A258568 -id:A258568 - cp's OEIS frontend

A258600 a(n) is the index m such that A036966(m) = prime(n)^3.

2, 4, 8, 13, 23, 29, 39, 45, 57, 75, 81, 99, 110, 117, 130, 149, 169, 176, 197, 209, 212, 236, 250, 270, 295, 309, 317, 328, 337, 354, 399, 414, 436, 445, 477, 483, 506, 529, 541, 563, 585, 591, 631, 635, 654, 657, 697, 747, 758, 765, 781, 803, 809, 845, 864

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

			.   n |  p |  a(n) | A036966(a(n)) = A030078(n) = p^3
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |             8
.   2 |  3 |     4 |            27
.   3 |  5 |     8 |           125
.   4 |  7 |    13 |           343
.   5 | 11 |    23 |          1331
.   6 | 13 |    29 |          2197
.   7 | 17 |    39 |          4913
.   8 | 19 |    45 |          6859
.   9 | 23 |    57 |         12167
.  10 | 29 |    75 |         24389
.  11 | 31 |    81 |         29791
.  12 | 37 |    99 |         50653
.  13 | 41 |   110 |         68921
.  14 | 43 |   117 |         79507
.  15 | 47 |   130 |        103823
.  16 | 53 |   149 |        148877
.  17 | 59 |   169 |        205379
.  18 | 61 |   176 |        226981
.  19 | 67 |   197 |        300763
.  20 | 71 |   209 |        357911
.  21 | 73 |   212 |        389017
.  22 | 79 |   236 |        493039
.  23 | 83 |   250 |        571787
.  24 | 89 |   270 |        704969
.  25 | 97 |   295 |        912673  .

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A258568, A000040, A030078, A036966, A258599, A258601, A258602, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258600 = (+ 1) . fromJust . (`elemIndex` a258568_list) . a000040

With[{m = 60}, c = Select[Range[Prime[m]^3], Min[FactorInteger[#][[;; , 2]]] > 2 &]; 1 + Flatten[FirstPosition[c, #] & /@ (Prime[Range[m]]^3)]] (* Amiram Eldar, Feb 07 2023 *)

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258600(n):
    c, m = 0, prime(n)**3
    for w in range(1,integer_nthroot(m,5)[0]+1):
        if all(d<=1 for d in factorint(w).values()):
            for y in range(1,integer_nthroot(z:=m//w**5,4)[0]+1):
                if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                    c += integer_nthroot(z//y**4,3)[0]
    return c # Chai Wah Wu, Sep 10 2024

A036966(a(n)) = A030078(n) = prime(n)^3.
A036966(m) mod prime(n) > 0 for m < a(n).
Also smallest number m such that A258568(m) = prime(n):
A258568(a(n)) = A000040(n) and A258568(m) != A000040(n) for m < a(n).

a(11)-a(55) and example corrected by Amiram Eldar, Feb 07 2023

A258567 a(1) = 1; thereafter a(n) = smallest prime factor of the powerful number A001694(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 2, 7, 2, 2, 3, 2, 2, 11, 5, 2, 2, 13, 2, 2, 2, 3, 3, 2, 2, 17, 2, 7, 19, 2, 2, 2, 3, 2, 2, 2, 23, 2, 5, 2, 3, 2, 3, 2, 2, 29, 2, 2, 31, 2, 2, 2, 2, 3, 3, 2, 2, 5, 2, 3, 11, 2, 37, 2, 2, 3, 2, 2, 41, 2, 2, 2, 43, 2, 2, 2, 3, 2, 2, 3

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001694, A020639, A258599, A258568, A258569, A258570, A258571.

Haskell
```
a258567 = a020639 . a001694
```

Table[If[Min[(f = FactorInteger[n])[[;; , 2]]] > 1 || n == 1, f[[1, 1]], Nothing], {n, 1, 3000}] (* Amiram Eldar, Jan 30 2023 *)

from math import isqrt
from sympy import mobius, integer_nthroot, primefactors
def A258567(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2, 3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return min(primefactors(bisection(f,n,n)),default=1) # Chai Wah Wu, Sep 10 2024

a(n) = A020639(A001694(n)).

a(A258599(n)) = A000040(n) and a(m) != A000040(n) for m < A258599(n).

Definition made more precise by N. J. A. Sloane, Apr 29 2024

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

A258600 a(n) is the index m such that A036966(m) = prime(n)^3.

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A258567 a(1) = 1; thereafter a(n) = smallest prime factor of the powerful number A001694(n).

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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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