A212127 -id:A212127 - cp's OEIS frontend

A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 127, 128, 131

Offset: 1

Antti Karttunen, May 24 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001065, A003415.
Cf. A212127, A212128 (subsequences), A344585 (complement).
Positions of nonnegative terms in A211991.
Differs from A212165 for the first time at n=121, where a(121) = 220, while A212165(121) = 223.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA344586(n) = (A003415(n) >= (sigma(n)-n));

A212128 Nonprimes whose arithmetic derivative equals the sum of its proper divisors.

Original entry on oeis.org

1, 12, 18, 220, 396, 287532

Offset: 1

Omar E. Pol, Dec 19 2012

a(7) > 10^12. - Giovanni Resta, Mar 11 2014

			The arithmetic derivative of 12 is equal to 16 (see A003415). On the other hand the sum of proper divisors of 12 is equal to 16 since 1+2+3+4+6 = 16, so 12 is in the sequence.

Cf. A018252, A000203, A001065, A003415, A211991, A212127.

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Select[Range[300000], ! PrimeQ[#] && dn[#] == DivisorSigma[1, #] - # &] (* T. D. Noe, Dec 27 2012 *)

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A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

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A212128 Nonprimes whose arithmetic derivative equals the sum of its proper divisors.

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