Find the number of 4 digit numbers so that when decomposed into prime factors, have the sum of prime factors equal to the sum of the exponents?


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Question Number 211250 by RojaTaniya last updated on 01/Sep/24

$F i n d t h e n u m b e r o f 4 d i g i t n u m b e r s$ $s o t h a t w h e n d e c o m p o s e d i n t o p r i m e$ $f a c t o r s, h a v e t h e s u m o f p r i m e f a c t o r s$ $e q u a l t o t h e s u m o f t h e e x p o n e n t s ?$
	Answered by mr W last updated on 03/Sep/24

	$s a y t h e n u m b e r i s N = p^{a} q^{b} r^{c} . . .$ $1000 ⩽ N ⩽ 9999$ $p, q, r, . . . \in P a n d p < q < r < . . .$ $a, b, c, . . . \in N$ $p + q + r + . . . = a + b + c + . . .$ ${(p^{a} q^{b} r^{c})}_{m i n} = p^{p + q + r - 2} q^{1} r^{1}$ ${(p^{a} q^{b} r^{c})}_{m a x} = p^{1} q^{1} r^{p + q + r - 2}$ $n o s o l u t i o n i f$ $p^{p + q + r - 2} q^{1} r^{1} > 9999 o r$ $p^{1} q^{1} r^{p + q + r - 2} < 1000$ $\underset{―}{c a s e 1 : N = p^{a}}$ $o n l y o n e s o l u t i o n : p = a = 5$ $s i n c e 3^{3} = 27 < 1000, 5^{5} = 3125, 7^{7} = 823543 > 9999$ $\underset{―}{c a s e 2 : N = p^{a} q^{b}}$ $2^{1} 3^{4} = 162 < 1000 \Rightarrow n o s o l u t i o n$ $2^{6} 5^{1} = 320$ $2^{3} 5^{4} = 5000, 2^{4} 5^{3} = 2000$ $2^{8} 7^{1} = 1792$ $2^{7} 7^{2} = 6272$ $2^{12} 11^{1} = 45056 > 9999 \Rightarrow n o s o l u t i o n$ $3^{7} 5^{1} = 10935 > 9999 \Rightarrow n o s o l u t i o n$ $\underset{―}{c a s e 3 : N = p^{a} q^{b} r^{c}}$ $2^{8} 3^{1} 5^{1} = 3840$ $2^{7} 3^{2} 5^{1} = 5760, 2^{7} 3^{1} 5^{2} = 9600, 2^{6} 3^{3} 5^{1} = 8640$ $2^{10} 3^{1} 7^{1} = 21504 > 9999 \Rightarrow n o s o l u t i o n$ $\underset{―}{c a s e 4 : N = p^{a} q^{b} r^{c} s^{d}}$ $2^{14} 3^{1} 5^{1} 7^{1} = 1720320 > 9999 \Rightarrow n o s o l u t i o$ $s u m m a r y :$ $t h e r e a r e 9 s u c h 4 - d i g i t n u m b e r s :$ $1792, 2000, 3125, 3840, 5000,$ $5760, 6272, 8640, 9600$
	Commented by Rasheed.Sindhi last updated on 03/Sep/24

	$E l e g a n t!$
	Commented by mr W last updated on 03/Sep/24

	$t h a n k s!$ $‘ ‘ b r u t e {f o r c e}^{″} a p p r o a c h, n o t s m a r t,$ $b u t e f f e c t i v e .$
	Commented by Rasheed.Sindhi last updated on 03/Sep/24

	$e^{x} c e l l e n t s i r!$
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