

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.


1



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



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); };
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.


KEYWORD

nonn,base


AUTHOR



STATUS

approved



