A336897 - OEIS

%I #23 Sep 20 2020 00:04:36

%S 3,7,11,24,46,29,376,134,73,158,13504,1388,718,734,373,758,328192,

%T 1667,55456,3602,123712,2063,4138,8324,68896,4442,3831808,3579392,

%U 8017,521408,66328,16622,8317,540608,1130368,18182,36412,73016,36604,9161,295264,9293,74488,74744,150256,37724,5357056,11489,348602368

%N Infinite sum of the natural numbers, compacted (see comments for an explanation).

%C The sequence is a subset of A038550, numbers that can be expressed as the sum of k>1 consecutive positive integers in only one way. If the successive terms of the present sequence are expressed as the sum of k>1 consecutive integers and added, the result will be 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11... (conjectured to extend ad infinitum).

%C Any sequence with this property has distinct positive terms. This is the lexicographically earliest sequence with this property.

%C The sequence behaves in a strange way: most of its terms are the sum of 2 or 3 consecutive integers, but sometimes huge "gaps" appear. The term a(65) = 34404982325248 is equal to 92911 + 92912 + 92913 + ... + 8295695 + 8295696 + 8295697; but a(66) = 16591397 is the sum of the next 2 consecutive terms only, 8295698 and 8295699. The term a(71) = 7893201574690816 is the sum of more than 10^8 consecutive integers(!):  8299930 + 8299931 + 8299932 + ... + 125917798.

%C Further indices of records are 77, 81, 89, 91, 101, 102, 106, 145, 149, 153, 157, 169, 173, 181, 191, 201, ... with a(201) ~ 10^562. - _M. F. Hasler_, Aug 29 2020

%H M. F. Hasler, <a href="/A336897/b336897.txt">Table of n, a(n) for n = 1..200</a>

%e The 1st term is 3 = 1 + 2.

%e The 2nd term is 7 = 3 + 4.

%e The 3rd term is 11 = 5 + 6.

%e The 4th term is 24 = 7 + 8 + 9.

%e The 5th term is 46 = 10 + 11 + 12 + 13.

%e The 6th term is 29 = 14 + 15, etc.

%o (PARI) (A336897_vec(N,s=0)=vector(N,n,my(o=s++);while(!is_A038550(o+=s++),);o)) (60) \\ slow for N >= 65. - _M. F. Hasler_, Aug 29 2020

%Y Cf. A038550 (numbers that can be expressed as the sum of k>1 consecutive integers in only one way).

%K nonn

%O 1,1

%A _Eric Angelini_ and _Jean-Marc Falcoz_, Aug 07 2020