A082772 -id:A082772 - cp's OEIS frontend

A082867 Numbers n such that there exists a proper divisor 1 < m < n where m divides n and m+1 divides n+1, but there is no divisor d such that d divides n, d+1 divides n+1 and d+2 divides n+2.

8, 15, 20, 24, 27, 32, 35, 39, 44, 48, 51, 56, 65, 68, 75, 80, 84, 87, 90, 92, 95, 99, 104, 111, 116, 119, 120, 125, 128, 132, 135, 140, 143, 144, 147, 152, 155, 159, 164, 168, 171, 175, 176, 185, 188, 189, 195, 200, 204, 207, 212, 216, 219, 224, 231, 236, 245

Offset: 1

Anne M. Donovan (anned3005(AT)aol.com), May 24 2003

			8 is a member because 2 divides 8, 3 divides 9 (and 4 does not divide 10).

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

Cf. A082772.

is(n)=my(t); fordiv(n,m, if(m==1, next); if((n+1)%(m+1)==0, if(m==n, return(t)); t=1; if((n+2)%(m+2)==0, return(0)))) \\ Charles R Greathouse IV, Jul 05 2022

Corrected by T. D. Noe, Oct 25 2006

A355490 Numbers of the form a+b+c = a^2 - b^2 - c^2 where a > b >= c > 0.

Original entry on oeis.org

8, 15, 20, 24, 27, 32, 35, 39, 44, 48, 49, 51, 54, 55, 56, 63, 64, 65, 68, 75, 80, 84, 87, 90, 92, 95, 98, 99, 104, 111, 114, 116, 119, 120, 123, 125, 128, 132, 135, 140, 143, 144, 147, 152, 153, 155, 159, 160, 164, 168, 170, 171, 174, 175, 176, 183, 184, 185, 188, 189, 195, 200, 203, 204, 207, 208, 209, 212, 215, 216, 219, 220, 224, 230, 231

Offset: 1

Mohammad Arab, Jul 04 2022

nonn

It seems that A082867 is a subsequence.

The first counterexample to the above is A082867(60) = 258. - Charles R Greathouse IV, Jul 05 2022

			8 is a term: 8 = 4+2+2 = 4^2 - 2^2 - 2^2.
15 is a term: 15 = 7+5+3 = 7^2 - 5^2 - 3^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A082867, A082772, A134582, A355491.

Solve[a==r^2-s^2-d^2 && 1<=r<=120 && 1<=s<=120 && 1<=d<=120 && 0<=a && r>s>=d && a==r+s+d, {a,r,s,d}, Integers]

list(lim)=my(v=List([8]));lim\=1;for(a=3,lim-2,my(a2=a^2);for(b=(sqrt(2*a^2+2*a+1)-1)\2,a-2,my(t=a2-b^2-a-b,s);if(issquare(4*t+1,&s) && (c=(s-1)/2)<=b && c<=b && a+b+c<=lim, listput(v,a+b+c)))); Set(v) \\ Charles R Greathouse IV, Jul 05 2022

a(57) = 184 inserted by Charles R Greathouse IV, Jul 05 2022

cp's OEIS Frontend

A082867 Numbers n such that there exists a proper divisor 1 < m < n where m divides n and m+1 divides n+1, but there is no divisor d such that d divides n, d+1 divides n+1 and d+2 divides n+2.

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