, 59049, 65536, 78125, 93312, 117649, 131072, 139968, 177147, 186624, 262144, 279936, 373248, 390625, 419904, 524288, 531441, 559872, 746496, 823543, 839808

Offset: 1

Benoit Cloitre, Apr 15 2002

a(m) mod prime(n) > 0 for m < A258603(n); a(A258600(n)) = A030516(n) = prime(n)^6. - Reinhard Zumkeller, Jun 06 2015

			2^7*3^6 = 93312 is a member (although not of A076470).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A036967, A036966, A001694. Different from A076470.

Cf. A030516, A258603.

import Data.Set (singleton, deleteFindMin, fromList, union)
a069493 n = a069493_list !! (n-1)
a069493_list = 1 : f (singleton z) [1, z] zs where
   f s q6s p6s'@(p6:p6s)
     | m < p6 = m : f (union (fromList $ map (* m) ps) s') q6s p6s'
     | otherwise = f (union (fromList $ map (* p6) q6s) s) (p6:q6s) p6s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030516_list
-- Reinhard Zumkeller, Jun 03 2015

Join[{1},Select[Range[900000],Min[FactorInteger[#][[All,2]]]>5&]] (* Harvey P. Dale, Mar 03 2018 *)

for(n=1,560000,if(n*sumdiv(n,d,isprime(d)/d^6)==floor(n*sumdiv(n,d,isprime(d)/d^6)),print1(n,",")))

from math import gcd
from sympy import factorint, integer_nthroot
def A069493(n):
    def f(x):
        c = n+x
        for y1 in range(1,integer_nthroot(x,11)[0]+1):
            if all(d<=1 for d in factorint(y1).values()):
                for y2 in range(1,integer_nthroot(z2:=x//y1**11,10)[0]+1):
                    if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
                        for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
                            if gcd(y3,y1)==1 and gcd(y3,y2)==1 and all(d<=1 for d in factorint(y3).values()):
                                for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
                                    if gcd(y4,y1)==1 and gcd(y4,y2)==1 and gcd(y4,y3)==1 and all(d<=1 for d in factorint(y4).values()):
                                        for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
                                            if gcd(y5,y1)==1 and gcd(y5,y2)==1 and gcd(y5,y3)==1 and gcd(y5,y4)==1 and all(d<=1 for d in factorint(y5).values()):
                                                c -= integer_nthroot(z5//y5**7,6)[0]
        return c
    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 10 2024

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^5*(p-1))) = 1.0334657852594050612296726462481884631303137561267151463866539131591664... - Amiram Eldar, Jul 09 2020

A258568 Smallest prime factors of 3-full numbers.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 5, 2, 2, 3, 2, 7, 2, 2, 5, 2, 3, 2, 2, 2, 2, 11, 2, 2, 2, 2, 3, 13, 7, 2, 2, 5, 3, 2, 2, 2, 2, 17, 2, 2, 2, 2, 3, 19, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 23, 2, 11, 2, 5, 2, 2, 7, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 29, 2, 2, 2, 3, 13, 31, 3, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (corrected by Amiram Eldar)

Cf. A036966, A020639, A258600, A258567, A258569, A258570, A258571.

Haskell
```
a258568 = a020639 . a036966
```

Table[If[Min[(f = FactorInteger[n])[[;; , 2]]] > 2 || n == 1, f[[1, 1]], Nothing], {n, 1, 30000}] (* Amiram Eldar, Feb 07 2023 *)

a(n) = A020639(A036966(n)).

a(A258600(n)) = A000040(n) and a(m) != A000040(n) for m < A258600(n).

Missing term a(78) inserted by Amiram Eldar, Feb 07 2023

cp's OEIS Frontend

A030078 Cubes of primes.

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A036966 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.

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A258599 a(n) is the index m such that A001694(m) = prime(n)^2.

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A258601 a(n) is the index m such that A036967(m) = prime(n)^4.

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A258602 a(n) is the index m such that A069492(m) = prime(n)^5.

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A258603 a(n) is the index m such that A069493(m) = prime(n)^6.

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