A061049 -id:A061049 - cp's OEIS frontend

A348762 a(n) = A000265((n-8)*(n+8)).

17, 9, 57, 5, 105, 33, 161, 3, 225, 65, 297, 21, 377, 105, 465, 1, 561, 153, 665, 45, 777, 209, 897, 15, 1025, 273, 1161, 77, 1305, 345, 1457, 3, 1617, 425, 1785, 117, 1961, 513, 2145, 35, 2337, 609, 2537, 165, 2745, 713, 2961, 3, 3185, 825, 3417, 221, 3657

Offset: 9

Simon Strandgaard, Oct 31 2021

nonn

Shares 495 initial terms with A061049. First difference is A061049(504)=62 vs. a(504)=31.

			a( 9) = A000265(( 9-8)*( 9+8)) = A000265( 17) = 17,
a(10) = A000265((10-8)*(10+8)) = A000265( 36) = 9,
a(11) = A000265((11-8)*(11+8)) = A000265( 57) = 57,
a(12) = A000265((12-8)*(12+8)) = A000265( 80) = 5,
a(13) = A000265((13-8)*(13+8)) = A000265(105) = 105.

Cf. A061049, A069834, A000265, A098849.

a[n_] := (n - 8)*(n + 8)/2^IntegerExponent[(n - 8)*(n + 8), 2]; Array[a, 53, 9] (* Amiram Eldar, Nov 22 2021 *)

A000265(n) = n >> valuation(n, 2);
a(n) = A000265((n-8)*(n+8));
[a(n)|n<-[9..27]]

def A348762(n):
    a, b = divmod(n*n-64, 2)
    while b == 0:
        a, b = divmod(a,2)
    return 2*a+b # Chai Wah Wu, Dec 05 2021

p (9..27).map { |n| x = (n-8)*(n+8); x /= 2 while x.even?; x }

a(n) = A000265(A098849(n-8)).

A177083 A006093(k)-fold repetition of A001248(k), k=1,2,3,..

Original entry on oeis.org

4, 9, 9, 25, 25, 25, 25, 49, 49, 49, 49, 49, 49, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169, 169

Offset: 1

Paul Curtz, Dec 09 2010

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.

Cf. A072055, A135177.

cp's OEIS Frontend

A348762 a(n) = A000265((n-8)*(n+8)).

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