A358675 - OEIS

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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

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OFFSET

1,1

LINKS

Table of n, a(n) for n=1..68.

Index entries for sequences related to primorial base

FORMULA

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

EXAMPLE

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.

PROG

(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) && (d<n) && 1==A329041sq((d*A003415(n/d)), (A003415(d)*(n/d))), return(0))); (1));

CROSSREFS

Composite positions of 1's in A358235.

Subsequence of A358674.

Cf. A003415, A049345.