A258599 -id:A258599 - cp's OEIS frontend

A001248 Squares of primes.

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

N. J. A. Sloane

Also 4, together with numbers n such that Sum_{d|n}(-1)^d = -A048272(n) = -3. - Benoit Cloitre, Apr 14 2002
Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - Farideh Firoozbakht, Feb 02 2005
Unique numbers having 3 divisors (1, their square root, themselves). - Alexandre Wajnberg, Jan 15 2006
Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - Lekraj Beedassy, Aug 17 2006
Subsequence of semiprimes A001358. - Lekraj Beedassy, Sep 06 2006
Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - Franz Vrabec, Sep 11 2008
Also numbers n such that phi(n) = n - sqrt(n). - Michel Lagneau, May 25 2012
For n > 1, n is the sum of numbers from A006254(n-1) to A168565(n-1). - Vicente Izquierdo Gomez, Dec 01 2012
A078898(a(n)) = 2. - Reinhard Zumkeller, Apr 06 2015
Let r(n) = (a(n) - 1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5. - Dimitris Valianatos, Feb 26 2019
Numbers k such that A051709(k) = 1. - Jianing Song, Jun 27 2021

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Marius Coman, On the special relation between the numbers of the form 505+ 1008k and the squares of primes, 2015.
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A000005, A000040, A001358, A001694, A002033, A005408, A005722, A006254.
Cf. A008864, A033273, A049001, A048272, A051709, A059956, A060800, A062799.
Cf. A078898, A082020, A085548, A087112, A095874, A168565, A192134, A211110.
Cf. A258599.
Subsequence of A000430, A001358, and of A190641.
Cf. A024450 (partial sums), A069482 (first differences).

a001248 n = a001248_list !! (n-1)
a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

[p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A001248:=n->ithprime(n)^2; seq(A001248(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)
Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)

forprime(p=2,1e3,print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011

A001248(n)=prime(n)^2  \\ M. F. Hasler, Sep 16 2012

from sympy import prime
def A001248(n): return prime(n)**2 # Chai Wah Wu, Aug 09 2024

[n^2 for n in prime_range(1,301)] # G. C. Greubel, May 02 2024

n such that A062799(n) = 2. - Benoit Cloitre, Apr 06 2002
A000005(a(n)^(k-1)) = A005408(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007
a(n) = A000040(n)^(3-1)=A000040(n)^2, where 3 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 3 or A002033(a(n)) = 2. - Juri-Stepan Gerasimov, Oct 10 2009
A033273(a(n)) = 3. - Juri-Stepan Gerasimov, Dec 07 2009
For n > 2: (a(n) + 17) mod 12 = 6. - Reinhard Zumkeller, May 12 2010
A192134(A095874(a(n))) = A005722(n) + 1. - Reinhard Zumkeller, Jun 26 2011
For n > 2: a(n) = 1 (mod 24). - Zak Seidov, Dec 07 2011
A211110(a(n)) = 2. - Reinhard Zumkeller, Apr 02 2012
a(n) = A087112(n,n). - Reinhard Zumkeller, Nov 25 2012
a(n) = prime(n)^2. - Jon E. Schoenfield, Mar 29 2015
Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - Daniel Suteu, Feb 06 2017
Sum_{n>=1} 1/a(n) = P(2) = 0.4522474200... (A085548). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 (A082020).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(2) = 6/Pi^2 (A059956). (End)

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(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 bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

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

Original entry on oeis.org

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

A258601 a(n) is the index m such that A036967(m) = prime(n)^4.

Original entry on oeis.org

2, 5, 10, 16, 28, 37, 55, 61, 80, 105, 113, 142, 163, 170, 190, 219, 249, 260, 286, 310, 318, 352, 374, 407, 448, 472, 482, 505, 511, 536, 614, 634, 672, 682, 740, 754, 783, 821, 842, 878, 916, 924, 984, 996, 1015, 1032, 1103, 1171, 1201, 1213, 1233, 1270, 1286, 1343, 1379

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

A036967(a(n)) = A030514(n) = prime(n)^4;
A036967(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258569(m) = prime(n):
A258569(a(n)) = A000040(n) and A258569(m) != A000040(n) for m < a(n).

