cp's OEIS Frontend

A356594 Numbers k for which there exists at least one pair of positive integers (x,y) such that k = x + y and k' = x' + y', and every such pair is coprime.

%I A356594 #52 Mar 23 2025 18:24:02
%S A356594 3,25,55,82,85,95,116,121,145,194,226,245,253,289,295,301,305,332,335,
%T A356594 343,362,391,407,418,422,446,455,493,529,535,548,583,611,671,731,745,
%U A356594 749,754,778,779,781,785,799,805,815,817,818,833,838,845,866,869,899,917,931,943,955,959,985,995,998
%N A356594 Numbers k for which there exists at least one pair of positive integers (x,y) such that k = x + y and k' = x' + y', and every such pair is coprime.
%C A356594 Subsequence of A212662.
%C A356594 a(1)=3 is the only prime term.
%C A356594 It is not known if this sequence is finite.
%C A356594 Every term in A212662 is a multiple of a term in this sequence (this could be considered its primitive sequence), and no term in this sequence divides another term of this sequence.
%C A356594 In general, we do not have (i+j)'=i'+j'; this is in contrast with the normal derivative, because the derivative of the sum of two functions is equal to the sum of the derivatives of the individual functions. The terms in this sequence are the ones for which there exist positive integers i and j with the properties illustrated above, thus building another common point between the arithmetic derivative and the normal derivative.
%o A356594 (PARI)
%o A356594 D(n)={x=factor(n);n*sum(i=1,matsize(x)[1],x[i,2]/x[i,1])}
%o A356594 Base(n)={for(i=1,n\2,if(D(n-i)+D(i)==D(n)&&gcd(i,n-i)==1,return(1)))}
%o A356594 Impure(n)={for(i=1,n\2,if(D(n-i)+D(i)==D(n)&&gcd(i,n-i)!=1,return(1)))}
%o A356594 IsTerm(n)={Base(n)&&!Impure(n)}
%o A356594 Bp(n)={for(i=2,n,if(IsTerm(i),print1(i, ", ")))}
%o A356594 Bp(1000)
%o A356594 (PARI) ard(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A356594 isok1(m) = for (k=1, m\2, if (ard(m-k)+ard(k) == ard(m), return(1))); \\ A212662
%o A356594 isok(m) = if (isok1(m), my(d=divisors(m)); for (k=1, #d, if((d[k]!=m) && isok1(d[k]), return(0))); return(1)); \\ _Michel Marcus_, Aug 28 2022
%Y A356594 Cf. A003415 (arithmetic derivative), A212662.
%K A356594 nonn
%O A356594 1,1
%A A356594 _Giosuè Cavallo_, Aug 14 2022