A168026 -id:A168026 - cp's OEIS frontend

A073337 Primes of the form 4k^2 - 10k + 7 with k positive.

3, 13, 31, 241, 307, 463, 757, 1123, 1723, 3307, 3541, 4831, 5113, 5701, 6007, 8011, 9901, 10303, 11131, 12433, 13807, 14281, 17293, 20023, 20593, 21757, 23563, 24181, 26083, 28057, 30103, 35911, 41413, 43891, 46441, 53593, 60271, 78121, 82657, 86143, 95791, 108571, 123553, 127807, 136531, 145543, 147073, 156421

Offset: 1

Zak Seidov, Aug 25 2002

Primes of the form k^2 + k + 1 with k odd and positive. - Peter Munn, Jan 27 2018

Primes of the form A000217(2*k) + A000217(2*k+2). - J. M. Bergot, May 09 2018

			3 is a term because for k=2, 4*k^2 - 10*k + 7 = 3 a prime.
7 is not a term because 7 can only be obtained with k=0 or k=5/2.

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A000217, A054554, A073338, A168026.

Subset of A002383.

Filtered(List([2..300],n->4*n^2-10*n+7),IsPrime); # Muniru A Asiru, Apr 15 2018

[a: n in [1..400] | IsPrime(a) where a is 4*n^2 - 10*n + 7]; // Vincenzo Librandi, Dec 23 2019

select(isprime, [seq(4*n^2-10*n+7 ,n=2..300)]); # Muniru A Asiru, Apr 15 2018

Select[Table[4 n^2 - 10 n + 7, {n, 1, 200}], PrimeQ] (* Vincenzo Librandi, Dec 23 2019 *)

select(isprime,vector(300,k,4*k^2 - 10*k + 7)) \\ Joerg Arndt, Feb 28 2018

a(n) = A054554(A073338(n)). - Elmo R. Oliveira, Apr 20 2025

Edited by Dean Hickerson, Aug 28 2002
a(1)=7 inserted and typo in Mathematica code corrected by Vincenzo Librandi, Dec 09 2011
Incorrect term 7 removed by Joerg Arndt, Feb 28 2018

A168022 Noncomposite numbers in the eastern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 2, 11, 53, 127, 233, 541, 743, 977, 1871, 3511, 4001, 4523, 5077, 9851, 11503, 12377, 14221, 16193, 19391, 20521, 21683, 22877, 24103, 29327, 30713, 33581, 42953, 55343, 57241, 63127, 67211, 80231, 84827, 91961, 101921, 104491, 123377

Offset: 0

Alonso del Arte, Nov 16 2009

Although 1 was not considered a prime number in Ulam's time, the March 1964 cover of Scientific American shows 1 highlighted in the same way as the primes.

G. C. Greubel, Table of n, a(n) for n = 0..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

Cf. A054552, all numbers of the form 4n^2 - 3n + 1. Primes of northeastern ray are in A073337. Noncomposites of the northern ray are in A168023. Noncomposites of the northwestern ray are in A168024. Noncomposites of the western ray are in A168025. Noncomposites of the southwestern ray are in A168026. Noncomposites of the southern ray are in A168027.

Select[Table[4 n^2 - 3 n + 1, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]

Positive numbers of the form 4n^2 - 3n + 1 with no more than two divisors.

A168023 Noncomposite numbers in the northern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 61, 139, 1009, 1279, 2281, 3109, 3571, 4591, 6361, 8419, 13399, 14341, 17359, 19531, 23029, 35251, 39901, 44839, 46549, 51871, 55579, 61381, 73849, 76039, 102241, 110059, 135241, 153469, 156619

Offset: 1

Alonso del Arte, Nov 16 2009

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

Cf. A054556, all numbers of the form 4n^2 - 9n + 6. Noncomposites of the eastern ray are in A168022. Primes of the northeastern ray are in A073337. Noncomposites of the northwestern ray are in A168024. Noncomposites of the western ray are in A168025. Noncomposites of the southwestern ray are in A168026. Noncomposites of the southern ray are in A168027.

Select[Table[4 n^2 - 9 n + 6, {n, 200}], Length[Divisors[ # ]] < 3 &]

Positive numbers of the form 4n^2 - 9n + 6 with no more than two divisors.

