OFFSET

1,1

COMMENTS

Consider the initial terms of numerator sequences (dropping initial zeros) of

3; A005563=N(1) ,

5,3; A061037=N(2) ,

7,16,1; A061039=N(3) ,

9,5,33,3; A061041=N(4) ,

11,24,39,56,3; A061043=N(5) ,

13,7,5,4,85,1; A061045=N(6) ,

15,32,51,72,95,120,3; A061047=N(7) ,

17,9,57,5,105,33,161,3; A061049=N(8) ,

19,40,7,88,115,16,175,208,1; N(9),

21,11,69,6,1,39,189,14,261,3; N(10),

23,48,75,104,135,168,203,240,279,320,3; N(11)

One must add the following associated (minimum) squares (taken from squared entries in A172038) to these values to reach the next possible square not larger than the entry itself:

1; N(1)

4,1; N(2)

9,9,0; N(3)

16,4,16,1; N(4)

25,25,25,25,1; N(5)

36,9,4,0,36,0; N(6)

49,49,49,49,49,49,1; N(7)

64,16,64,4,64,16,64,1, ; N(8)

Only if the index of N(.) is a prime we obtain a string of equal consecutive terms in these complementary rows: 4, 9, 25, 49, 121, 169..

The current sequence lists the consecutive complementary squares, A001248, in the rows with prime index, including their multiplicity (which is A006093).

This generates a link between the primes and the Rydberg-Ritz spectrum of the hydrogen atom.

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Dec 09 2010

STATUS

approved