cp's OEIS Frontend

A020700 Numbers k such that k + sum of its prime factors = (k+1) + sum of its prime factors.

%I A020700 #24 Feb 26 2023 17:54:55
%S A020700 7,14,63,80,224,285,351,363,475,860,902,1088,1479,2013,2023,3478,3689,
%T A020700 3925,5984,6715,8493,9456,13224,15520,17227,18569,19502,20490,21804,
%U A020700 24435,24476,27335,31899,32390,35815,37406,37582,41876,49468,50609,54137,57239
%N A020700 Numbers k such that k + sum of its prime factors = (k+1) + sum of its prime factors.
%C A020700 If p, (3/2)*(p+1), (3/2)*(p^2+p)+1 and (3/2)*(p^2+1)+2*p are all prime, then (3/2)*p*(3*p^2+4*p+3) is a term. The Generalized Bunyakovsky Conjecture implies that there are infinitely many of these. - _Robert Israel_, Apr 15 2022
%H A020700 Robert Israel, <a href="/A020700/b020700.txt">Table of n, a(n) for n = 1..2500</a>
%e A020700 A075254(7) = 7+7 = 14 and A075254(8) = 8+2+2+2 = 14, so 7 is in the sequence.
%t A020700 SequencePosition[Table[n+Total[Times@@@FactorInteger[n]],{n,58000}],{x_,x_}][[;;,1]] (* _Harvey P. Dale_, Feb 26 2023 *)
%o A020700 (PARI) A075254(n) = my(f = factor(n)); n + sum(i=1, #f~, f[i,1]*f[i,2]);
%o A020700 isok(n) = A075254(n) == A075254(n+1); \\ _Michel Marcus_, Jun 05 2014
%o A020700 (Python)
%o A020700 from sympy import factorint
%o A020700 from itertools import count, islice
%o A020700 def sopf(n): return sum(p*e for p, e in factorint(n).items())
%o A020700 def agen(): # generator of terms
%o A020700     sopfkplus1 = 2
%o A020700     for k in count(2):
%o A020700         sopfk, sopfkplus1 = sopfkplus1, sopf(k+1)
%o A020700         if k + sopfk == k + 1 + sopfkplus1: yield k
%o A020700 print(list(islice(agen(), 42))) # _Michael S. Branicky_, Apr 15 2022
%Y A020700 Cf. A020905, A228126.
%K A020700 easy,nonn
%O A020700 1,1
%A A020700 _Enoch Haga_
%E A020700 More terms from _Michel Marcus_, Jun 05 2014