A258601 - OEIS

%I #20 Sep 11 2024 00:33:18

%S 2,5,10,16,28,37,55,61,80,105,113,142,163,170,190,219,249,260,286,310,

%T 318,352,374,407,448,472,482,505,511,536,614,634,672,682,740,754,783,

%U 821,842,878,916,924,984,996,1015,1032,1103,1171,1201,1213,1233,1270,1286,1343,1379

%N a(n) is the index m such that A036967(m) = prime(n)^4.

%C A036967(a(n)) = A030514(n) = prime(n)^4;

%C A036967(m) mod prime(n) > 0 for m < a(n);

%C also smallest number m such that A258569(m) = prime(n):

%C A258569(a(n)) = A000040(n) and A258569(m) != A000040(n) for m < a(n).

%H Andrew Howroyd, <a href="/A258601/b258601.txt">Table of n, a(n) for n = 1..1000</a>

%e .   n |  p |  a(n) | A036967(a(n)) = A030514(n) = p^4

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

%e .   1 |  2 |     2 |            16

%e .   2 |  3 |     5 |            81

%e .   3 |  5 |    10 |           625

%e .   4 |  7 |    16 |          2401

%e .   5 | 11 |    28 |         14641

%e .   6 | 13 |    37 |         28561

%e .   7 | 17 |    55 |         83521

%e .   8 | 19 |    61 |        130321

%e .   9 | 23 |    80 |        279841

%e .  10 | 29 |   105 |        707281

%e .  11 | 31 |   113 |        923521

%e .  12 | 37 |   142 |       1874161

%e .  13 | 41 |   163 |       2825761

%e .  14 | 43 |   170 |       3418801

%e .  15 | 47 |   190 |       4879681

%e .  16 | 53 |   219 |       7890481

%e .  17 | 59 |   249 |      12117361

%e .  18 | 61 |   260 |      13845841

%e .  19 | 67 |   286 |      20151121

%e .  20 | 71 |   310 |      25411681

%e .  21 | 73 |   318 |      28398241

%e .  22 | 79 |   352 |      38950081

%e .  23 | 83 |   374 |      47458321

%e .  24 | 89 |   407 |      62742241

%e .  25 | 97 |   448 |      88529281

%o (Haskell)

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

%o a258601 = (+ 1) . fromJust . (`elemIndex` a258569_list) . a000040

%o (Python)

%o from math import gcd

%o from sympy import prime, integer_nthroot, factorint

%o def A258601(n):

%o     c, m = 0, prime(n)**4

%o     for u in range(1,integer_nthroot(m,7)[0]+1):

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

%o             for w in range(1,integer_nthroot(a:=m//u**7,6)[0]+1):

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

%o                     for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):

%o                         if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):

%o                             c += integer_nthroot(z//y**5,4)[0]

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

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

%o a(n)=#Gen(prime(n)^4,4) \\ _Andrew Howroyd_, Sep 10 2024

%Y Cf. A258569, A000040, A030514, A036967, A258599, A258600, A258602, 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