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A277272 a(n+1) is the smallest number not already in the sequence whose sum of prime factors (with repetition) shares a factor with the sum of the prime factors of a(n); a(1)=2.

%I A277272 #28 Feb 21 2024 10:30:46
%S A277272 2,4,8,3,9,14,20,24,26,5,6,21,15,16,18,25,30,32,33,7,10,12,38,27,35,
%T A277272 36,39,42,44,46,51,49,50,55,57,11,28,40,45,48,54,62,60,64,65,66,69,13,
%U A277272 22,56,63,74,68,70,72,77,78,81,84,85,87,91,86,92,95,93,17,52,88,99,145,98,100,94,115,102
%N A277272 a(n+1) is the smallest number not already in the sequence whose sum of prime factors (with repetition) shares a factor with the sum of the prime factors of a(n); a(1)=2.
%C A277272 Inspired by A064413 (the EKG sequence). Conjecture: The sequence is a permutation of the natural numbers >1, in which composite terms appear once only (in irregular fashion) and the primes arise in their natural order. Immediately following the appearance of any prime p>3 the next term is q*(2^r) where q is the largest prime less than p, and r is the unique integer such that p=q+(2*r).
%C A277272 The above conjecture fails at a(95)=23, which appears before a(162)=19. (Also, the term appearing immediately after 37 is 218=2*109; the term immediately after 97 is 1335=3*5*89.) - _Jon E. Schoenfield_, Nov 05 2016
%H A277272 Jon E. Schoenfield, <a href="/A277272/b277272.txt">Table of n, a(n) for n = 1..10000</a>
%H A277272 Jon E. Schoenfield, <a href="/A277272/a277272.txt">Magma program for generating a b-file</a>
%e A277272 a(2)=4 because sopf(4)=4 and is the smallest number (other than 2) to share a factor (2) with sopf(2)=2.
%e A277272 a(3)=8 since sopf(8)=6 and is the smallest number (other than 2 and 4) to share a factor (2) with sopf(4).
%t A277272 f[n_] := Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ n]; a = {2}; Do[k = 2; While[Nand[IntersectingQ[f[Total@ f@ k], f[Total@ f@ a[[n - 1]]]], ! MemberQ[a, k]], k++]; AppendTo[a, k], {n, 2, 76}]; a (* _Michael De Vlieger_, Oct 08 2016 *)
%o A277272 (Python)
%o A277272 from math import gcd
%o A277272 from sympy import factorint
%o A277272 from functools import cache
%o A277272 from itertools import count, islice
%o A277272 @cache
%o A277272 def sopfr(n): return sum(p*e for p,e in factorint(n).items())
%o A277272 def agen(): # generator of terms
%o A277272     aset, an, mink = {2}, 2, 3
%o A277272     while True:
%o A277272         yield an
%o A277272         s = sopfr(an)
%o A277272         an = next(k for k in count(mink) if k not in aset and gcd(sopfr(k), s)!=1)
%o A277272         aset.add(an)
%o A277272         while mink in aset: mink += 1
%o A277272 print(list(islice(agen(), 76))) # _Michael S. Branicky_, Feb 21 2024
%Y A277272 Cf. A008472 (sopf), A064413.
%K A277272 nonn
%O A277272 1,1
%A A277272 _David James Sycamore_, Oct 08 2016