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