A387527 a(n) = largest k such that, starting at n, the k pairs (n,n+1), (n+1,n+2), ... (n+k-1,n+k) have the "Distinct Prime Exponents" (DPE) property.

1, 0, 2, 1, 0, 0, 3, 2, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0

Offset

1,3

Comments

Two positive integers x and y have the "Distinct Prime Exponents" (or DPE) property if their prime factorizations x = Product p_i^e_i and y = Product q_i^f_i are such that the exponent sets {e_i} and {f_i} are disjoint.

For example, if x = 24 = 2^3*3^1 and 28 = 2^2*7^1, {e_i} = {1,3} and {f_i} = {1,2} are not disjoint, so 24 and 28 do not have the DPE property.

On the other hand, if x = 24 = 2^3*3^1 and y = 25 = 5^2, {e_i} = {1,3} and {f_i} = {2} are disjoint, so 24 and 25 do have the DPE.

At the present time, only one number n out to at least 10^12 has a(n) >= 4, namely a(24) = 4.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A387528 (a(n) = 1), A359749 (a(n) >= 1), A387529 (a(n) = 2), A387530 and A140104 (a(n) >= 2), A387531 and A388967 (a(n) >= 3).

Sequence in context: A280605 A362427 A218253 * A037872 A037854 A362633

Adjacent sequences: A387524 A387525 A387526 * A387528 A387529 A387530

Keyword

nonn

Author

Scott R. Shannon and N. J. A. Sloane, Oct 02 2025

Status

approved