login
Integers that can be expressed as the sum of two or more positive consecutive numbers (the largest being even) AND also as the sum of two or more positive consecutive numbers (the largest being odd).
1

%I #15 May 02 2023 03:03:52

%S 9,15,18,21,27,30,33,35,36,39,42,45,49,51,54,55,57,60,63,66,69,70,72,

%T 75,77,78,81,84,87,90,91,93,95,98,99,102,105,108,110,111,114,115,117,

%U 119,120,121,123,126,129,132,133,135,138,140,141,143,144,147,150,153,154

%N Integers that can be expressed as the sum of two or more positive consecutive numbers (the largest being even) AND also as the sum of two or more positive consecutive numbers (the largest being odd).

%C Intersection of A177732 and A177731.

%C From _Robert Israel_, May 02 2023: (Start)

%C Numbers k with odd divisors d_1, d_2 >= 2 such that k + (d_1+1)/2 is odd and

%C k + (d_2+1)/2 is even.

%C Contains no primes, powers of 2 or products of a prime and a power of 2.

%C Contains odd semiprime p*q iff at least one of p and q == 3 (mod 4).

%C (End)

%H Robert Israel, <a href="/A177733/b177733.txt">Table of n, a(n) for n = 1..10000</a>

%e 9 is in the sequence because 2+3+4=9=4+5.

%e 15 is in the sequence because 7+8=15=1+2+3+4+5.

%p filter:= proc(n) local a,b,x,y,todd,teven;

%p todd:= false; teven:= false;

%p for a in select(type,numtheory:-divisors(n),odd) minus {1} do

%p b:= 2*n/a;

%p x:= (a+b+1)/2;

%p if x::odd then todd:= true; if teven then return true fi

%p else teven:= true; if todd then return true fi

%p fi od:

%p false

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, May 01 2023

%t z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1; z=200;lst2={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst2,c]],{b,a-1,1,-1}],{a,2,z,2}]; Intersection[lst1,lst2]

%Y Cf. A000217, A177713.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, May 12 2010