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

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

• nonn

Comments

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).

Examples

			.   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  .
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A258570, A000040, A050997, A069492, A258599, A258600, A258601, A258603.

Programs

Haskell

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258602 = (+ 1) . fromJust . (`elemIndex` a258570_list) . a000040

PARI

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^5,5) \\ Andrew Howroyd, Sep 10 2024

Python

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

Extensions

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024