			.   n |  p |  a(n) | A036967(a(n)) = A030514(n) = p^4
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            16
.   2 |  3 |     5 |            81
.   3 |  5 |    10 |           625
.   4 |  7 |    16 |          2401
.   5 | 11 |    28 |         14641
.   6 | 13 |    37 |         28561
.   7 | 17 |    55 |         83521
.   8 | 19 |    61 |        130321
.   9 | 23 |    80 |        279841
.  10 | 29 |   105 |        707281
.  11 | 31 |   113 |        923521
.  12 | 37 |   142 |       1874161
.  13 | 41 |   163 |       2825761
.  14 | 43 |   170 |       3418801
.  15 | 47 |   190 |       4879681
.  16 | 53 |   219 |       7890481
.  17 | 59 |   249 |      12117361
.  18 | 61 |   260 |      13845841
.  19 | 67 |   286 |      20151121
.  20 | 71 |   310 |      25411681
.  21 | 73 |   318 |      28398241
.  22 | 79 |   352 |      38950081
.  23 | 83 |   374 |      47458321
.  24 | 89 |   407 |      62742241
.  25 | 97 |   448 |      88529281

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

Cf. A258569, A000040, A030514, A036967, A258599, A258600, A258602, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258601 = (+ 1) . fromJust . (`elemIndex` a258569_list) . a000040

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^4,4) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258601(n):
    c, m = 0, prime(n)**4
    for u in range(1,integer_nthroot(m,7)[0]+1):
        if all(d<=1 for d in factorint(u).values()):
            for w in range(1,integer_nthroot(a:=m//u**7,6)[0]+1):
                if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                    for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                        if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                            c += integer_nthroot(z//y**5,4)[0]
    return c # Chai Wah Wu, Sep 10 2024

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

A258602 a(n) is the index m such that A069492(m) = prime(n)^5.

Original entry on oeis.org

2, 5, 12, 20, 37, 45, 68, 82, 106, 142, 154, 196, 219, 234, 260, 305, 342, 360, 407, 434, 451, 496, 528, 573, 635, 668, 681, 720, 737, 770, 885, 919, 966, 984, 1065, 1087, 1139, 1193, 1228, 1283, 1331, 1348, 1440, 1455, 1484, 1509, 1624, 1731, 1767, 1789

Offset: 1

Reinhard Zumkeller, Jun 06 2015

nonn

A069492(a(n)) = A050997(n) = prime(n)^5;
A069492(m) mod prime(n) > 0 for m < a(n);
also smallest number m such that A258570(m) = prime(n):
A258570(a(n)) = A000040(n) and A258570(m) != A000040(n) for m < a(n).

			.   n |  p |  a(n) | A069492(a(n)) = A050997(n) = p^5
. ----+----+-------+---------------------------------
.   1 |  2 |     2 |            32
.   2 |  3 |     5 |           243
.   3 |  5 |    12 |          3125
.   4 |  7 |    20 |         16807
.   5 | 11 |    37 |        161051
.   6 | 13 |    45 |        371293
.   7 | 17 |    68 |       1419857
.   8 | 19 |    82 |       2476099
.   9 | 23 |   106 |       6436343
.  10 | 29 |   142 |      20511149
.  11 | 31 |   154 |      28629151
.  12 | 37 |   196 |      69343957
.  13 | 41 |   219 |     115856201
.  14 | 43 |   234 |     147008443
.  15 | 47 |   260 |     229345007
.  16 | 53 |   305 |     418195493
.  17 | 59 |   342 |     714924299
.  18 | 61 |   360 |     844596301
.  19 | 67 |   407 |    1350125107
.  20 | 71 |   434 |    1804229351
.  21 | 73 |   451 |    2073071593
.  22 | 79 |   496 |    3077056399
.  23 | 83 |   528 |    3939040643
.  24 | 89 |   573 |    5584059449
.  25 | 97 |   635 |    8587340257  .

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

Cf. A258570, A000040, A050997, A069492, A258599, A258600, A258601, A258603.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a258602 = (+ 1) . fromJust . (`elemIndex` a258570_list) . a000040

\\ Gen(limit,k) defined in A036967.
a(n)=#Gen(prime(n)^5,5) \\ Andrew Howroyd, Sep 10 2024

from math import gcd
from sympy import prime, integer_nthroot, factorint
def A258602(n):
    c, m = 0, prime(n)**5
    for t in range(1,integer_nthroot(m,9)[0]+1):
        if all(d<=1 for d in factorint(t).values()):
            for u in range(1,integer_nthroot(s:=m//t**9,8)[0]+1):
                if gcd(t,u)==1 and all(d<=1 for d in factorint(u).values()):
                    for w in range(1,integer_nthroot(a:=s//u**8,7)[0]+1):
                        if gcd(u,w)==1 and gcd(t,w)==1 and all(d<=1 for d in factorint(w).values()):
                            for y in range(1,integer_nthroot(z:=a//w**7,6)[0]+1):
                                if gcd(w,y)==1 and gcd(u,y)==1 and gcd(t,y)==1 and all(d<=1 for d in factorint(y).values()):
                                    c += integer_nthroot(z//y**6,5)[0]
    return c # Chai Wah Wu, Sep 10 2024

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A001248 Squares of primes.

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A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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

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A258601 a(n) is the index m such that A036967(m) = prime(n)^4.

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A258602 a(n) is the index m such that A069492(m) = prime(n)^5.

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

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