A053664 Smallest number m such that m == i (mod prime(i)) for all 1<=i<=n.

1, 5, 23, 53, 1523, 29243, 299513, 4383593, 188677703, 5765999453, 5765999453, 2211931390883, 165468170356703, 8075975022064163, 361310530977154973, 20037783573808880093, 1779852341342071295513, 40235059344426324076913

Offset: 1

Author

Joe K. Crump (joecr(AT)carolina.rr.com), Feb 16 2000

Keywords

• nonn
• easy
• nice

Comments

Suggested by Chinese Remainder Theorem.

Examples

			a(3) = 23 because this is the smallest number m such that m == 1 (mod 2), m == 2 (mod 3) and m == 3 (mod 5).
a(4) = 53 because 53 - 1 is divisible by 2, 53 - 2 is divisible by 3, 53 - 3 is divisible by 5 and 53 - 4 is divisible by 7.
		

References

• Niven and Zuckerman, An Introduction to the Theory of Numbers, John Wiley, 1966, p. 40
• Paulo Ribenboim, The New Book of Prime Numbers Records, Springer 1996, p. 33

Links

• Nick Hobson and Robert G. Wilson v, Table of n, a(n) for n = 1..350 (first 100 terms from Nick Hobson)
• Project Euler, Problem 552: Chinese leftovers II

Crossrefs

Cf. A192363.

Programs

Mathematica

f[n_] := ChineseRemainder[ Range[n], Prime[Range[n]]]; Array[f, 20]

PARI

for(n=1,20,m=1; while(sum(i=1,n,abs(m%prime(i)-i))>0,m++); print1(m,","))

PARI

x=Mod(1, 1); for(i=1, 18, x=chinese(x, Mod(i, prime(i))); print1(component(x, 2), ", ")) /* Nick Hobson (nickh(AT)qbyte.org), Jan 08 2007 */

Python

from sympy.ntheory.modular import crt
from sympy import prime
def A053664(n): return int(crt([prime(i) for i in range(1,n+1)],list(range(1,n+1)))[0]) # Chai Wah Wu, May 01 2023

Extensions

Additional comments from Luis A. Rodriguez (luiroto(AT)yahoo.com), Apr 23 2002

Edited by N. J. A. Sloane and Robert G. Wilson v, May 03 2002