A258599 a(n) is the index m such that A001694(m) = prime(n)^2.

2, 4, 6, 10, 16, 20, 28, 31, 39, 48, 51, 65, 71, 75, 84, 94, 107, 110, 120, 129, 133, 145, 152, 163, 180, 187, 191, 199, 202, 212, 238, 246, 258, 261, 282, 286, 297, 309, 319, 330, 342, 344, 366, 372, 377, 382, 407, 431, 440, 443, 450, 463, 468, 487, 498

Offset: 1

Author

Reinhard Zumkeller, Jun 06 2015

Keywords

• nonn

Examples

			.   n |  p |  a(n) | A001694(a(n)) = A001248(n) = p^2
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             4
.   2 |  3 |     4 |             9
.   3 |  5 |     6 |            25
.   4 |  7 |    10 |            49
.   5 | 11 |    16 |           121
.   6 | 13 |    20 |           169
.   7 | 17 |    28 |           289
.   8 | 19 |    31 |           361
.   9 | 23 |    39 |           529
.  10 | 29 |    48 |           841
.  11 | 31 |    51 |           961
.  12 | 37 |    65 |          1369
.  13 | 41 |    71 |          1681
.  14 | 43 |    75 |          1849
.  15 | 47 |    84 |          2209
.  16 | 53 |    94 |          2809
.  17 | 59 |   107 |          3481
.  18 | 61 |   110 |          3721
.  19 | 67 |   120 |          4489
.  20 | 71 |   129 |          5041
.  21 | 73 |   133 |          5329
.  22 | 79 |   145 |          6241
.  23 | 83 |   152 |          6889
.  24 | 89 |   163 |          7921
.  25 | 97 |   180 |          9409  .
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Haskell

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258599 = (+ 1) . fromJust . (`elemIndex` a258567_list) . a000040

Mathematica

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 *)

Python

from math import isqrt
from sympy import prime, integer_nthroot, factorint
def A258599(n):
    m = prime(n)**2
    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

Formula

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

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

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

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