A272858 -id:A272858 - cp's OEIS frontend

A272818 Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization.

1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 30375, 36000, 48600, 51840, 54675, 69120, 84375, 121500, 134456, 169344, 174960, 192000, 225000, 240000, 247808, 337500, 340736, 395136, 435456, 451584, 703125, 750141, 781250, 787320, 823543, 857304, 885735

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

For p prime, p^p satisfy the condition, hence A051674 (and also A048102) is a subsequence. Moreover, if p and q are primes and i and j are positive integers, if p^i * q^j verify the condition, then the same is true for p^j * q^i. So 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. In addition, any number having no more than two distinct prime factors (apart their multiplicity) is a term iff it belongs also to A272858.

			885735 = 3^11 * 5 is included because (3+5) + 3*5 = (11+1) + 11*1.
2^10 * 3^6 * 19^2 is included because (2+3+19)+ 2*3*19 = (10+6+2)+ 10*6*2.

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..1413 (terms < 10^19, first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A122406, A272858, A272859.

Select[Range[10^6], Total@ First@ # + Times @@ First@ # == Total@ Last@ # + Times @@ Last@ # &@ Transpose@ FactorInteger@ # &] (* Michael De Vlieger, May 08 2016 *)

spp(v) = vecsum(v) + prod(k=1, #v, v[k]);
isok(n) = my(f = factor(n)); spp(f[,1]) == spp(f[,2]); \\ Michel Marcus, May 08 2016

def d(n):
    v = factor(n)
    d1 = sum(w[0] for w in v) + prod(w[0] for w in v)
    d2 = sum(w[1] for w in v) + prod(w[1] for w in v)
    return d1 == d2
[k for k in (1..10000) if d(k)]

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A272818 Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization.

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