cp's OEIS Frontend

A191749 Numbers not the sum of a smaller number and its prime factors (with multiplicity).

%I A191749 #47 Aug 14 2025 17:54:08
%S A191749 1,2,3,5,7,9,12,13,16,18,20,21,25,27,28,30,32,37,43,44,45,48,49,50,52,
%T A191749 57,60,61,64,66,67,68,70,73,75,77,78,80,81,85,87,90,91,92,97,100,101,
%U A191749 102,104,108,110,112,115,117,126,129,130,132,133,135,137,139,144,145
%N A191749 Numbers not the sum of a smaller number and its prime factors (with multiplicity).
%C A191749 If a number is not squarefree, then its repeated prime factors are added as many times as the exponent indicates (e.g., the sum of prime factors of 8 is 6 since 8 = 2 * 2 * 2 and 2 + 2 + 2 = 6).
%C A191749 No even semiprime (A100484) can be in this sequence, since, if nothing else, it is the sum of a prime number and that prime number's only prime factor (itself).
%H A191749 David A. Corneth, <a href="/A191749/b191749.txt">Table of n, a(n) for n = 1..10000</a>
%e A191749 3 is in the sequence since neither 1 + sopfr(1) nor 2 + sopfr(2) add up to 3 (instead these equal 2 and 4 respectively).
%e A191749 Because 2 + sopfr(2) = 4, the number 4 is not in this sequence.
%t A191749 pfAddSeq[start_, max_] := NestWhileList[# + Plus@@Times@@@FactorInteger@# &, start, # < max &]; Complement[Range[200], Flatten[Table[Drop[pfAddSeq[n, 200], 1], {n, 2, 200}]]] (* corrected by _Amiram Eldar_, Aug 14 2025 *)
%o A191749 (PARI) upto(n) = {
%o A191749 	v = vector(n);
%o A191749 	for(i = 2, n,
%o A191749 		c = i + sopfr(i);
%o A191749 		if(c <= n,
%o A191749 			v[c] = 1));
%o A191749 	select(x -> x == 0, v, 1)}
%o A191749 sopfr(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i,1] * f[i,2])} \\ _David A. Corneth_, Aug 14 2025
%Y A191749 Cf. A096461, A192896 (only a(1) of those sequences can be in this sequence). Cf. also A001414. Analogous to A005114.
%K A191749 nonn
%O A191749 1,2
%A A191749 _Alonso del Arte_, Jul 13 2011
%E A191749 2 inserted by and more terms from _David A. Corneth_, Aug 14 2025