A212409 Numbers m such that m, m' and m'' are in arithmetic progression, where m' and m'' are the first and second arithmetic derivatives of m.

2, 4, 8, 27, 54, 156, 380, 476, 782, 861, 1053, 1976, 2542, 2565, 3125, 3213, 3368, 6250, 8732, 13338, 13724, 22734, 42716, 46136, 51640, 56156, 56444, 58941, 64796, 67196, 92637, 115198, 121875, 251516, 261598, 288333, 296875, 311418, 348570, 371875, 379053

Offset

1,1

Comments

A051674 is a subsequence of this sequence.

If we consider also the third derivative, the numbers for which m''' - m'' = m'' - m' = m' - m are 4, 27, 380, 2565, 3125, 296875, 696764, 823543, ...

Solution of m'' - 2m' + m = 0. - Giorgio Balzarotti, Nov 11 2013

Links

Donovan Johnson, Table of n, a(n) for n = 1..300

Example

For m=8732, we have m'=9116, m''=9500, and 8732 - 9116 = 9116 - 9500 = -384, so 8732 is a term.
For m=115198, we have m'=58559, m''=1920, and 115198 - 58559 = 58559 - 1920 = 56639, so 115198 is a term.

Maple

with(numtheory);
A212409:= proc(n)
local a, b, i, p, pfs;
for i from 1 to n do
  pfs:=ifactors(i)[2];  a:=i*add(op(2, p)/op(1, p), p=pfs) ;
  pfs:=ifactors(a)[2];  b:=a*add(op(2, p)/op(1, p), p=pfs) ;
  if b-a=a-i then print(i); fi;
od;
end:
A212409(1000000);

Crossrefs

Cf. A003415, A051674.

Sequence in context: A196265 A112285 A037170 * A006399 A326293 A324328

Adjacent sequences: A212406 A212407 A212408 * A212410 A212411 A212412

Keyword

nonn

Author

Paolo P. Lava, May 15 2012

Status

approved