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A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i). 3
1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 28800, 30375, 34560, 36000, 38880, 48600, 54675, 84375, 92160, 96000, 121500, 134456, 153600, 169344, 217728, 218700, 225000, 247808, 262440, 296352, 300000, 337500, 340736, 387072, 395136, 489888, 666792, 703125, 750141, 781250, 823543, 857304, 885735 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Similarly, 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 A272859 at least for the terms of A114129.

LINKS

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..6058 (terms < 10^18, first 100 terms from G. Coppoletta)

FORMULA

If N is a positive integer and N = Product_{i=1..k} (p_i)^e_i is its prime factorization, then N is in A272858 iff Product_{i=1..k} (1 + p_i) = Product_{i=1..k} (1 + e_i).

For a number with three different prime factors N = p1^e1 * p2^e2 * p3^e3, the defining condition can be expressed as: p1 + p2 + p3 + p1*p2 + p1*p3 + p2*p3 + p1*p2*p3 = e1 + e2 + e3 + e1*e2 + e1*e3 + e2*e3 + e1*e2*e3.

EXAMPLE

92160 is included because 92160 = 2^11 * 3^2 * 5 and (2+1)*(3+1)*(5+1) = (11+1)*(2+1)*(1+1).

MATHEMATICA

ok[n_] := Block[{p, e}, {p, e} = Transpose@ FactorInteger@ n; Times @@ (1+p) == Times @@ (1+e)]; Select[Range[10^6], ok] (* Giovanni Resta, May 08 2016 *)

PROG

(Sage) def d(n):

    v=factor(n)[:]

    d1=prod(1+v[j][0] for j in range(len(v)))

    d2=prod(1+v[j][1] for j in range(len(v)))

    return d1-d2

[k for k in (1..1000000) if d(k)==0]

(PARI) is(n)=my(f=factor(n)); prod(i=1, #f~, f[i, 1]+1)==prod(i=1, #f~, f[i, 2]) \\ Charles R Greathouse IV, Sep 08 2016

CROSSREFS

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272859.

Sequence in context: A227866 A180576 A158186 * A272818 A272859 A054412

Adjacent sequences:  A272855 A272856 A272857 * A272859 A272860 A272861

KEYWORD

nonn

AUTHOR

Giuseppe Coppoletta, May 08 2016

STATUS

approved

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Last modified July 22 05:47 EDT 2019. Contains 325213 sequences. (Running on oeis4.)