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mathematical analysis > calculus > sequence > integer sequence > Zeisel number

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Zeisel number

A Zeisel number, named after Helmut Zeisel, is a square-free integer k with at least three prime factors which fall into the pattern
${\\displaystyle p_{x}=ap_{x-1}+b}$
where a and b are some integer constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest. For the purpose of determining Zeisel numbers, ${\\displaystyle p_{0}=1}$ . The first few Zeisel numbers are
105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979, 294409, 336611, 353977, 448585, 507579, 982513, 1012121, 1073305, 1242709, 1485609, 2089257, 2263811, 2953711, … (sequence A051015 in the OEIS).
To give an example, 1729 is a Zeisel number with the constants a = 1 and b = 6, its factors being 7, 13 and 19, falling into the pattern
${\\displaystyle {\egin{aligned}p_{1}=7,&{}\\quad p_{1}=1p_{0}+6\\\\p_{2}=13,&{}\\quad p_{2}=1p_{1}+6\\\\p_{3}=19,&{}\\quad p_{3}=1p_{2}+6\\end{aligned}}}$
1729 is an example for Carmichael numbers of the kind ${\\displaystyle (6n+1)(12n+1)(18n+1)}$ , which satisfies the pattern ${\\displaystyle p_{x}=ap_{x-1}+b}$ with a= 1 and b = 6n, so that every Carmichael number of the form (6n+1)(12n+1)(18n+1) is a Zeisel number.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Zeisel_number)

nombre de Zeisel

French

URI

http://data.loterre.fr/ark:/67375/PSR-CKP72PV4-6

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