cp's OEIS Frontend

A381060 Numbers t which are the sum of some subset of the values of k satisfying the equation (t - floor((t - k)/k)) mod k = 0 (t > 1, 1 <= k < t).

%I A381060 #6 Mar 10 2025 18:12:43
%S A381060 23,29,39,41,53,59,65,71,77,79,83,89,99,101,107,111,113,119,125,137,
%T A381060 143,149,155,161,167,173,179,185,191,197,199,209,221,227,233,239,245,
%U A381060 251,257,263,269,279,281,287,293,299,305,311,317,323,329,335,339,341,349,353,359,365,371
%N A381060 Numbers t which are the sum of some subset of the values of k satisfying the equation (t - floor((t - k)/k)) mod k = 0 (t > 1, 1 <= k < t).
%C A381060 The sequence is based on the triangle in A380305, which is a variant of the triangle in A048158. Thus, the elements of this sequence are counterparts of pseudoperfect numbers (A005835), such as the elements of A375595 are counterparts of abundant numbers (A005101).
%C A381060 The sequence includes all elements of A380153.
%C A381060 The first even element of this sequence is a(768) = 4094.
%e A381060 23 is in this sequence because the only k's < 23 satisfying the equation (23 - floor((23 - k)/k)) mod k = 0 are: 1, 5, 7, 11, hence: 5+7+11 = 23.
%e A381060 29 is in this sequence because the only k's < 29 satisfying the equation (29 - floor((29 - k)/k)) mod k = 0 are: 1, 2, 3, 5, 9, 14, hence: 1+2+3+9+14 = 29 and 1+5+9+14 = 29.
%e A381060 47 is not in this sequence because the only k's < 47 satisfying the equation (47 - floor((47 - k)/k)) mod k = 0  are: 1, 3, 7, 11, 15, 23 and no subset of these numbers adds to 47.
%o A381060 (Maxima)
%o A381060 (kill(all), s(y):=(f(i,j):=mod(i-floor((i-j)/j),j),s:0,x:1,
%o A381060       for k:1 thru floor(y/2) do
%o A381060               (if f(y,k)=0 then
%o A381060               (s:s+k, B[x]:k, x:x+1)),
%o A381060 B:setify(makelist(B[r],r,1,x-1)), s),
%o A381060 n:1, for t:2 thru 1000 do
%o A381060               (if s(t)>=t  then
%o A381060                      (for b:2 while b<=x-1 and e#t do
%o A381060                      (C:args(powerset(B,b)),
%o A381060                             for h:1 while h<=length(C) and e#t do
%o A381060                             (e:apply("+" , args(C[h])),
%o A381060                                     if e=t then
%o A381060                                     (print(n , " " , t), n:n+1))))));
%Y A381060 Cf. A005835, A048158, A375595, A380153, A380305.
%K A381060 nonn
%O A381060 1,1
%A A381060 _Lechoslaw Ratajczak_, Feb 12 2025