A097273 Least integer with each "mod 2 prime signature".

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 27, 30, 32, 36, 45, 48, 54, 60, 64, 72, 81, 90, 96, 105, 108, 120, 128, 135, 144, 162, 180, 192, 210, 216, 225, 240, 243, 256, 270, 288, 315, 324, 360, 384, 405, 420, 432, 450, 480, 486, 512, 540, 576, 630, 648, 675, 720

Offset: 1

Author

Ray Chandler, Aug 22 2004

Keywords

• nonn

Comments

For n = 2^e_0 * p_1^e_1 * ... * p_n^e_n where p_i is odd prime and e_1 >= e_2 >= ... >= e_n, define "mod 2 prime signature" to be ordered prime exponents (e_0,e_1,...,e_n).

Least integer with a given mod 2 prime signature is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.

A097272 sorted and duplicates removed.

Numbers k such that A097272(k) = k.

Verified up to a(68) = 972, 2*a(n) is also the order of a dihedral group D such that the lattice of normal subgroups of D is not isomorphic to the lattice of normal subgroups of any dihedral group of order less than 2*a(n). - Miles Englezou, May 18 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A097272, A097274, A097275.

Cf. A025487, A070826, A147516.

Programs

Mathematica

lpsQ[n_] := n==1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1,1]] == Prime[Length[f] + 1]); Select[Range[1000], lpsQ[# / 2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 23 2024 *)

Formula

Sum_{n>=1} 1/a(n) = 2 * Product_{n>=2} 1/(1 - 1/A070826(n)) = 3.2482341898... . - Amiram Eldar, Jul 23 2024

Extensions

Offset corrected by Amiram Eldar, Jul 23 2024