A258601 -id:A258601 - cp's OEIS frontend

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

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

A036967 4-full numbers: if a prime p divides k then so does p^4.

Original entry on oeis.org

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 2592, 3125, 3888, 4096, 5184, 6561, 7776, 8192, 10000, 10368, 11664, 14641, 15552, 15625, 16384, 16807, 19683, 20000, 20736, 23328, 28561, 31104, 32768, 34992

Offset: 1

N. J. A. Sloane

a(m) mod prime(n) > 0 for m < A258601(n); a(A258601(n)) = A030514(n) = prime(n)^4. - Reinhard Zumkeller, Jun 06 2015

E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 300 terms from T. D. Noe)

A030514 is a subsequence.

Cf. A001694, A036966, A046101, A258601.

import Data.Set (singleton, deleteFindMin, fromList, union)
a036967 n = a036967_list !! (n-1)
a036967_list = 1 : f (singleton z) [1, z] zs where
   f s q4s p4s'@(p4:p4s)
     | m < p4 = m : f (union (fromList $ map (* m) ps) s') q4s p4s'
     | otherwise = f (union (fromList $ map (* p4) q4s) s) (p4:q4s) p4s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030514_list
-- Reinhard Zumkeller, Jun 03 2015

Join[{1},Select[Range[35000],Min[Transpose[FactorInteger[#]][[2]]]>3&]] (* Harvey P. Dale, Jun 05 2012 *)

is(n)=n==1 || vecmin(factor(n)[,2])>3 \\ Charles R Greathouse IV, Sep 17 2015

M(v,u,lim)=vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))
Gen(lim,k)={my(v=[1]); forprime(p=2, sqrtnint(lim, k), v=M(v, concat([1], vector(logint(lim,p)-k+1,e,p^(e+k-1))), lim));v}
Gen(35000,4) \\ Andrew Howroyd, Sep 10 2024

from sympy import factorint
A036967_list = [n for n in range(1,10**5) if min(factorint(n).values(),default=4) >= 4] # Chai Wah Wu, Aug 18 2021

from math import gcd
from sympy import integer_nthroot, factorint
def A036967(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 u in range(1,integer_nthroot(x,7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1,integer_nthroot(a:=x//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
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.1488462139214317030108176090790939019972506733993367867997411290952527... - Amiram Eldar, Jul 09 2020

More terms from Erich Friedman

Corrected by Vladeta Jovovic, Aug 17 2002

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

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).

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

A258569 Smallest prime factors of 4-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 2, 2, 5, 3, 2, 2, 2, 3, 7, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 11, 2, 5, 2, 7, 3, 2, 2, 2, 13, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 5, 2, 2, 17, 2, 2, 2, 7, 2, 19, 2, 2, 3, 2, 2, 11, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 23, 2, 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 Georg Fischer)

Cf. A036967, A020639, A258601, A258567, A258568, A258570, A258571.

Haskell
```
a258569 = a020639 . a036967
```

Reap[Sow[1]; Do[f = FactorInteger[k]; If[Min[f[[All, 2]]] >= 4, Sow[f[[1, 1]]]], {k, 2, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 29 2020 *)

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 3, print1(f[1, 1], ", ")));} \\ Amiram Eldar, Sep 09 2024

a(n) = A020639(A036967(n));

a(A258601(n)) = A000040(n) and a(m) != A000040(n) for m < A258601(n).

cp's OEIS Frontend

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

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A036967 4-full numbers: if a prime p divides k then so does p^4.

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

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

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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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