A258602 -id:A258602 - cp's OEIS frontend

A050997 Fifth powers of primes.

32, 243, 3125, 16807, 161051, 371293, 1419857, 2476099, 6436343, 20511149, 28629151, 69343957, 115856201, 147008443, 229345007, 418195493, 714924299, 844596301, 1350125107, 1804229351, 2073071593, 3077056399, 3939040643, 5584059449, 8587340257, 10510100501

Offset: 1

Eric W. Weisstein

Numbers k such that A062799(k) = 5.

Let r(n) = (a(n)+1)/(a(n)-1) if a(n) mod 4 = 3, (a(n)-1)/(a(n)+1) otherwise; then Product_{n>=1} r(n) = (31/33) * (244/242) * (3124/3126) * (16808/16806) * ... = 246016/259875. - Dimitris Valianatos, Mar 09 2020

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, MathWorld: Prime Power.
Index to sequences related to prime signature

Cf. A000040, A001248, A030078, A030514, A085965, A131992, A131993, A013663, A157291.

Cf. A258602.

a050997 = (^ 5) . a000040
a050997_list = map (^ 5) a000040_list
-- Reinhard Zumkeller, Jun 03 2015

[p^5: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Array[Prime[ # ]^5 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

PARI
```
vector(66,n,prime(n)^5)
```

A056595(a(n)) = 3. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(5) = 0.0357550174... (A085965). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(5)/zeta(10) (A157291).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(5) = 1/A013663. (End)

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).

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

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

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).

cp's OEIS Frontend

A050997 Fifth powers of primes.

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

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

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

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

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