A036967 -id:A036967 - cp's OEIS frontend

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

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

A360841 4-full numbers (A036967) sandwiched between twin primes.

Original entry on oeis.org

2592, 139968, 995328, 37340352, 63700992, 99574272, 169869312, 414720000, 1399680000, 4076863488, 10871635968, 17714700000, 22781250000, 25312500000, 35888419872, 51840000000, 82012500000, 98802571392, 135304020000, 136136700000, 170749797552, 174960000000, 196730062848

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			2592 = 2^5 * 3^4 is a term since it is 4-full and 2591 and 2593 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..1520 (terms below 3*10^19)
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Intersection of A014574 and A036967.

Subsequence of A113839 and A360840.

Select[6*Range[2*10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 3 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 3;

A176273 Partial sums of A036967.

Original entry on oeis.org

1, 17, 49, 113, 194, 322, 565, 821, 1333, 1958, 2687, 3711, 5007, 7055, 9242, 11643, 14235, 17360, 21248, 25344, 30528, 37089, 44865, 53057, 63057, 73425, 85089, 99730, 115282, 130907, 147291, 164098, 183781, 203781, 224517, 247845, 276406

Offset: 1

Jonathan Vos Post, Apr 14 2010

			a(11) = 1 + 16 + 32 + 64 + 81 + 128 + 243 + 256 + 512 + 625 + 729 = 2687.

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

Cf. A000583, A001694, A036966, A036967.

Accumulate @ Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 3 &] (* Amiram Eldar, Feb 07 2023 *)

a(n) = Sum_{i=1..n} A036967(i).

A030514 a(n) = prime(n)^4.

Original entry on oeis.org

16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, 2825761, 3418801, 4879681, 7890481, 12117361, 13845841, 20151121, 25411681, 28398241, 38950081, 47458321, 62742241, 88529281, 104060401, 112550881, 131079601, 141158161

Offset: 1

Jeff Burch

Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the n-th prime). - Alexandre Wajnberg, Jan 15 2006
Subsequence of A036967. - Reinhard Zumkeller, Feb 05 2008
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
The general product formula for even s is: product_{p = A000040} (p^s-1)/(p^s+1) = 2*Bernoulli(2s)/( binomial(2s, s)*Bernoulli^2(s)), where the infinite product is over all primes. Here, with s = 4, product_{n = 1, 2, ...} (a(n)-1)/(a(n)+1) = 6/7. In A030516, where s = 6, the product of the ratios is 691/715. For s = 8, the 8th row in A120458, the corresponding product of ratios is 7234/7293. - R. J. Mathar, Feb 01 2009
Except for the first three terms, all others are congruent to 1 mod 240. - Robert Israel, Aug 29 2014

R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.

Cf. A258601.

a030514 = (^ 4) . a000040
a030514_list = map (^ 4) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

[NthPrime(n)^4: n in [1..100] ]; // Vincenzo Librandi, Apr 22 2011

map(p -> p^4, select(isprime,[2,seq(2*i+1,i=1..100)])); # Robert Israel, Aug 29 2014

Array[Prime[#]^4 &, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^4 \\ Charles R Greathouse IV, Mar 21 2013

[p**4 for p in prime_range(100)]
# Zerinvary Lajos, May 15 2007

a(n) = A000040(n)^(5-1) = A000040(n)^4, where 5 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 5. - Alexandre Wajnberg, Jan 15 2006
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(4) = 0.0769931397... (A085964). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(4)/zeta(8) = 105/Pi^4 (A157290).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(4) = 90/Pi^4 (A215267). (End)

Description corrected by Eric W. Weisstein

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

A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

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, 216, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336

Offset: 1

Eric W. Weisstein

nonn

Also called cubeful numbers, but this term is ambiguous and is best avoided.
Numbers n such that A007427(n) = sum(d|n,mu(d)*mu(n/d)) == 0. - Benoit Cloitre, Apr 17 2002
The convention in the OEIS is that squareful, cubeful, biquadrateful (A046101), ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full (A036966), 4-full (A036967) are used for Golomb's notion of powerful numbers (A001694, see references there), when each prime factor occurs to a power > 1. - M. F. Hasler, Feb 12 2008. Added by N. J. A. Sloane, Apr 25 2023:  This suggestion has not been a success. It is hopeless to try to make a distinction between "cubeful" and "cubefull". To avoid ambiguity, do not use either term, but instead say exactly what you mean.
Also solutions to equation tau_{-2}(n)=0, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Amiram Eldar, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cubefree.

