A336897 Infinite sum of the natural numbers, compacted (see comments for an explanation).

3, 7, 11, 24, 46, 29, 376, 134, 73, 158, 13504, 1388, 718, 734, 373, 758, 328192, 1667, 55456, 3602, 123712, 2063, 4138, 8324, 68896, 4442, 3831808, 3579392, 8017, 521408, 66328, 16622, 8317, 540608, 1130368, 18182, 36412, 73016, 36604, 9161, 295264, 9293, 74488, 74744, 150256, 37724, 5357056, 11489, 348602368

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Aug 07 2020

Keywords

• nonn

Comments

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).

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

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.

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

Examples

			The 1st term is 3 = 1 + 2.
The 2nd term is 7 = 3 + 4.
The 3rd term is 11 = 5 + 6.
The 4th term is 24 = 7 + 8 + 9.
The 5th term is 46 = 10 + 11 + 12 + 13.
The 6th term is 29 = 14 + 15, etc.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..200

Crossrefs

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

Programs

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