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

%I A258603 #18 Sep 11 2024 00:34:25
%S A258603 2,6,13,22,45,58,87,102,135,181,199,252,287,306,342,401,461,479,536,
%T A258603 583,602,665,712,776,860,911,932,975,997,1051,1212,1258,1331,1356,
%U A258603 1479,1502,1580,1651,1705,1784,1856,1879,2013,2037,2093,2113,2272,2438,2484,2510
%N A258603 a(n) is the index m such that A069493(m) = prime(n)^6.
%C A258603 A069493(a(n)) = A030516(n) = prime(n)^6;
%C A258603 A069493(m) mod prime(n) > 0 for m < a(n);
%C A258603 also smallest number m such that A258571(m) = prime(n):
%C A258603 A258571(a(n)) = A000040(n) and A258571(m) != A000040(n) for m < a(n).
%H A258603 Andrew Howroyd, <a href="/A258603/b258603.txt">Table of n, a(n) for n = 1..1000</a>
%e A258603 .   n |  p |  a(n) | A069493(a(n)) = A030516(n) = p^6
%e A258603 . ----+----+-------+---------------------------------
%e A258603 .   1 |  2 |     2 |            64
%e A258603 .   2 |  3 |     6 |           729
%e A258603 .   3 |  5 |    13 |         15625
%e A258603 .   4 |  7 |    22 |        117649
%e A258603 .   5 | 11 |    45 |       1771561
%e A258603 .   6 | 13 |    58 |       4826809
%e A258603 .   7 | 17 |    87 |      24137569
%e A258603 .   8 | 19 |   102 |      47045881
%e A258603 .   9 | 23 |   135 |     148035889
%e A258603 .  10 | 29 |   181 |     594823321
%e A258603 .  11 | 31 |   199 |     887503681
%e A258603 .  12 | 37 |   252 |    2565726409
%e A258603 .  13 | 41 |   287 |    4750104241
%e A258603 .  14 | 43 |   306 |    6321363049
%e A258603 .  15 | 47 |   342 |   10779215329
%e A258603 .  16 | 53 |   401 |   22164361129
%e A258603 .  17 | 59 |   461 |   42180533641
%e A258603 .  18 | 61 |   479 |   51520374361
%e A258603 .  19 | 67 |   536 |   90458382169
%e A258603 .  20 | 71 |   583 |  128100283921
%e A258603 .  21 | 73 |   602 |  151334226289
%e A258603 .  22 | 79 |   665 |  243087455521
%e A258603 .  23 | 83 |   712 |  326940373369
%e A258603 .  24 | 89 |   776 |  496981290961
%e A258603 .  25 | 97 |   860 |  832972004929  .
%o A258603 (Haskell)
%o A258603 import Data.List (elemIndex); import Data.Maybe (fromJust)
%o A258603 a258603 = (+ 1) . fromJust . (`elemIndex` a258571_list) . a000040
%o A258603 (Python)
%o A258603 from math import gcd
%o A258603 from sympy import prime, integer_nthroot, factorint
%o A258603 def A258603(n):
%o A258603     c, m = 0, prime(n)**6
%o A258603     for y1 in range(1,integer_nthroot(m,11)[0]+1):
%o A258603         if all(d<=1 for d in factorint(y1).values()):
%o A258603             for y2 in range(1,integer_nthroot(z2:=m//y1**11,10)[0]+1):
%o A258603                 if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
%o A258603                     for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
%o A258603                         if all(gcd(y3,x)==1 for x in (y1,y2)) and all(d<=1 for d in factorint(y3).values()):
%o A258603                             for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
%o A258603                                 if all(gcd(y4,x)==1 for x in (y1,y2,y3)) and all(d<=1 for d in factorint(y4).values()):
%o A258603                                     for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
%o A258603                                         if all(gcd(y5,x)==1 for x in (y1,y2,y3,y4)) and all(d<=1 for d in factorint(y5).values()):
%o A258603                                             c += integer_nthroot(z5//y5**7,6)[0]
%o A258603     return c # _Chai Wah Wu_, Sep 10 2024
%o A258603 (PARI) \\ Gen(limit,k) defined in A036967.
%o A258603 a(n)=#Gen(prime(n)^6,6) \\ _Andrew Howroyd_, Sep 10 2024
%Y A258603 Cf. A258571, A000040, A030516, A069493, A258599, A258600, A258601, A258602.
%K A258603 nonn
%O A258603 1,1
%A A258603 _Reinhard Zumkeller_, Jun 06 2015
%E A258603 a(11) onwards corrected by _Chai Wah Wu_ and _Andrew Howroyd_, Sep 10 2024