%I #18 Sep 11 2024 00:34:25

%S 2,6,13,22,45,58,87,102,135,181,199,252,287,306,342,401,461,479,536,

%T 583,602,665,712,776,860,911,932,975,997,1051,1212,1258,1331,1356,

%U 1479,1502,1580,1651,1705,1784,1856,1879,2013,2037,2093,2113,2272,2438,2484,2510

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

%C A069493(a(n)) = A030516(n) = prime(n)^6;

%C A069493(m) mod prime(n) > 0 for m < a(n);

%C also smallest number m such that A258571(m) = prime(n):

%C A258571(a(n)) = A000040(n) and A258571(m) != A000040(n) for m < a(n).

%H Andrew Howroyd, <a href="/A258603/b258603.txt">Table of n, a(n) for n = 1..1000</a>

%e . n | p | a(n) | A069493(a(n)) = A030516(n) = p^6

%e . ----+----+-------+---------------------------------

%e . 1 | 2 | 2 | 64

%e . 2 | 3 | 6 | 729

%e . 3 | 5 | 13 | 15625

%e . 4 | 7 | 22 | 117649

%e . 5 | 11 | 45 | 1771561

%e . 6 | 13 | 58 | 4826809

%e . 7 | 17 | 87 | 24137569

%e . 8 | 19 | 102 | 47045881

%e . 9 | 23 | 135 | 148035889

%e . 10 | 29 | 181 | 594823321

%e . 11 | 31 | 199 | 887503681

%e . 12 | 37 | 252 | 2565726409

%e . 13 | 41 | 287 | 4750104241

%e . 14 | 43 | 306 | 6321363049

%e . 15 | 47 | 342 | 10779215329

%e . 16 | 53 | 401 | 22164361129

%e . 17 | 59 | 461 | 42180533641

%e . 18 | 61 | 479 | 51520374361

%e . 19 | 67 | 536 | 90458382169

%e . 20 | 71 | 583 | 128100283921

%e . 21 | 73 | 602 | 151334226289

%e . 22 | 79 | 665 | 243087455521

%e . 23 | 83 | 712 | 326940373369

%e . 24 | 89 | 776 | 496981290961

%e . 25 | 97 | 860 | 832972004929 .

%o (Haskell)

%o import Data.List (elemIndex); import Data.Maybe (fromJust)

%o a258603 = (+ 1) . fromJust . (`elemIndex` a258571_list) . a000040

%o (Python)

%o from math import gcd

%o from sympy import prime, integer_nthroot, factorint

%o def A258603(n):

%o c, m = 0, prime(n)**6

%o for y1 in range(1,integer_nthroot(m,11)[0]+1):

%o if all(d<=1 for d in factorint(y1).values()):

%o for y2 in range(1,integer_nthroot(z2:=m//y1**11,10)[0]+1):

%o if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):

%o for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):

%o if all(gcd(y3,x)==1 for x in (y1,y2)) and all(d<=1 for d in factorint(y3).values()):

%o for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):

%o if all(gcd(y4,x)==1 for x in (y1,y2,y3)) and all(d<=1 for d in factorint(y4).values()):

%o for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):

%o if all(gcd(y5,x)==1 for x in (y1,y2,y3,y4)) and all(d<=1 for d in factorint(y5).values()):

%o c += integer_nthroot(z5//y5**7,6)[0]

%o return c # _Chai Wah Wu_, Sep 10 2024

%o (PARI) \\ Gen(limit,k) defined in A036967.

%o a(n)=#Gen(prime(n)^6,6) \\ _Andrew Howroyd_, Sep 10 2024

%Y Cf. A258571, A000040, A030516, A069493, A258599, A258600, A258601, A258602.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Jun 06 2015

%E a(11) onwards corrected by _Chai Wah Wu_ and _Andrew Howroyd_, Sep 10 2024