A168027 Noncomposite numbers in the southern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 23, 163, 281, 431, 613, 827, 2003, 2377, 3221, 3691, 6521, 7877, 10151, 10973, 11827, 12713, 17623, 18701, 23333, 24571, 25841, 27143, 28477, 38711, 43577, 45263, 48731, 50513, 65921, 72227, 81083, 85703, 95327, 97813, 102881, 124433

Offset: 0

Alonso del Arte, Nov 16 2009

G. C. Greubel, Table of n, a(n) for n = 0..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

Cf. A033951, all numbers of the form 4n^2 + 3n + 1. 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 southwestern ray are in A168026. There are no primes on the southeastern ray, which, being A016754, are the odd squares, and thus none of them are prime.

Select[Table[4 n^2 + 3 n + 1, {n, 0, 199}], Length[Divisors[ # ]] < 3 &]

Positive numbers of the form 4n^2 + 3n + 1 with no more than two divisors.

A168024 Noncomposite numbers in the northwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

Original entry on oeis.org

1, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857

Offset: 0

Alonso del Arte, Nov 16 2009

G. C. Greubel, Table of n, a(n) for n = 0..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

Essentially the same sequence as A002496, A121326, A163588.

Cf. A053755, all numbers of the form 4n^2 + 1. 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 (the same as this sequence but without the initial 1). Noncomposites of the western ray are in A168025. Noncomposites of the southwestern ray are in A168026. Noncomposites of the southern ray are in A168027.

Select[Table[4 n^2 + 1, {n, 0, 99}], Length[Divisors[ # ]] < 3 &]

Positive numbers of the form 4n^2 + 1 with no more than two divisors.

A187677 Primes of the form 8k^2 + 6k - 1 for positive k.

Original entry on oeis.org

13, 43, 89, 151, 229, 433, 701, 859, 1033, 1223, 1429, 1889, 2143, 2699, 3001, 3319, 4003, 4751, 5563, 7873, 10009, 11173, 11779, 12401, 13693, 17203, 18719, 19501, 21943, 25423, 27259, 28201, 30133, 31123, 33151, 36313, 38501, 39619, 41903, 46663, 49139, 51679

Offset: 1

Alonso del Arte, Mar 21 2011

In a variant of the Ulam spiral in which only odd numbers are entered, some primes still line up along some diagonals but not others. Without the even numbers, primes can also line up in horizontal and diagonal lines. This sequence comes from an upwards vertical line which starts with 13.
Primes of A091823. - Klaus Purath, Jan 03 2021
This is a subsequence of A162761. - Davide Rotondo, Jun 14 2025

Alonso del Arte, Ulam spiral (2009). [EmdrGreg's comment suggested the odd number spiral variant.]
OEIS Wiki, Ulam spiral

Cf. A073337 and A168026 are diagonals of the usual Ulam spiral which have some of the same primes as this vertical line.

Cf. A091823, A162761.

[ a: n in [0..2500] | IsPrime(a) where a is 8*n^2 + 6*n - 1 ]; // Vincenzo Librandi, Apr 24 2011

Select[Table[2((2n - 1)^2 - n) - 1, {n, 100}], PrimeQ]

lista(nn) = my(list=List(), p); for (n=1, nn, if(isprime(p=8*n^2+6*n-1), listput(list, p))); Vec(list); \\ Michel Marcus, Jun 14 2025

a(n) = 2((2n - 1)^2 - n) - 1 (or, find the number in the corresponding spot in the better-known Ulam spiral, double it and subtract 1).

The polynomial 8n^2 - 10n + 1 produces the same primes.

cp's OEIS Frontend

A073337 Primes of the form 4*k^2 - 10*k + 7 with k positive.

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A168022 Noncomposite numbers in the eastern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

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A168023 Noncomposite numbers in the northern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

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A168027 Noncomposite numbers in the southern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

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A168024 Noncomposite numbers in the northwestern ray of the Ulam spiral as oriented on the March 1964 cover of Scientific American.

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A187677 Primes of the form 8*k^2 + 6*k - 1 for positive k.

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A073337 Primes of the form 4k^2 - 10k + 7 with k positive.

A187677 Primes of the form 8k^2 + 6k - 1 for positive k.