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Numbers k such that A376294(k) < k.
2

%I #39 Oct 26 2024 22:58:30

%S 32,64,81,121,125,128,169,243,256,289,343,361,512,529,625,729,841,864,

%T 961,972,1024,1152,1250,1296,1331,1369,1458,1536,1600,1681,1728,1849,

%U 1875,1944,2000,2025,2048,2187,2197,2209,2304,2401,2500,2560,2592,2662,2744,2809,2916,3087,3125,3136

%N Numbers k such that A376294(k) < k.

%C There are infinitely many numbers in this sequence, since the growth of powers of small primes far outpaces the growth of their digits when concatenated.

%C First differs from A195330 at 320 which is a term there but not here.

%H Haines Hoag, <a href="/A376254/b376254.txt">Table of n, a(n) for n = 1..20000</a>

%e 32 is a term since 32=2^5 and 25<32.

%e 1152 is a term since 1152=2^7*3^2 and 27*32=864, and 864<1152.

%t f[p_, e_] := 10^IntegerLength[e]*p + e; q[1] = False; q[k_] := Times @@ f @@@ FactorInteger[k] < k; Select[Range[3200], q] (* _Amiram Eldar_, Sep 26 2024 *)

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def ok(n): return prod(int(str(p)+str(e)) for p, e in factorint(n).items()) < n

%o print([k for k in range(1, 3200) if ok(k)]) # _Michael S. Branicky_, Sep 27 2024

%Y Cf. A376294, A195330.

%K nonn,base

%O 1,1

%A _Haines Hoag_, Sep 17 2024