A258599 - OEIS

%I #17 Sep 10 2024 14:21:18

%S 2,4,6,10,16,20,28,31,39,48,51,65,71,75,84,94,107,110,120,129,133,145,

%T 152,163,180,187,191,199,202,212,238,246,258,261,282,286,297,309,319,

%U 330,342,344,366,372,377,382,407,431,440,443,450,463,468,487,498

%N a(n) is the index m such that A001694(m) = prime(n)^2.

%H Reinhard Zumkeller, <a href="/A258599/b258599.txt">Table of n, a(n) for n = 1..1000</a>

%F A001694(a(n)) = A001248(n) = prime(n)^2.

%F A001694(m) mod prime(n) > 0 for m < a(n).

%F Also smallest number m such that A258567(m) = prime(n):

%F A258567(a(n)) = A000040(n) and A258567(m) != A000040(n) for m < a(n).

%e .   n |  p |  a(n) | A001694(a(n)) = A001248(n) = p^2

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

%e .   1 |  2 |     2 |             4

%e .   2 |  3 |     4 |             9

%e .   3 |  5 |     6 |            25

%e .   4 |  7 |    10 |            49

%e .   5 | 11 |    16 |           121

%e .   6 | 13 |    20 |           169

%e .   7 | 17 |    28 |           289

%e .   8 | 19 |    31 |           361

%e .   9 | 23 |    39 |           529

%e .  10 | 29 |    48 |           841

%e .  11 | 31 |    51 |           961

%e .  12 | 37 |    65 |          1369

%e .  13 | 41 |    71 |          1681

%e .  14 | 43 |    75 |          1849

%e .  15 | 47 |    84 |          2209

%e .  16 | 53 |    94 |          2809

%e .  17 | 59 |   107 |          3481

%e .  18 | 61 |   110 |          3721

%e .  19 | 67 |   120 |          4489

%e .  20 | 71 |   129 |          5041

%e .  21 | 73 |   133 |          5329

%e .  22 | 79 |   145 |          6241

%e .  23 | 83 |   152 |          6889

%e .  24 | 89 |   163 |          7921

%e .  25 | 97 |   180 |          9409  .

%t With[{m = 60}, c = Select[Range[Prime[m]^2], Min[FactorInteger[#][[;; , 2]]] > 1 &]; 1 + Flatten[FirstPosition[c, #] & /@ (Prime[Range[m]]^2)]] (* _Amiram Eldar_, Feb 07 2023 *)

%o (Haskell)

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

%o a258599 = (+ 1) . fromJust . (`elemIndex` a258567_list) . a000040

%o (Python)

%o from math import isqrt

%o from sympy import prime, integer_nthroot, factorint

%o def A258599(n):

%o     m = prime(n)**2

%o     return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # _Chai Wah Wu_, Sep 10 2024

%Y Cf. A258567, A000040, A001248, A001694, A258600, A258601, A258602, A258603.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Jun 06 2015