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

%I A258602 #19 Sep 11 2024 00:33:57
%S A258602 2,5,12,20,37,45,68,82,106,142,154,196,219,234,260,305,342,360,407,
%T A258602 434,451,496,528,573,635,668,681,720,737,770,885,919,966,984,1065,
%U A258602 1087,1139,1193,1228,1283,1331,1348,1440,1455,1484,1509,1624,1731,1767,1789
%N A258602 a(n) is the index m such that A069492(m) = prime(n)^5.
%C A258602 A069492(a(n)) = A050997(n) = prime(n)^5;
%C A258602 A069492(m) mod prime(n) > 0 for m < a(n);
%C A258602 also smallest number m such that A258570(m) = prime(n):
%C A258602 A258570(a(n)) = A000040(n) and A258570(m) != A000040(n) for m < a(n).
%H A258602 Andrew Howroyd, <a href="/A258602/b258602.txt">Table of n, a(n) for n = 1..1000</a>
%e A258602 .   n |  p |  a(n) | A069492(a(n)) = A050997(n) = p^5
%e A258602 . ----+----+-------+---------------------------------
%e A258602 .   1 |  2 |     2 |            32
%e A258602 .   2 |  3 |     5 |           243
%e A258602 .   3 |  5 |    12 |          3125
%e A258602 .   4 |  7 |    20 |         16807
%e A258602 .   5 | 11 |    37 |        161051
%e A258602 .   6 | 13 |    45 |        371293
%e A258602 .   7 | 17 |    68 |       1419857
%e A258602 .   8 | 19 |    82 |       2476099
%e A258602 .   9 | 23 |   106 |       6436343
%e A258602 .  10 | 29 |   142 |      20511149
%e A258602 .  11 | 31 |   154 |      28629151
%e A258602 .  12 | 37 |   196 |      69343957
%e A258602 .  13 | 41 |   219 |     115856201
%e A258602 .  14 | 43 |   234 |     147008443
%e A258602 .  15 | 47 |   260 |     229345007
%e A258602 .  16 | 53 |   305 |     418195493
%e A258602 .  17 | 59 |   342 |     714924299
%e A258602 .  18 | 61 |   360 |     844596301
%e A258602 .  19 | 67 |   407 |    1350125107
%e A258602 .  20 | 71 |   434 |    1804229351
%e A258602 .  21 | 73 |   451 |    2073071593
%e A258602 .  22 | 79 |   496 |    3077056399
%e A258602 .  23 | 83 |   528 |    3939040643
%e A258602 .  24 | 89 |   573 |    5584059449
%e A258602 .  25 | 97 |   635 |    8587340257  .
%o A258602 (Haskell)
%o A258602 import Data.List (elemIndex); import Data.Maybe (fromJust)
%o A258602 a258602 = (+ 1) . fromJust . (`elemIndex` a258570_list) . a000040
%o A258602 (Python)
%o A258602 from math import gcd
%o A258602 from sympy import prime, integer_nthroot, factorint
%o A258602 def A258602(n):
%o A258602     c, m = 0, prime(n)**5
%o A258602     for t in range(1,integer_nthroot(m,9)[0]+1):
%o A258602         if all(d<=1 for d in factorint(t).values()):
%o A258602             for u in range(1,integer_nthroot(s:=m//t**9,8)[0]+1):
%o A258602                 if gcd(t,u)==1 and all(d<=1 for d in factorint(u).values()):
%o A258602                     for w in range(1,integer_nthroot(a:=s//u**8,7)[0]+1):
%o A258602                         if gcd(u,w)==1 and gcd(t,w)==1 and all(d<=1 for d in factorint(w).values()):
%o A258602                             for y in range(1,integer_nthroot(z:=a//w**7,6)[0]+1):
%o A258602                                 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()):
%o A258602                                     c += integer_nthroot(z//y**6,5)[0]
%o A258602     return c # _Chai Wah Wu_, Sep 10 2024
%o A258602 (PARI) \\ Gen(limit,k) defined in A036967.
%o A258602 a(n)=#Gen(prime(n)^5,5) \\ _Andrew Howroyd_, Sep 10 2024
%Y A258602 Cf. A258570, A000040, A050997, A069492, A258599, A258600, A258601, A258603.
%K A258602 nonn
%O A258602 1,1
%A A258602 _Reinhard Zumkeller_, Jun 06 2015
%E A258602 a(11) onwards corrected by _Chai Wah Wu_ and _Andrew Howroyd_, Sep 10 2024