A333055 - OEIS

%I #9 Mar 07 2020 03:51:39

%S 26,104,189,231,242,243,344,374,663,664,735,776,782,874,903,1015,1029,

%T 1095,1106,1112,1161,1208,1269,1335,1374,1544,1625,1809,1832,1917,

%U 1952,1970,2055,2133,2241,2247,2264,2343,2344,2504,2655,2696,2726,2781,2874,2936

%N Numbers k such that k and k+1 have different (ordered) prime signatures and d(k) = d(k+1), where d(k) is the number of divisors of k (A000005).

%C Apparently most of the numbers k such that k and k+1 have the same number of divisors (A005237) also have the same prime signature, i.e., they are also terms of A052213 which is a subsequence of A005237.

%C For example, up to 10^8 there are 9593611 terms in A005237, of them only 1573778 (about 16.4%) are not in A052213. This sequence in the complement of A052213 with respect to A005237.

%H Amiram Eldar, <a href="/A333055/b333055.txt">Table of n, a(n) for n = 1..10000</a>

%e 26 is a term since 26 = 2 * 13 and 27 = 3^3 have different prime signatures, and d(26) = d(27) = 4.

%t Select[Range[3000], DivisorSigma[0, #] == DivisorSigma[0, #+1] && Sort[FactorInteger[#][[;;,2]]] != Sort[FactorInteger[#+1][[;;,2]]] &]

%Y Cf. A000005, A005237, A052213, A124010, A333056, A333057.

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 06 2020