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%I #14 Nov 28 2022 17:22:31
%S 8,9,10,15,16,20,21,22,25,28,30,33,34,35,39,44,46,49,50,51,55,56,57,
%T 58,65,66,68,69,77,81,82,84,85,87,91,92,93,94,95,102,106,108,111,112,
%U 115,116,118,119,120,121,123,125,128,129,133,136,138,141,142,143,145,147,148,155,156,159,160,161
%N Numbers k such that for all nontrivial factorizations of k as x*y, 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 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F {k | k is composite and A358235(k) = 1}.
%e 16 has two nontrivial factorizations into two factors, 2*8 and 4*4. For both of these, the sums (2*A003415(8))+(A003415(2)+8) = 24+8 ("400" + "110") and (4*A003415(4))+(A003415(4)*4) = 16+16 ("220" + "220") generate carries in the primorial base (as 2 and 4 are the max. digits allowed in the second and third rightmost positions, see A049345), therefore 16 is included in this sequence.
%o (PARI) isA358675(n) = ((n>1)&&!isprime(n)&&(1==A358235(n)));
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
%o A329041sq(row,col) = A327936(A276086(row)*A276086(col));
%o isA358675(n) = if(1==n || isprime(n), 0, fordiv(n, d, if((d>1) && (d<n) && 1==A329041sq((d*A003415(n/d)), (A003415(d)*(n/d))), return(0))); (1));
%Y Composite positions of 1's in A358235.
%Y Subsequence of A358674.
%Y Cf. A003415, A049345.
%K nonn,base
%O 1,1
%A _Antti Karttunen_, Nov 26 2022
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