A071834 -id:A071834 - cp's OEIS frontend

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

Jaroslav Krizek, Nov 13 2017

nonn

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.

			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.

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

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

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

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

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

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

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

A295076 Numbers n > 1 such that n and sigma(n) have the same smallest prime factor.

Original entry on oeis.org

6, 10, 12, 14, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 62, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140, 142, 146, 148

Offset: 1

Jaroslav Krizek, Nov 13 2017

nonn

Supersequence of A088829; this sequence contains also odd numbers: 441, 1521, 3249, 3969, 8649, 11025, ...
Even terms of A000396 (perfect numbers) are a subsequence.
Subsequence of A295078.
Numbers n such that A020639(n) = A020639(sigma(n)).
Numbers n such that A020639(n) = A071189(n).

			30 = 2*3*5 and sigma(30) = 72 = 2^3*3^2 hence 30 is in the sequence.

Cf. A000203, A020639, A071189, A088829, A295078.

Cf. A071834 (numbers n such that n and sigma(n) have the same largest prime factor).

[n: n in [2..1000000] | Minimum(PrimeDivisors(SumOfDivisors(n))) eq Minimum(PrimeDivisors(n))]

select(t -> min(numtheory:-factorset(t))=min(numtheory:-factorset(numtheory:-sigma(t))), [$2..1000]); # Robert Israel, Nov 14 2017

Rest@ Select[Range@ 150, SameQ @@ Map[FactorInteger[#][[1, 1]] &, {#, DivisorSigma[1, #]}] &] (* Michael De Vlieger, Nov 13 2017 *)

isok(n) = factor(n)[1,1] == factor(sigma(n))[1,1]; \\ Michel Marcus, Nov 14 2017

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

cp's OEIS Frontend

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

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