A049296 First differences of A008364. Also first differences of reduced residue system (RRS) for 4th primorial number, A002110(4)=210.

10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10, 2, 4, 2, 4, 6, 2

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• easy
• nice

Comments

First differences of reduced residue systems modulo primorial numbers are essentially palindromic + 1 separator term (2). The palindromic part starts and ends with p_(n+1)-1 for the n-th primorial number.

This sequence has period A005867(4)=A000010(A002110(4))=48. The 0th, first, 2nd and 3rd similar difference sequences are as follows: {1},{2},{4,2},{6,4,2,4,2,4,6,2} obtained from reduced residue systems of consecutive primorials.

Difference sequence of the "4th diatomic sequence" - A. de Polignac (1849), J. Dechamps (1907).

References

• Dickson L. E., History of the Theory of Numbers, Vol. 1, p. 439, Chelsea, 1952.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005867, A008364, A002110, A001223.

Programs

Haskell

a049296 n = a049296_list !! (n-1)
a049296_list = zipWith (-) (tail a008364_list) a008364_list
-- Reinhard Zumkeller, Jan 06 2013

Mathematica

t1=Table[ GCD[ w, 210 ], {w, 1, 210} ] /t2=Flatten[ Position[ t1, 1 ] ] /t3=Mod[ RotateLeft[ t2 ]-t2, 210 ]
Differences[Select[Range[600],GCD[#,210]==1&]] (* Harvey P. Dale, Jan 13 2012 *)

Extensions

Corrected by Frederic Devaux, Feb 02 2007