A299795 -id:A299795 - cp's OEIS frontend

A073904 Smallest multiple k*n of n having n divisors.

1, 2, 9, 8, 625, 12, 117649, 24, 36, 80, 25937424601, 60, 23298085122481, 448, 2025, 384, 48661191875666868481, 180, 104127350297911241532841, 240, 35721, 11264, 907846434775996175406740561329, 360, 10000, 53248, 26244, 1344

Offset: 1

Amarnath Murthy, Aug 18 2002

Smallest refactorable number, m, such that m=k*n has n divisors. - Robert G. Wilson v, Oct 31 2005

			Smallest multiple a(n)=k*n; a(1)=1*1, a(2)=1*2, a(3)=3*3, a(4)=2*4, a(5)=125*5, a(6)=2*6, ... having d(k*n)=n divisors; d(1)=1, d(2)=2, d(3^2)=3, d(2^3)=4, d(5^4)=5, d(2^2*3)=3*2=6, ...

Jon E. Schoenfield, Table of n, a(n) for n = 1..388 (first 115 terms from Carole Dubois)
Carole Dubois, semi-Logarithmic pin plot of A073904(n)

Cf. A076931, A050376, A005179, A037992, A050376, A111172, A299795.
Cf. A033950 (refactorable numbers, also known as tau numbers).
Cf. A110821 (SuperRefactorable numbers).

f[n_] := Block[{k = 1}, If[ PrimeQ[n], n^(n - 1), While[d = DivisorSigma[0, k*n]; d != n, k++ ]; k*n]]; Table[ f[n], {n, 28}] (* Robert G. Wilson v *)

If p is a prime then a(p) = p^(p-1). If n = p^2 then a(n) = 2^(p-1)*p^(p-1).
a(p^r) = (2*3*5*...*p_r)^(p-1) for r < p <= p_r. a(p^r) = (2*3*...*p_(r-1))^(p-1)*p^(p-1) for p > p_r. Else a(p^r) = ...? for r >= p. Problem a(2^r) = ...? Cf. A005179(p^n)=(2*3*...*p_n)^(p-1) for p_n < 2^p. - Thomas Ordowski, Aug 20 2005
a(p^r) = (2*3...*p_(r-1)*p)^(p-1) for p > p_r; else a(p^r) = (2*3...*p...*p_m)^(p-1)*p^(p^k-p) for p <= p_r and p_m < 2^p, where m=r-k+1 for smallest k such that p^k > r, so k=floor(log(r)/log(p))+1 and p > log(p_m)/log(2). Examples: If k=1 then a(p^r) = (2*3*...*p_r)^(p-1) for r < p <= p_r. If p=2 then a(2^r) = (2*3*...*p_m)*2^(2^k-2) for r < 5. For instance, let r=4 so k=3, m=2 and a(2^4)=384. - Thomas Ordowski, Aug 22 2005
If p is a prime and n=p^r then a(p^r) = (s_1*s_2*...*s_r)^(p-1) where (s_r) is a permutation of the (ascending sequence) numbers of the form q^(p^j) for every prime q and j>=0; permutation such that s_(p^j)=p^(p^j) and shifted remainder. For example, if p=3 then (s_r): 3, 2, 3^3, 5, 7, 2^3, 11, 13, 3^9, 17, 19, ... so a(3^r) = (3*2*27*5*...*s_r)^2. - Thomas Ordowski, Aug 29 2005
If n=2^r then a(2^r) is the product of the first r members of the A109429 sequence. - Thomas Ordowski, Aug 29 2005
a(n) = n * A076931(n). - Thomas Ordowski, Oct 07 2005
a(4) = 8; a(2*prime(n)) = A299795(n), for n>1. - Bernard Schott, Nov 06 2022

a(12) corrected by Thomas Ordowski, Aug 18 2005
Further corrections from Thomas Ordowski, Oct 07 2005
a(21), a(27) & a(28) from Robert G. Wilson v, Oct 31 2005

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Peter Luschny, Mar 03 2018

nonn

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.

An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

Peter Luschny, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Indices of the nonzero values of A300333.

Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

A371075 Fixed points of A374941.

Original entry on oeis.org

12, 30, 80, 324, 448, 888, 11264, 53248, 1114112, 3684352, 4980736, 18055168, 96468992

Offset: 1

Tanmaya Mohanty, Mar 10 2024

			12 is a term because A374941(12) = 12.

Cf. A000396, A005101, A299795, A374941.

f(n) = my(d=divisors(n)); sum(i=2, #d-1, if (isprime(d[i]), d[i], f(d[i])));
isok(k) = f(k) == k; \\ Michel Marcus, Mar 10 2024

from sympy import divisors, isprime
from functools import cache
@cache
def f(n): return sum(di if isprime(di) else f(di) for di in divisors(n)[1:-1])
def ok(n): return n == f(n)
print([k for k in range(1, 1000) if ok(k)]) # Michael S. Branicky, Mar 31 2024 after Michel Marcus

Equals the ordered set {k: A374941(k) = k}.

a(13) from Michael S. Branicky, Mar 31 2024

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A073904 Smallest multiple k*n of n having n divisors.

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A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

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A371075 Fixed points of A374941.

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