A168026 Noncomposite numbers in the southwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

1, 7, 43, 73, 157, 211, 421, 601, 1483, 2551, 2971, 3907, 4423, 6163, 6481, 8191, 12211, 19183, 22651, 26407, 27061, 28393, 31153, 35533, 37057, 37831, 42643, 47743, 55933, 60763, 71023, 74257, 77563, 83233, 84391, 98911, 110557, 113233

Offset: 1

Author

Alonso del Arte, Nov 16 2009

Keywords

• easy
• nonn

Comments

From Peter Munn, Mar 17 2018: (Start)

Noncomposites of the form k^2 + k + 1 with k even and nonnegative (and the same values occur with k odd and negative). Equivalently, noncomposites of the form 4k^2 + 2k + 1 with k >= 0, or 4k^2 - 6k + 3 with k > 0.

A073337 lists those of the form k^2 + k + 1 with k odd and positive, and this is equivalently those of the form 4k^2 - 2k + 1 with k > 0.

(End)

Numbers that are the sum of A000217(2*k-3) + A000217(2*k-1) that result in either unity or a prime, for k,n >= 1. For k,n >= 0, a(n+1) = 4*k*2 + 2*k + 1 will give the same results. - J. M. Bergot, May 07 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Alonso del Arte, Ulam spiral (2009). [Note that "East" and "West" in this video match the cover of Scientific American, but "North" and "South" are switched.]
• BackIssues.com, Scientific American March 1964 back issue
• MathWorld, Prime Spiral
• Scientific American, March 1964 cover
• Wikipedia, Ulam Spiral

Crossrefs

Cf. A054569, all numbers of the form 4k^2 - 6k + 3 with k > 0. Noncomposites of the eastern ray are in A168022. Primes of the northeastern ray are in A073337. Noncomposites of the northern ray are in A168023. Primes of the northwestern ray are in A121326. Noncomposites of the western ray are in A168025. Noncomposites of the southern ray are in A168027.

Programs

Mathematica

Select[Table[4 n^2 - 6 n + 3, {n, 200}], Length[Divisors[ # ]] < 3 &]

PARI

lista(nn) = {print1(1, ", "); for(k=1, nn, if(isprime(p=4*k^2-6*k+3), print1(p, ", ")));} \\ Altug Alkan, Mar 22 2018

Formula

Numbers of the form 4k^2 - 6k + 3 with k > 0 and no more than two divisors. [corrected by Peter Munn, Mar 17 2018]