A069493 -id:A069493 - cp's OEIS frontend

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

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

A360843 6-full numbers (A069493) sandwiched between twin primes.

Original entry on oeis.org

139968, 98802571392, 174960000000, 889223142528, 1594323000000, 2348273369088, 19144761127488, 28697814000000, 56358560858112, 84537841287168, 150289495621632, 186624000000000, 328341017826432, 369056250000000, 392147405854848, 578415690713088, 597871125000000

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			139968 = 2^6 * 3^7 is a term since it is 6-full and 139967 and 139969 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..74 (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 A069493.

Subsequence of A113839, A360840, A360841 and A360842.

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

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 5;

A076470 Perfect powers m^k where k >= 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, 117649, 131072, 177147, 262144, 279936, 390625, 524288, 531441, 823543, 1000000, 1048576, 1594323, 1679616, 1771561

Offset: 1

Robert G. Wilson v, Oct 14 2002

nonn

A necessary but not sufficient condition is that if p|n when at least p^6|n.

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

Cf. A001597, A076467, A076468, A076469.
Different from A069493.
Cf. A002117, A013661, A013662, A013663.

a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 1953124}]; a

from sympy import mobius, integer_nthroot
def A076470(n):
    def f(x): return int(n+9+x-(sum(integer_nthroot(x,d)[0] for d in (6,10,15))<<1)-sum(integer_nthroot(x,d)[0] for d in (8,9,12,20,25))+sum(mobius(k)*(sum(integer_nthroot(x,k*i)[0] for i in range(1,6))-5) for k in range(6,x.bit_length())))
    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 14 2024

Sum_{n>=1} 1/a(n) = 5 - zeta(2) - zeta(3) - zeta(4) - zeta(5) + Sum_{k>=2} mu(k)*(5 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k) - zeta(5*k)) = 1.03342597171... . - Amiram Eldar, Dec 03 2022

A258571 Smallest prime factors of 6-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 3, 2, 5, 2, 7, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 3, 2, 2, 7, 2, 2, 2, 2, 2, 2, 3, 2, 11, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 13, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 11, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000 (terms 1..191 from Reinhard Zumkeller)

Cf. A069493, A020639, A258603, A258567, A258568, A258569, A258570.

Haskell
```
a258571 = a020639 . a069493
```

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 5, print1(f[1, 1], ", "))); } \\ Amiram Eldar, Sep 12 2024

a(n) = A020639(A069493(n)).

a(A258603(n)) = A000040(n) and a(m) != A000040(n) for m < A258603(n).

A360844 a(n) is the least k-full number that is sandwiched between twin primes.

Original entry on oeis.org

4, 432, 2592, 139968, 139968, 174960000000, 56358560858112, 84537841287168, 578415690713088, 578415690713088, 1141260857376768, 61628086298345472, 61628086298345472, 61628086298345472, 322850407500000000000000000000, 322850407500000000000000000000, 62518864539857068333550694039552

Offset: 2

Amiram Eldar, Feb 23 2023

nonn

k-full number is a number m such that if a prime p divides m then so does p^k. All the exponents in the canonical prime factorization of a k-full number are not smaller than k.
a(2)-a(15) are the terms below 3*10^19. Except for a(7) = 174960000000, they are all 3-smooth numbers (A003586, and thus they are terms of A027856). Are there other terms that are not 3-smooth?
a(168) = 2^176 * 3^173 * 7^168 is the first term that is not 5-smooth. - Bert Dobbelaere, Feb 24 2023

			The first 3 terms, their factorizations and the corresponding twin primes are:
  n |   a(n)  prime factorization  A051904(a(n))  {a(n)-1, a(n)+1}
  ----------------------------------------------------------------
  2 |     4                  2^2              2             {3, 5}
  3 |   432            2^4 * 3^3              3         {431, 433}
  4 |  2592            2^5 * 3^4              4       {2591, 2593}

Bert Dobbelaere, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Cf. A001097, A001694, A014574, A036966, A036967, A069492, A069493.
Cf. A113839, A360840, A360841, A360842, A360843.
Cf. A027856, A003586, A051904.

More terms from Bert Dobbelaere, Feb 24 2023

cp's OEIS Frontend

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

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A360843 6-full numbers (A069493) sandwiched between twin primes.

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A076470 Perfect powers m^k where k >= 6.

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A258571 Smallest prime factors of 6-full numbers, a(1)=1.

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Haskell

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Formula

A360844 a(n) is the least k-full number that is sandwiched between twin primes.

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Extensions