cp's OEIS Frontend

A358674 Numbers k for which there is a factorization of k into such a pair of natural numbers x and y, that the sum (x * y') + (x' * y) will generate at least one carry when the addition is done in the primorial base. Here n' stands for A003415(n), the arithmetic derivative of n.

%I A358674 #15 Nov 28 2022 17:22:25
%S A358674 8,9,10,15,16,20,21,22,24,25,28,30,32,33,34,35,36,39,40,42,44,45,46,
%T A358674 48,49,50,51,52,54,55,56,57,58,60,64,65,66,68,69,72,75,76,77,78,80,81,
%U A358674 82,84,85,87,88,90,91,92,93,94,95,96,98,100,102,104,105,106,108,110,111,112,114,115,116,118,119
%N A358674 Numbers k for which there is a factorization of k into such a pair of natural numbers x and y, that the sum (x * y') + (x' * y) will generate at least one carry when the addition is done in the primorial base. Here n' stands for A003415(n), the arithmetic derivative of n.
%H A358674 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A358674 {k | A358235(k) < A038548(k)}.
%e A358674 Refer to the examples in A358235 to see why 24 is included in this sequence.
%o A358674 (PARI) isA358674(n) = !A358672(n);
%o A358674 (PARI)
%o A358674 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A358674 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A358674 A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
%o A358674 A329041sq(row,col) = A327936(A276086(row)*A276086(col));
%o A358674 isA358674(n) = { fordiv(n, d, if(d>(n/d), return(0)); if(1<A329041sq((d*A003415(n/d)), (A003415(d)*(n/d))), return(1))); (0); };
%Y A358674 Positions of zeros in A358672.
%Y A358674 Cf. A003415, A038548, A276086, A329041, A358235, A358673 (complement).
%Y A358674 Subsequences: A016754 (without its initial 1), A358675.
%K A358674 nonn,base
%O A358674 1,1
%A A358674 _Antti Karttunen_, Nov 26 2022