A358675 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.

8, 9, 10, 15, 16, 20, 21, 22, 25, 28, 30, 33, 34, 35, 39, 44, 46, 49, 50, 51, 55, 56, 57, 58, 65, 66, 68, 69, 77, 81, 82, 84, 85, 87, 91, 92, 93, 94, 95, 102, 106, 108, 111, 112, 115, 116, 118, 119, 120, 121, 123, 125, 128, 129, 133, 136, 138, 141, 142, 143, 145, 147, 148, 155, 156, 159, 160, 161

Offset: 1

Author

Antti Karttunen, Nov 26 2022

Keywords

• nonn
• base

Examples

			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.
		

Links

• Index entries for sequences related to primorial base

Crossrefs

Composite positions of 1's in A358235.

Subsequence of A358674.

Cf. A003415, A049345.

Programs

PARI

isA358675(n) = ((n>1)&&!isprime(n)&&(1==A358235(n)));

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
A329041sq(row,col) = A327936(A276086(row)*A276086(col));
isA358675(n) = if(1==n || isprime(n), 0, fordiv(n, d, if((d>1) && (dA329041sq((d*A003415(n/d)), (A003415(d)*(n/d))), return(0))); (1));

Formula

{k | k is composite and A358235(k) = 1}.