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A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors.

%I A295078 #30 Sep 08 2022 08:46:20
%S A295078 6,28,40,84,120,140,224,234,270,420,468,496,672,756,936,1080,1120,
%T A295078 1170,1372,1488,1550,1638,1782,1862,2176,2340,2480,2574,3100,3250,
%U A295078 3276,3360,3472,3564,3724,3744,3780,4116,4464,4598,4650,4680,5148,5456,5586,6048,6200
%N A295078 Numbers n > 1 such that n and sigma(n) have the same smallest and simultaneously largest prime factors.
%C A295078 All even perfect numbers are terms.
%C A295078 Conjecture: A007691 (multiply-perfect numbers) is a subsequence.
%C A295078 Note that an odd perfect number (if it exists) would be a counterexample to the conjecture. - _Robert Israel_, Jan 08 2018
%C A295078 Intersection of A071834 and A295076.
%C A295078 Numbers n such that A020639(n) = A020639(sigma(n)) and simultaneously A006530(n) = A006530(sigma(n)).
%C A295078 Numbers n such that A020639(n) = A071189(n) and simultaneously A006530(n) = A071190(n).
%C A295078 Supersequence of A027598.
%H A295078 Jaroslav Krizek, <a href="/A295078/b295078.txt">Table of n, a(n) for n = 1..1000</a>
%e A295078 40 = 2^3*5 and sigma(40) = 90 = 2*3^2*5 hence 40 is in the sequence.
%e A295078 The first odd term is 29713401 = 3^2 * 23^2 * 79^2; sigma(29713401) = 45441669 = 3*7^3*13*43*79.
%p A295078 filter:= proc(n) local f, s; uses numtheory;
%p A295078   f:= factorset(n);
%p A295078   s:= factorset(sigma(n));
%p A295078   min(f) = min(s) and max(f)=max(s)
%p A295078 end proc:
%p A295078 select(filter, [$2..10^4]); # _Robert Israel_, Jan 08 2018
%t A295078 Rest@ Select[Range@ 6200, SameQ @@ Map[{First@ #, Last@ #} &@ FactorInteger[#][[All, 1]] &, {#, DivisorSigma[1, #]}] &] (* _Michael De Vlieger_, Nov 13 2017 *)
%o A295078 (Magma) [n: n in [2..10000] | Minimum(PrimeDivisors(n)) eq Minimum(PrimeDivisors(SumOfDivisors(n))) and Maximum(PrimeDivisors(n)) eq Maximum(PrimeDivisors(SumOfDivisors(n)))]
%o A295078 (PARI) isok(n) = if (n > 1, my(fn = factor(n)[,1], fs = factor(sigma(n))[,1]); (vecmin(fn) == vecmin(fs)) && (vecmax(fn) == vecmax(fs))); \\ _Michel Marcus_, Jan 08 2018
%Y A295078 Cf. A000203, A006530, A007691, A020639, A027598, A071189, A071190, A295076.
%K A295078 nonn
%O A295078 1,1
%A A295078 _Jaroslav Krizek_, Nov 13 2017
%E A295078 Added condition n>1 to definition. Corrected b-file. - _N. J. A. Sloane_, Feb 03 2018