Complement of A004709.
Subsequences: A000578 and A030078.
Cf. A001694, A036966, A046101, A088453.

a046099 n = a046099_list !! (n-1)
a046099_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

isA046099 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) >= 3 then
            return true;
        end if;
    end do:
    false ;
end proc:
for n from 1 do
    if isA046099(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015

lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>2, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *)

is(n)=n>7 && vecmax(factor(n)[,2])>2 \\ Charles R Greathouse IV, Sep 17 2015

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) != n
print(list(filter(ok, range(1, 337)))) # Michael S. Branicky, Aug 16 2021

from sympy import mobius, integer_nthroot
def A046099(n):
    def f(x): return n+sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A212793(a(n)) = 0. - Reinhard Zumkeller, May 27 2012

Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(3*s)-1))/zeta(3*s). - Amiram Eldar, Dec 27 2022

More terms from Vladimir Joseph Stephan Orlovsky, Aug 15 2008

Edited by N. J. A. Sloane, Jul 27 2009

A046101 Biquadrateful numbers.

Original entry on oeis.org

16, 32, 48, 64, 80, 81, 96, 112, 128, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 640, 648, 656, 672, 688, 704

Offset: 1

Eric W. Weisstein

nonn

The convention in the OEIS is that squareful, cubeful (A046099), biquadrateful, ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full (A036966), 4-full (A036967) are used for Golomb's notion of powerful numbers (A001694, see references there), when each prime factor occurs to a power > 1. - M. F. Hasler, Feb 12 2008
Also solutions to equation tau_{-3}(n)=0, where tau_{-3} is A007428. - Enrique Pérez Herrero, Jan 19 2013
Sum_{n>0} 1/a(n)^s = Zeta(s) - Zeta(s)/Zeta(4s). - Enrique Pérez Herrero, Jan 21 2013
A051903(a(n)) > 3. - Reinhard Zumkeller, Sep 03 2015
The asymptotic density of this sequence is 1 - 1/zeta(4) = 1 - 90/Pi^4 = 0.076061... - Amiram Eldar, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046100, A046099, A036967, A001694, A051903, A215267.

a046101 n = a046101_list !! (n-1)
a046101_list = filter ((> 3) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015

with(NumberTheory):
isBiquadrateful := n -> is(denom(Radical(n) / LargestNthPower(n, 2)) <> 1):
select(isBiquadrateful, [`$`(1..704)]);  # Peter Luschny, Jul 12 2022

lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>3, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *)
Select[Range[1000],Max[Transpose[FactorInteger[#]][[2]]]>3&] (* Harvey P. Dale, May 25 2014 *)

is(n)=n>9 && vecmax(factor(n)[,2])>3 \\ Charles R Greathouse IV, Sep 03 2015

from sympy import mobius, integer_nthroot
def A046101(n):
    def f(x): return n+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 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

A069492 5-full numbers: if a prime p divides k then so does p^5.

Original entry on oeis.org

1, 32, 64, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 7776, 8192, 15552, 15625, 16384, 16807, 19683, 23328, 31104, 32768, 46656, 59049, 62208, 65536, 69984, 78125, 93312, 100000, 117649, 124416, 131072, 139968, 161051

Offset: 1

Benoit Cloitre, Apr 15 2002

a(m) mod prime(n) > 0 for m < A258602(n); a(A258602(n)) = A050997(n) = prime(n)^5. - Reinhard Zumkeller, Jun 06 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms n=148..10000 corrected by Andrew Howroyd)

Cf. A036967, A036966, A001694.
Cf. A050997.
Cf. A258602.

import Data.Set (singleton, deleteFindMin, fromList, union)
a069492 n = a069492_list !! (n-1)
a069492_list = 1 : f (singleton z) [1, z] zs where
   f s q5s p5s'@(p5:p5s)
     | m < p5 = m : f (union (fromList $ map (* m) ps) s') q5s p5s'
     | otherwise = f (union (fromList $ map (* p5) q5s) s) (p5:q5s) p5s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a050997_list
-- Reinhard Zumkeller, Jun 03 2015

for(n=1,250000,if(n*sumdiv(n,d,isprime(d)/d^5)==floor(n*sumdiv(n,d,isprime(d)/d^5)),print1(n,",")))

\\ Gen(limit,k) defined in A036967.
Gen(170000, 5) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import integer_nthroot, factorint
def A069492(n):
    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
    def f(x):
        c = n+x
        for t in range(1,integer_nthroot(x,9)[0]+1):
            if all(d<=1 for d in factorint(t).values()):
                for u in range(1,integer_nthroot(s:=x//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
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^4*(p-1))) = 1.0695724994489739263413712783666538355049945684326048537289707764272637... - Amiram Eldar, Jul 09 2020

cp's OEIS Frontend

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

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A360841 4-full numbers (A036967) sandwiched between twin primes.

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A176273 Partial sums of A036967.

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A030514 a(n) = prime(n)^4.

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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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A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

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