

A168026


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


7



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
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OFFSET

1,2


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*k3) + A000217(2*k1) 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


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]


MATHEMATICA

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


PROG

(PARI) lista(nn) = {print1(1, ", "); for(k=1, nn, if(isprime(p=4*k^26*k+3), print1(p, ", "))); } \\ Altug Alkan, Mar 22 2018


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.
Sequence in context: A139832 A139848 A052029 * A142102 A297306 A247949
Adjacent sequences: A168023 A168024 A168025 * A168027 A168028 A168029


KEYWORD

easy,nonn


AUTHOR

Alonso del Arte, Nov 16 2009


STATUS

approved



