Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

bibkey: sloane2017a008590 authors: N. J. A. Sloane; Lechoslaw Ratajczak year: 2017 title: “OEIS A008590, Multiples of 8” doi: null url: https://oeis.org/A008590 claim: “%N Multiples of 8. %C From Lechoslaw Ratajczak, Sep 03 2017: (Start) Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers. a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even). a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even). a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even). The conjecture was checked for 5*10^4 consecutive integers. (End)” strata_touched:

  • D5/S3/ArithSums/RatajczakGcdSumParityCharacterization license: citation-only triage: anchor

OEIS A008590

The NAME of A008590 states:

Multiples of 8.

The Ratajczak CONJECTURE is printed as:

From Lechoslaw Ratajczak, Sep 03 2017: (Start)

Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers.

a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even).

a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even).

a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even).

The conjecture was checked for 5*10^4 consecutive integers. (End)

The AUTHOR line is:

N. J. A. Sloane

The AUTHOR line carries no date; the year 2017 records the date of Ratajczak’s conjecture. The example at m = 24 fixes gcd_2(k,m) as gcd(k,m) divided by the least prime factor of the gcd, and lcd_2(k,m) as that least prime factor. Only the classification for m >= 2 is proved. The literal statement fails at the empty-sum value m = 1, which is disclosed rather than adjudicated.

Verified locator

  • URL: https://oeis.org/A008590
  • NAME (verbatim): Multiples of 8.
  • COMMENT attribution (verbatim): From Lechoslaw Ratajczak, Sep 03 2017: (Start)
  • COMMENT conjecture (verbatim): Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers.
  • COMMENT examples (verbatim): a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even). a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even). a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even).
  • COMMENT check (verbatim): The conjecture was checked for 5*10^4 consecutive integers. (End)
  • AUTHOR (verbatim): N. J. A. Sloane