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A120321 RF(7): refactorable numbers with 7 as smallest prime factor.

%I A120321 #12 Aug 01 2024 09:16:37
%S A120321 117649,208422380089,567869252041,2839760855281,5534900853769,
%T A120321 17416274304961,69980368892329,104413920565969,301855146292441,
%U A120321 558845013849409,743702041351801,1268163904241521,2607614922465721
%N A120321 RF(7): refactorable numbers with 7 as smallest prime factor.
%C A120321 Numbers that are odd squares, 7 is their smallest prime factor, and are refactorable.
%C A120321 See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 7^(7-1)=117649 is the first element. Other elements would also be 7^6*17^6 or 7^16*17^6. Here are the prime factorizations for the first 49 elements of RF7: (7^6), (7^6)*(11^6), (7^6)*(13^6), (7^6)*(17^6), (7^6)*(19^6), (7^6)*(23^6), (7^6)*(29^6), (7^6)*(31^6), (7^6)*(37^6), (7^6)*(41^6), (7^6)*(43^6), (7^6)*(47^6), (7^6)*(53^6), (7^6)*(59^6), (7^6)*(61^6), (7^6)*(67^6), (7^6)*(71^6), (7^6)*(73^6), (7^6)*(79^6), (7^6)*(83^6), (7^6)*(89^6), (7^12)*(13^6), (7^6)*(97^6), (7^6)*(101^6), (7^6)*(103^6), (7^6)*(107^6), (7^6)*(109^6), (7^6)*(113^6), (7^6)*(127^6), (7^6)*(131^6), (7^6)*(137^6), (7^6)*(139^6), (7^6)*(11^6)*(13^6), (7^6)*(149^6), (7^6)*(151^6), (7^6)*(157^6), (7^6)*(163^6), (7^6)*(167^6), (7^6)*(13^12), (7^6)*(173^6), (7^6)*(179^6), (7^6)*(181^6), (7^6)*(11^6)*(17^6), (7^6)*(191^6), (7^6)*(193^6), (7^6)*(197^6), (7^6)*(199^6), (7^6)*(11^6)*(19^6), (7^6)*(211^6).
%H A120321 Amiram Eldar, <a href="/A120321/b120321.txt">Table of n, a(n) for n = 1..10000</a>
%e A120321 a(1) = 7^(7-1) = 117649.
%p A120321 with(numtheory); p:=7: RF7:=[p^(p-1)]: P:=[seq(ithprime(i),i=2..pi(p)-1)]; for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if andmap(z -> k mod z <> 0, P) then for s from 2 to p-1 by 2 do #accelerate creation n:=7^6*k^s; t:=tau(n); if not n in RF7 and (n mod t = 0) then RF7:=[op(RF7),n]; print(ifactor(n)); fi; od; fi; od od; RF7:=sort(RF7);
%Y A120321 Intersection of A033950 and A084968.
%Y A120321 Cf. A036896, A036897.
%K A120321 nonn
%O A120321 1,1
%A A120321 _Walter Kehowski_, Jun 21 2006