A258571 -id:A258571 - cp's OEIS frontend

A258603 a(n) is the index m such that A069493(m) = prime(n)^6.

2, 6, 13, 22, 45, 58, 87, 102, 135, 181, 199, 252, 287, 306, 342, 401, 461, 479, 536, 583, 602, 665, 712, 776, 860, 911, 932, 975, 997, 1051, 1212, 1258, 1331, 1356, 1479, 1502, 1580, 1651, 1705, 1784, 1856, 1879, 2013, 2037, 2093, 2113, 2272, 2438, 2484, 2510

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

A069493(a(n)) = A030516(n) = prime(n)^6;
A069493(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258571(m) = prime(n):
A258571(a(n)) = A000040(n) and A258571(m) != A000040(n) for m < a(n).

			.   n |  p |  a(n) | A069493(a(n)) = A030516(n) = p^6
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            64
.   2 |  3 |     6 |           729
.   3 |  5 |    13 |         15625
.   4 |  7 |    22 |        117649
.   5 | 11 |    45 |       1771561
.   6 | 13 |    58 |       4826809
.   7 | 17 |    87 |      24137569
.   8 | 19 |   102 |      47045881
.   9 | 23 |   135 |     148035889
.  10 | 29 |   181 |     594823321
.  11 | 31 |   199 |     887503681
.  12 | 37 |   252 |    2565726409
.  13 | 41 |   287 |    4750104241
.  14 | 43 |   306 |    6321363049
.  15 | 47 |   342 |   10779215329
.  16 | 53 |   401 |   22164361129
.  17 | 59 |   461 |   42180533641
.  18 | 61 |   479 |   51520374361
.  19 | 67 |   536 |   90458382169
.  20 | 71 |   583 |  128100283921
.  21 | 73 |   602 |  151334226289
.  22 | 79 |   665 |  243087455521
.  23 | 83 |   712 |  326940373369
.  24 | 89 |   776 |  496981290961
.  25 | 97 |   860 |  832972004929  .

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A258571, A000040, A030516, A069493, A258599, A258600, A258601, A258602.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258603 = (+ 1) . fromJust . (`elemIndex` a258571_list) . a000040

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^6,6) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258603(n):
    c, m = 0, prime(n)**6
    for y1 in range(1,integer_nthroot(m,11)[0]+1):
        if all(d<=1 for d in factorint(y1).values()):
            for y2 in range(1,integer_nthroot(z2:=m//y1**11,10)[0]+1):
                if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
                    for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
                        if all(gcd(y3,x)==1 for x in (y1,y2)) and all(d<=1 for d in factorint(y3).values()):
                            for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
                                if all(gcd(y4,x)==1 for x in (y1,y2,y3)) and all(d<=1 for d in factorint(y4).values()):
                                    for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
                                        if all(gcd(y5,x)==1 for x in (y1,y2,y3,y4)) and all(d<=1 for d in factorint(y5).values()):
                                            c += integer_nthroot(z5//y5**7,6)[0]
    return c # Chai Wah Wu, Sep 10 2024

a(11) onwards corrected by Chai Wah Wu and Andrew Howroyd, Sep 10 2024

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

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).

cp's OEIS Frontend

A258603 a(n) is the index m such that A069493(m) = prime(n)^6.

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

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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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