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A336897
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Infinite sum of the natural numbers, compacted (see comments for an explanation).
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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Cf. A038550 (numbers that can be expressed as the sum of k>1 consecutive integers in only one way).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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