cp's OEIS Frontend

A370268 Intersection of A189715 and A370267.

%I A370268 #6 Feb 15 2024 08:46:20
%S A370268 1,4,6,7,9,10,15,16,22,24,25,28,31,33,36,40,42,49,54,55,58,60,63,64,
%T A370268 70,73,79,81,87,88,90,96,97,100,103,105,106,112,118,121,124,127,132,
%U A370268 135,144,145,150,151,154,159,160,166,168,169,175,177,186,193,196,198,199,202,214,216,217,220,223,225,231,232,240,241,247
%N A370268 Intersection of A189715 and A370267.
%C A370268 A189715 and A370267 are closely related in that they may be generated by the same process, but starting from numbers of the form 6m+1 and 8m+1 respectively - see A370267 for details.
%C A370268 Independent definition: numbers with an even number of prime factors not of the form 3m+1 and an even number of prime factors not of the form 8m+-1 (counting repetitions).
%C A370268 The sequence starts with the first 72 nonzero numbers of the form x^2 + 6y^2 (see A002481). After the absence of 0, this sequence next differs from A002481 by including 247, 391, 442, ... . From these early intermittent differences, the densities of the two sequences diverge progressively, driven by the absence from A002481 of many of the squarefree composite numbers that are present here though their prime factors are not. (Both sequences are closed under multiplication.) Asymptotic densities are 1/4 and 0 respectively.
%C A370268 Likewise, if we list the even terms halved, we find a similar relationship to the nonzero terms of A002480. The first 66 terms match, then we find we have generated intermittent extra terms: 221, 299, 323, ... .
%C A370268 Numbers whose squarefree part is congruent to {1,7} mod 24, {10,22} mod 48, {15,33} mod 72, or {6,42} mod 144. (Each congruence describes a coset of A334832 under A059897(.,.) as described in A334832. This sequence corresponds to the subgroup of the quotient group generated by {6,7,10}.)
%F A370268 {a(n) : n >= 1} = {A059897(i,j*k) : i in A334832, j in {1,7}, k in {1,6,10,15}}.
%o A370268 (PARI) isok(k) = {c = core(k); c%24 == 1 || c%24 == 7 || c%48 == 10 || c%48 == 22 || c%72 == 15 || c%72 == 33 || c%144 == 6 || c%144 == 42}
%Y A370268 Intersection of A189715 and A370267.
%Y A370268 A002481\{0}, A334832 are subsequences.
%Y A370268 Cf. A002480, A042999, A045309, A059897.
%K A370268 nonn,easy
%O A370268 1,2
%A A370268 _Peter Munn_, Feb 13 2024