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

1,1

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.

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A344588 A247111 A071834 * A055196 A323752 A120624

Adjacent sequences: A295075 A295076 A295077 * A295079 A295080 A295081

Keyword

nonn

Author

Jaroslav Krizek, Nov 13 2017

Extensions

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

Status

approved