A283418 -id:A283418 - cp's OEIS frontend

A373316 Numbers k such that k and k+2 are both primitive abundant numbers.

18, 102, 364, 366, 474, 532, 642, 834, 1036, 1146, 1182, 1374, 1504, 1696, 1876, 1986, 2210, 2584, 2994, 3052, 3126, 3556, 4396, 4542, 4564, 5032, 5514, 5572, 5574, 5622, 6232, 6412, 6522, 6976, 7026, 7206, 7912, 7924, 8202, 8596, 8706, 9654, 9714

Offset: 1

Abhiram R Devesh, May 31 2024

nonn

			18 = 2*3*3 is an abundant number, but its proper divisors are 1, 2, 3, 6 and 9, none of which are abundant.
18 + 2 = 20 = 2*2*5 is an abundant number, but its proper divisors are 1, 2, 4, 5 and 10, none of which are abundant.
Thus, both 18 and 20 are primitive abundant numbers, so 18 is in the sequence.

Abhiram R Devesh, Table of n, a(n) for n = 1..208

Cf. A091191, A283418.

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1); primAbQ[n_] := primAbQ[n] = (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2; Select[Range[2, 10^4], primAbQ[#] && primAbQ[# + 2] &] (* Amiram Eldar, Jul 20 2024 *)

A380933 Numbers k such that k and k+1 are both in A380929.

Original entry on oeis.org

121643775, 157390064, 161019495, 275734304, 584899875, 1493214975, 1614323655, 2043708975, 3081783375, 3118599224, 3426851295, 3902652495, 3947893424, 5849043375, 11731509855, 12138531615, 13008843224, 14598032624, 17588484584, 19782621495, 20191564575, 20759209064

Offset: 1

Amiram Eldar, Feb 08 2025

Numbers k such that A380845(k) > 2*k and A380845(k+1) > 2*(k+1).

			121643775 is a term since A380845(121643775) = 244722015 > 2 * 121643775 = 243287550, and A380845(121643776) = 256456081 > 2 * 121643776 = 243287552.

Amiram Eldar, Table of n, a(n) for n = 1..32

Subsequence of A096399 and A380929.
Cf. A380845, A380932.
Similar sequences: A283418, A318167, A327635, A327942, A331412, A333951, A357608, A364727, A364861.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k];
seq[lim_] := Module[{s = {}}, Do[If[q[k], If[q[k-1], AppendTo[s, k-1]]; If[q[k+1], AppendTo[s, k]]], {k, 3, lim, 2}]; s];
seq[3*10^8]

isab(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}
list(lim) = forstep(k = 3, lim, 2, if(isab(k), if(isab(k-1), print1(k-1, ", ")); if(isab(k+1), print1(k, ", "))));

cp's OEIS Frontend

A373316 Numbers k such that k and k+2 are both primitive abundant numbers.

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