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A334261 Numbers m that are solutions of the arithmetic differential equation 2m" - m' - 4 = 0.

%I A334261 #57 Jan 09 2024 16:56:27
%S A334261 4,9,21,25,33,49,57,69,85,93,121,129,133,145,169,177,205,213,217,237,
%T A334261 249,253,265,289,309,361,393,417,445,469,489,493,505,517,529,553,565,
%U A334261 573,597,633,669,685,697,753,781,793,813,817,841,865,889,913,933,949,961,973,985,993
%N A334261 Numbers m that are solutions of the arithmetic differential equation 2m" - m' - 4 = 0.
%C A334261 Contains all the squared primes, plus additional terms.
%C A334261 At least some of these additional terms that are not divisible by 3 are given by the product of the smallest prime in a prime triplet, and the largest prime in that triplet, this certainly being true until the term 11413.
%H A334261 Nathan Mabey, <a href="/A334261/b334261.txt">Table of n, a(n) for n = 1..20263</a>
%e A334261 4 is a term since 4'' = 4, 4' = 4, and 2*4 - 4 - 4 = 0.
%o A334261 (Python)
%o A334261 from sympy import factorint
%o A334261 def d(n): return sum(n*e//p for p, e in factorint(n).items())
%o A334261 def ok(m): return m > 1 and 2*d(d(m)) - d(m) == 4
%o A334261 print([k for k in range(1, 999) if ok(k)]) # _Michael S. Branicky_, Aug 15 2022
%o A334261 (PARI) d(n) = if (n>0, my(f=factor(n)); sum(i=1, #f~, n*f[i, 2]/f[i, 1]), 0);
%o A334261 isok(m) = -2*d(d(m)) + d(m) + 4 == 0; \\ _Michel Marcus_, Apr 21 2020
%Y A334261 Cf. A003415 (n'), A068346 (n"), A001248 (squared primes), A368701 (characteristic function).
%Y A334261 Cf. also A189941.
%K A334261 easy,nonn
%O A334261 1,1
%A A334261 _Nathan Mabey_, Apr 20 2020