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A272818 Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(Sage) def d(n):

    v=factor(n)[:]; L=len(v)

    d1=sum(v[j][0] for j in range(L))+prod(v[j][0] for j in range(L))

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

    return d1-d2

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

(PARI) 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

CROSSREFS

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

Sequence in context: A180576 A158186 A272858 * A272859 A054412 A122405

Adjacent sequences:  A272815 A272816 A272817 * A272819 A272820 A272821

KEYWORD

nonn

AUTHOR

Giuseppe Coppoletta, May 08 2016

STATUS

approved

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Last modified July 19 12:32 EDT 2019. Contains 325159 sequences. (Running on oeis4.)