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.

8, 9, 10, 15, 16, 20, 21, 22, 24, 25, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 64, 65, 66, 68, 69, 72, 75, 76, 77, 78, 80, 81, 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

Offset: 1

Author

Antti Karttunen, Nov 26 2022

Keywords

• nonn
• base

Examples

			Refer to the examples in A358235 to see why 24 is included in this sequence.
		

Links

• Index entries for sequences related to primorial base

Crossrefs

Positions of zeros in A358672.

Cf. A003415, A038548, A276086, A329041, A358235, A358673 (complement).

Subsequences: A016754 (without its initial 1), A358675.

Programs

PARI

isA358674(n) = !A358672(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));
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); };

Formula

{k | A358235(k) < A038548(k)}.