A333055 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).

26, 104, 189, 231, 242, 243, 344, 374, 663, 664, 735, 776, 782, 874, 903, 1015, 1029, 1095, 1106, 1112, 1161, 1208, 1269, 1335, 1374, 1544, 1625, 1809, 1832, 1917, 1952, 1970, 2055, 2133, 2241, 2247, 2264, 2343, 2344, 2504, 2655, 2696, 2726, 2781, 2874, 2936

Offset

1,1

Comments

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.

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.

Links

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A065013 A031434 A173089 * A244633 A042320 A042322

Adjacent sequences: A333052 A333053 A333054 * A333056 A333057 A333058

Keyword

nonn

Author

Amiram Eldar, Mar 06 2020

Status

approved