login
A376294
The product of n's prime powers, with the base and exponent concatenated.
1
1, 21, 31, 22, 51, 651, 71, 23, 32, 1071, 111, 682, 131, 1491, 1581, 24, 171, 672, 191, 1122, 2201, 2331, 231, 713, 52, 2751, 33, 1562, 291, 33201, 311, 25, 3441, 3591, 3621, 704, 371, 4011, 4061, 1173, 411, 46221, 431, 2442, 1632, 4851, 471, 744, 72, 1092, 5301
OFFSET
1,2
COMMENTS
Multiplicative with a(p^e) = the concatenation of p and e.
A prime which occurs just once in the factorization of n is taken as exponent 1 so that for instance n = 7 = 7^1 becomes 71.
It is unknown whether there are any fixed points a(n) = n beyond n=1.
FORMULA
For primes p, a(p) = 10p + 1.
EXAMPLE
For n=5, a(5) = 51 since 5=5^1.
For n=20, a(20) = 22*51 = 1122 since 20=2^2*5^1.
MAPLE
a:= n-> mul(parse(cat(i[])), i=ifactors(n)[2]):
seq(a(n), n=1..51); # Alois P. Heinz, Sep 19 2024
MATHEMATICA
f[p_, e_] := 10^IntegerLength[e] * p + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2024 *)
PROG
(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([b, e]=f[i, ]); b*10^(1+logint(e, 10))+e)} \\ Andrew Howroyd, Sep 19 2024
(Python)
from math import prod
from sympy import factorint
def a(n): return prod(int(str(p)+str(e)) for p, e in factorint(n).items())
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Sep 27 2024
CROSSREFS
Cf. A376254 (indices of a(n) < n), A287483 (indices of local maxima).
Sequence in context: A116116 A079394 A112375 * A067599 A342838 A261322
KEYWORD
nonn,base,mult,new
AUTHOR
Haines Hoag, Sep 19 2024
STATUS
approved