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

6, 28, 40, 84, 120, 140, 224, 234, 270, 420, 468, 496, 672, 756, 936, 1080, 1120, 1170, 1372, 1488, 1550, 1638, 1782, 1862, 2176, 2340, 2480, 2574, 3100, 3250, 3276, 3360, 3472, 3564, 3724, 3744, 3780, 4116, 4464, 4598, 4650, 4680, 5148, 5456, 5586, 6048, 6200

Offset: 1

Author

Jaroslav Krizek, Nov 13 2017

Keywords

• nonn

Comments

All even perfect numbers are terms.

Conjecture: A007691 (multiply-perfect numbers) is a subsequence.

Note that an odd perfect number (if it exists) would be a counterexample to the conjecture. - Robert Israel, Jan 08 2018

Intersection of A071834 and A295076.

Numbers n such that A020639(n) = A020639(sigma(n)) and simultaneously A006530(n) = A006530(sigma(n)).

Numbers n such that A020639(n) = A071189(n) and simultaneously A006530(n) = A071190(n).

Supersequence of A027598.

Examples

			40 = 2^3*5 and sigma(40) = 90 = 2*3^2*5 hence 40 is in the sequence.
The first odd term is 29713401 = 3^2 * 23^2 * 79^2; sigma(29713401) = 45441669 = 3*7^3*13*43*79.
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000203, A006530, A007691, A020639, A027598, A071189, A071190, A295076.

Programs

Magma

[n: n in [2..10000] | Minimum(PrimeDivisors(n)) eq Minimum(PrimeDivisors(SumOfDivisors(n))) and Maximum(PrimeDivisors(n)) eq Maximum(PrimeDivisors(SumOfDivisors(n)))]

Maple

filter:= proc(n) local f, s; uses numtheory;
  f:= factorset(n);
  s:= factorset(sigma(n));
  min(f) = min(s) and max(f)=max(s)
end proc:
select(filter, [$2..10^4]); # Robert Israel, Jan 08 2018

Mathematica

Rest@ Select[Range@ 6200, SameQ @@ Map[{First@ #, Last@ #} &@ FactorInteger[#][[All, 1]] &, {#, DivisorSigma[1, #]}] &] (* Michael De Vlieger, Nov 13 2017 *)

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

Extensions

Added condition n>1 to definition. Corrected b-file. - N. J. A. Sloane, Feb 03 2018