A069492 -id:A069492 - cp's OEIS frontend

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

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

A360842 5-full numbers (A069492) sandwiched between twin primes.

Original entry on oeis.org

139968, 995328, 63700992, 4076863488, 17714700000, 82012500000, 98802571392, 174960000000, 445240556352, 641194278912, 889223142528, 1059917571072, 1594323000000, 1663012435968, 2348273369088, 3333709317312, 5717741400000, 16260080320512, 19144761127488, 28697814000000

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			139968 = 2^6 * 3^7 is a term since it is 5-full and 139967 and 139969 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..250 (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 A069492.

Subsequence of A113839, A360840 and A360841.

Select[6*Range[2*10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 4 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 4;

A258570 Smallest prime factors of 5-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, 2, 2, 5, 2, 7, 3, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 7, 2, 2, 2, 11, 3, 2, 2, 2, 2, 2, 2, 13, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 7, 2, 2, 2, 2, 2, 2, 2, 17, 2, 3, 2, 2, 11, 2, 5, 2, 2, 2, 2, 2, 3, 19, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

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

Cf. A069492, A020639, A258602, A258567, A258568, A258569, A258571.

Haskell
```
a258570 = a020639 . a069492
```

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 4, print1(f[1, 1], ", "))); } \\ Amiram Eldar, Sep 12 2024

a(n) = A020639(A069492(n)).

a(A258602(n)) = A000040(n) and a(m) != A000040(n) for m < A258602(n).

A358250 Numbers whose square has a number of divisors coprime to 210.

Original entry on oeis.org

1, 32, 64, 243, 256, 512, 729, 2048, 3125, 6561, 7776, 15552, 15625, 16384, 16807, 19683, 23328, 32768, 46656, 62208, 100000, 117649, 124416, 161051, 177147, 186624, 200000, 209952, 262144, 371293, 373248, 390625, 419904, 497664, 500000, 537824, 629856, 759375

Offset: 1

Michael De Vlieger, Dec 03 2022

nonn

210 is the product of the smallest 4 primes.
Numbers k such that gcd(d(k^2), 210) = 1, where d(k) is the number of divisors of k (A000005).
Also numbers with no exponents = 1 mod 3, 2 mod 5, or 3 mod 7; also numbers whose square has a number of divisors coprime to 105. - Charles R Greathouse IV, Dec 08 2022

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Subsequence of A069492 and hence of A036967, A036966, and A001694.
Subsequence of other sequences of numbers k such that gcd(d(k^2), m) = 1: A350014 (m=6), A354179 (m=30).
Cf. A000005, A000290, A008364.

With[{nn = 2^20}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], CoprimeQ[DivisorSigma[0, #^2], 210] &]]

is(n,f=factor(n))=if(n<32, return(n==1)); my(t=f[,2]%105, N=19200959813818273241621521446046); for(i=1,#t, if(bittest(N,t[i]), return(0))); 1 \\ Charles R Greathouse IV, Dec 08 2022

Sum_{n>=1} 1/a(n) = Product_{p prime} (Sum_{k=2..210, gcd(k-1,210)=1} p^(k/2))/(p^105-1) = 1.05981355805... . - Amiram Eldar, Dec 06 2022

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

A369306 The number of cubefree divisors d of n such that n/d is also cubefree.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 1, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 0, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 2, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 0, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 19 2024

The analogous sequence with squarefree divisors (the number of squarefree divisors d of n such that n/d is also squarefree) is abs(A007427(n)).

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

Cf. A000005, A004709, A005117, A007427, A069492, A073184.

Cf. A001620, A002117, A244115.

f[p_,e_] := Switch[e, 1, 2, 2, 3, 3, 2, 4, 1, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> [2, 3, 2, 1, 0][min(x, 5)], factor(n)[,2]));

Multiplicative with a(p) = 2, a(p^2) = 3, a(p^3) = 2, a(p^4) = 1, and a(p^e) = 0 for e >= 5.
a(n) >= 0, with equality if and only if n is a 5-full number (A069492) larger than 1.
a(n) = 1 if and only if n is the 4th power of a squarefree number (A005117).
a(n) <= A000005(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)^2/zeta(3*s)^2.
Sum_{k=1..n} a(k) ~ (n/zeta(3)^2) * (log(n) + 2*gamma - 1 - 6*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Python

Extensions

A360842 5-full numbers (A069492) sandwiched between twin primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A258570 Smallest prime factors of 5-full numbers, a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

PARI

Formula

A358250 Numbers whose square has a number of divisors coprime to 210.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A369306 The number of cubefree divisors d of n such that n/d is also cubefree.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula