A335254 Numbers k such that the abundance (A033880) of k is equal to the deficiency (A033879) of k+1.

672, 523776, 19327369215

Offset: 1

Author

Amiram Eldar, May 28 2020

Keywords

• nonn
• hard
• bref
• more

Comments

Equivalently, k and k+1 have the same absolute value of abundance (or deficiency) with opposite signs.

Equivalently, s(k) + s(k+1) = k + (k+1), where s(k) is the sum of proper divisors of k (A001065).

If k is a 3-perfect number (A005820) and k+1 is a prime, then k is in the sequence. Of the 6 known 3-perfect numbers only 672 and 523776 have this property.

a(4) > 10^11, if it exists.

a(4) > 10^13, if it exists. - Giovanni Resta, May 30 2020

Examples

			672 is a term since A033880(672) = sigma(672) - 2*672 = 2016 - 1344 = 672, and A033879(673) = 2*673 - sigma(673) = 1346 - 674 = 672.
		

Crossrefs

Cf. A000203, A001065, A005820, A033879, A033880, A330901, A335253.

Programs

Mathematica

ab[n_] := DivisorSigma[1, n] - 2*n; Select[Range[6 * 10^5], ab[#] == -ab[# + 1] &]