A258567 -id:A258567 - cp's OEIS frontend

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

Reinhard Zumkeller, Jun 06 2015

nonn

			.   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  .

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

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

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

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 *)

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

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).

A258568 Smallest prime factors of 3-full numbers.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 5, 2, 2, 3, 2, 7, 2, 2, 5, 2, 3, 2, 2, 2, 2, 11, 2, 2, 2, 2, 3, 13, 7, 2, 2, 5, 3, 2, 2, 2, 2, 17, 2, 2, 2, 2, 3, 19, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 23, 2, 11, 2, 5, 2, 2, 7, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 29, 2, 2, 2, 3, 13, 31, 3, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (corrected by Amiram Eldar)

Cf. A036966, A020639, A258600, A258567, A258569, A258570, A258571.

Haskell
```
a258568 = a020639 . a036966
```

Table[If[Min[(f = FactorInteger[n])[[;; , 2]]] > 2 || n == 1, f[[1, 1]], Nothing], {n, 1, 30000}] (* Amiram Eldar, Feb 07 2023 *)

a(n) = A020639(A036966(n)).

a(A258600(n)) = A000040(n) and a(m) != A000040(n) for m < A258600(n).

Missing term a(78) inserted by Amiram Eldar, Feb 07 2023

A258569 Smallest prime factors of 4-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 2, 2, 5, 3, 2, 2, 2, 3, 7, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 11, 2, 5, 2, 7, 3, 2, 2, 2, 13, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 5, 2, 2, 17, 2, 2, 2, 7, 2, 19, 2, 2, 3, 2, 2, 11, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 23, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (some terms corrected by Georg Fischer)

Cf. A036967, A020639, A258601, A258567, A258568, A258570, A258571.

Haskell
```
a258569 = a020639 . a036967
```

Reap[Sow[1]; Do[f = FactorInteger[k]; If[Min[f[[All, 2]]] >= 4, Sow[f[[1, 1]]]], {k, 2, 10^6}]][[2, 1]] (* Jean-François Alcover, Sep 29 2020 *)

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 3, print1(f[1, 1], ", ")));} \\ Amiram Eldar, Sep 09 2024

a(n) = A020639(A036967(n));

a(A258601(n)) = A000040(n) and a(m) != A000040(n) for m < A258601(n).

A258570 Smallest prime factors of 5-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, 2, 2, 5, 2, 7, 3, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 7, 2, 2, 2, 11, 3, 2, 2, 2, 2, 2, 2, 13, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 7, 2, 2, 2, 2, 2, 2, 2, 17, 2, 3, 2, 2, 11, 2, 5, 2, 2, 2, 2, 2, 3, 19, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (some terms corrected by Andrew Howroyd)

Cf. A069492, A020639, A258602, A258567, A258568, A258569, A258571.

Haskell
```
a258570 = a020639 . a069492
```

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 4, print1(f[1, 1], ", "))); } \\ Amiram Eldar, Sep 12 2024

a(n) = A020639(A069492(n)).

a(A258602(n)) = A000040(n) and a(m) != A000040(n) for m < A258602(n).

A258571 Smallest prime factors of 6-full numbers, a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 3, 2, 5, 2, 7, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 3, 2, 2, 7, 2, 2, 2, 2, 2, 2, 3, 2, 11, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 13, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 11, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000 (terms 1..191 from Reinhard Zumkeller)

Cf. A069493, A020639, A258603, A258567, A258568, A258569, A258570.

Haskell
```
a258571 = a020639 . a069493
```

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 5, print1(f[1, 1], ", "))); } \\ Amiram Eldar, Sep 12 2024

a(n) = A020639(A069493(n)).

a(A258603(n)) = A000040(n) and a(m) != A000040(n) for m < A258603(n).

cp's OEIS Frontend

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

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A258568 Smallest prime factors of 3-full numbers.

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A258569 Smallest prime factors of 4-full numbers, a(1)=1.

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A258570 Smallest prime factors of 5-full numbers, a(1)=1.

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A258571 Smallest prime factors of 6-full numbers, a(1)=1.

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