A048102 -id:A048102 - cp's OEIS frontend

A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6144, 6272, 10976, 12500, 21600, 30375, 36000, 48600, 54675, 77760, 84375, 114688, 116640, 121500, 134456, 138240, 169344, 173056, 225000, 229376, 247808, 337500, 354294, 384000, 395136, 600000, 653184, 655360, 703125, 750141, 823543, 857304, 913952, 979776

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Moreover, due to the multiplicativity of the arithmetic function sigma, A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e., any permutation of its prime signature) gives also a term.

The condition defining this sequence coincides with the condition in A272858 at least for the terms of A114129.

			173056 is included because 173056 = 2^10 * 13^2 and sigma(2*13) = sigma(10*2).
653184 is included because 653184 = 2^7 * 3^6 * 7 and sigma(2*3*7) = sigma(7*6*1).

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272858.

Select[Range[10^6], First@ # == Last@ # &@ Map[DivisorSigma[1, Times @@ #] &, Transpose@ FactorInteger@ #] &] (* Michael De Vlieger, May 12 2016 *)

A272859 = []
for n in (1..10000):
    v = factor(n)
    if prod(1 + w[0] for w in v) == sigma(prod(w[1] for w in v)): A272859.append(n)
print(A272859)

A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 16, 27, 100, 121, 18, 169, 196, 225, 8, 289, 108, 361, 50, 441, 484, 529, 144, 125, 676, 3, 98, 841, 900, 961, 64, 1089, 1156, 1225, 54, 1369, 1444, 1521, 400, 1681, 1764, 1849, 242, 675, 2116, 2209, 72, 343, 500, 2601, 338, 2809, 12, 3025

Offset: 1

Amiram Eldar, Nov 17 2023

A permutation of the positive integers. 1 is the only fixed point.

a(n) is a powerful number (A001694) if and only if n is not in A100717.

Cf. A000040, A001694, A005117, A007947, A048102, A072873, A100717, A103889, A130508, A342090.

Similar sequences: A011262, A011264.

f[p_, e_] := p^(e + 1 + If[Mod[e, p] == 0, -p, 0]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + 1 + if(!(f[i,2]%f[i,1]), -f[i,1])));}

a(2^e) = 2^A103889(e).
a(3^e) = 3^A130508(e).
A007947(a(n)) = A007947(n).
a(A051674(n)) = A000040(n).
a(n) is squarefree (A005117) if and only if n is in A048102.
a(A048102(n)) = A007947(A048102(n)).
a(n) == 0 (mod n) if and only if n is not in A342090.
a(n) | n if and only if n is in A072873.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 1/p + 1/(1 + p) - (p-1)/(p^p * (1 + p^p))) = 0.660264348361... .

A113853 Numbers whose prime factors are raised to the powers of themselves.

Original entry on oeis.org

108, 12500, 84375, 337500, 3294172, 22235661, 88942644, 2573571875, 10294287500, 69486440625, 277945762500, 1141246682444, 7703415106497, 30813660425988, 891598970659375, 1211500426369012, 3566395882637500, 8177627877990831, 24073172207803125, 32710511511963324

Offset: 1

Cino Hilliard, Jan 25 2006

Does not include A000312: Number of labeled mappings from n points to themselves (endofunctions): n^n.

			108 = 2^2*3^3. 2 is raised to the power of itself and 3 is raised to the power of itself.

Cf. A000312, A048102, A051674, A094289.

p = Drop[Subsets@Prime@Range@7, 8]; Take[ Sort[Times @@@ (p^p)], 18] (* Robert G. Wilson v, Jan 26 2006 *)

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^p) - Sum_{p prime} 1/p^p - 1 = 0.009354434361... - Amiram Eldar, Oct 13 2020

Corrected and extended by Robert G. Wilson v, Jan 26 2006

Offset corrected and more terms added by Amiram Eldar, Oct 13 2020

A276372 Numbers n such that, in the prime factorization of n, the list of the exponents is a rotation of the list of the prime factors.

Original entry on oeis.org

1, 4, 27, 72, 108, 800, 3125, 6272, 12500, 30375, 36000, 48600, 84375, 247808, 337500, 395136, 750141, 823543, 857304, 1384448, 3294172, 22235661, 24532992, 37879808, 53782400, 88942644, 122500000, 161980416, 171478296, 189267968, 235782657, 600112800, 1313046875, 2155524696

Offset: 1

Robert C. Lyons, Aug 31 2016

nonn

			4 is in the sequence because the prime factorization of 4 is 2^2, and the list of exponents (i.e., [2]) is a rotation of the list of prime factors (i.e., [2]).
36000 is in the sequence because the prime factorization of 36000 is 2^5 * 3^2 * 5^3, and the list of exponents (i.e., [5, 2, 3]) is a rotation of the list of prime factors (i.e., [2, 3, 5]).
84 is not in the sequence because the prime factorization of 84 is 2^2 * 3^1 * 7^1, and the list of exponents (i.e., [2, 1, 1]) is not a rotation of the list of prime factors (i.e., [2, 3, 7]).
21600 is not in the sequence because the prime factorization of 21600 is 2^5 * 3^3 * 5^2, and the list of exponents (i.e., [5, 3, 2]) is not a rotation of the list of prime factors (i.e., [2, 3, 5]).

Subsequence of A122406 and of A056166. A048102 is a subsequence.

Cf. A008475, A008478, A054411, A054412, A082949, A113855.

Select[Range[10^6], Function[w, Total@ Boole@ Map[First@ w == # &, RotateLeft[Last@ w, #] & /@ Range[Length@ Last@ w]] > 0]@ Transpose@ FactorInteger@ # &] (* Michael De Vlieger, Sep 01 2016 *)

def in_seq( n ):
    if n == 1: return True
    pf = list( factor( n ) )
    primes    = [ t[0] for t in pf ]
    exponents = [ t[1] for t in pf ]
    if primes[0] in exponents:
        i = exponents.index(primes[0])
        exp_rotated = exponents[i : ] + exponents[0 : i]
        return primes == exp_rotated
    else:
        return False
print([n for n in range(1, 10000000) if in_seq(n)])

# Much faster program that generates the solutions rather than searching for them.
from sage.misc.misc import powerset
primes = primes_first_n(9)
max_prime = primes[-1]
solutions = set([1])
max_solution = min(2^max_prime * max_prime^2, max_prime^max_prime)
for subset in powerset(primes):
    subset_list = list(subset)
    for i in range(0, len(subset_list)):
        exponents = subset_list[i : ] + subset_list[0 : i]
        product = 1
        for j in range(0, len(subset_list)):
            product = product * subset_list[j]^exponents[j]
        if product <= max_solution:
            solutions.add(product)
print(sorted(solutions))

cp's OEIS Frontend

A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

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A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.

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A113853 Numbers whose prime factors are raised to the powers of themselves.

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A276372 Numbers n such that, in the prime factorization of n, the list of the exponents is a rotation of the list of the prime factors.

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Mathematica

Sage

Sage