A121326 -id:A121326 - cp's OEIS frontend

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

Alonso del Arte, Nov 16 2009

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

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. 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.

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

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

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

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.

A214517 Differences between the numbers n such that 4n^2 + 1 is prime.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 1, 5, 2, 7, 1, 5, 4, 5, 3, 2, 8, 3, 2, 2, 1, 2, 2, 6, 2, 3, 2, 5, 3, 2, 2, 10, 1, 2, 7, 3, 3, 2, 5, 3, 2, 2, 3, 5, 2, 8, 3, 4, 6, 7, 5, 17, 1, 5, 2, 3, 7, 5, 3, 2, 2, 10, 1, 2, 2, 8, 3, 20, 4, 6, 7, 3, 4, 5, 20, 1, 4, 1, 4, 10, 3, 3, 2, 3

Offset: 1

T. D. Noe, Aug 06 2012

nonn

Sequence A001912 has the values of n. This sequence is the first differences of A001912.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A121326 (primes of the form 1+4*n^2), A001912 (values of n).

Differences[Select[Range[100], PrimeQ[1 + 4*#^2] &, 101]]

a(n) = A214516(n)/2 for n > 1.

A090687 Primes of the form 6*k^2 + 1.

Original entry on oeis.org

7, 97, 151, 487, 601, 727, 2647, 3457, 4057, 7351, 9127, 9601, 11617, 12697, 14407, 15607, 17497, 20887, 21601, 29401, 33751, 37447, 39367, 42337, 47527, 53017, 54151, 71287, 77977, 80737, 84967, 95257, 102967, 110977, 126151, 142297, 151687

Offset: 1

Cino Hilliard, Dec 18 2003

Subset of A002476. See also A121326. The values of k for which 6*k^2 + 1 is prime are 1, 4, 5, 9, 10, 11, 21, 24, 26, 35, 39, 40, 44, 46, 49, 51, 54, 59, 60, 70, 75, 79, 81, 84, 89, 94, 95, 109, 114, 116, 119, 126, 131, 136, 145, 154, ... - Jonathan Vos Post, Aug 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A002476, A090686.

[a: n in [0..600] | IsPrime(a) where a is 6*n^2+1]; // Vincenzo Librandi, Dec 03 2011

Select[Table[6n^2+1,{n,0,1000}],PrimeQ] (* Vincenzo Librandi, Dec 03 2011 *)

mx2pmp(n) = { for(x=1,n, y = 6*x^2+1; if(isprime(y),print1(y",")) ) }

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 14 2008

A123669 Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0; or -1 if no such prime exists.

Original entry on oeis.org

5, 17, 37, -1, 101

Offset: 1

Alexander Adamchuk, Nov 15 2006

a(n) = -1 for n = {4, 16, 32, 64, 108, 256, 500, ...}.
All primes of the form 4*n^2 + 1 belong to a(n). They are listed in A121326(n).
Last digit of a(5k)>0 is 1.
Last digit of a(n)>0 is 7 if k is not congruent to 0 mod 5, except a(1) = 5.
All currently known a(n) for 6a(7)-a(8) = {197, 257}. a(10) = 401.
a(12)-a(18) = {577, 677, 614657, 185302018885184100000000000000000000000000000001, -1, 1336337, 1297}.
a(20) = 1601. a(22)-a(24) = {197352587024076973231046657, 4477457, 5308417}.
a(27)-a(28) = {2917, 3137}. a(32)-a(33) = {-1, 4357}.
a(37)-a(38) = {5477, 1238846438084943599707227160577}. a(40)-a(42) = {40960001, 45212177, 7057}.
a(44)-a(45) = {59969537, 8101}. a(47)-a(48) = {8837, 2708192040014184559945134363758220403329915059847434832829218817}.
a(51) = 355149324327687480512960334807820417442703411649746143408158197478603636302066719166373229531510062746472251495292613758147362817.
a(53) = 126247697.
a(55)-a(60) = {12101, 375817263084708503965641077546115954135779496817219617550715846657, 662148260948741787228316709317924977225312314678010411233675575297, 13457, 193877777, 14401}.
a(62)-a(67) = {153777, 15877, -1, 16901, 303595777, 17957}.
a(70)-a(71) = {384160001, 406586897}. a(73) = 21317.
a(75)-a(82) = {22501, 284936905588473857, 562448657, 24337, 150838912030874130174020868290707457, 25601, 2564253345083631031816684000763319514758972657894465952263290175258003723567069899841752707150583949000981132009709206360818037538528413351937, 723394817}.
a(85) = 28901. a(87)-a(88) = {916636177, 30977}. a(90) = 32401.
a(92) = 33857.
a(94)-a(95) = {2435149272410363768730097404205858817, 4791383378576850493153910080681360672521575296790233332710625780023370220270083429409686634957161195934369337557766908660231890537157173340981965932463779247224064100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001}.
a(97) = 1416468497. a(99) = 1536953617.
a(n) is currently unknown for n = {6, 9, 11, 19, 21, 25, 26, 29, 30, 31, 34, 35, 36, 39, 43, 46, 49, 50, 52, 54, 61, 68, 69, 72, 74, 83, 84, 86, 89, 91, 93, 96, 98, 100, ...}.

Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A121326.

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.

A121834 Primes p of the form 4n^2 + 1 such that 4p^2+1 is also prime.

Original entry on oeis.org

5, 37, 677, 1297, 2917, 8837, 13457, 50177, 147457, 156817, 246017, 341057, 414737, 746497, 1136357, 1726597, 1833317, 2119937, 2802277, 2808977, 3013697, 3549457, 3865157, 3896677, 4104677, 4384837, 5354597, 5410277, 5779217, 6031937, 6635777, 7001317

Offset: 1

Zak Seidov, Aug 28 2006

nonn

Intersection of A001912 and A121326. Except for the first term all other terms are == 7 (mod 10). Also all the primes 4*p^2+1 are == 7 (mod 10). - Zak Seidov, Mar 05 2015

Cf. A002496, A052291, A121326.

fpQ[n_]:=PrimeQ[n]&&PrimeQ[4n^2+1]; Select[4Range[2000]^2+1,fpQ] (* Harvey P. Dale, Nov 07 2016 *)

isA121834(n)={ if( (n-1) %4, return(0) ; ) ; if( issquare((n-1)/4), if( isprime(4*n^2+1), return(1), return(0) ), return(0) ; ) ; } { for(i=1,1000000, p=prime(i) ; if( isA121834(p), print1(p,",") ; ) ; ) ; } /* R. J. Mathar, Sep 01 2006*/

More terms from R. J. Mathar, Sep 01 2006

A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 10, 17, 20, 23, 44, 50, 56, 65, 76, 106, 144, 165, 173

Offset: 1

T. D. Noe, Aug 06 2012

			a(1) = 1 because 4*1^2 + 1 = 5 and 4*2^2 + 1 = 17 are primes.
a(2) = 2 because 4*3^2 + 1 = 37 is prime, 4*4^2 + 1 = 65 is composite, and 4*5^2 + 1 = 101 is prime.
a(3) = 5 because 4*13^2 + 1 is prime, 4*n^2 + 1 is composite for n = 14..17, and 4*18^2 + 1 is prime.

Cf. A121326 (primes of the form 1+4*n^2), A001912 (values of n).

Cf. A214517 (differences), A214519 (where record differences occur).

n = 1; last = 1; t = {1}; While[Length[t] < 15, n++; p = 1 + 4*n^2; If[PrimeQ[p], If[n - last > t[[-1]], AppendTo[t, n - last]]; last = n]]; t

A214519 Least number m such that 4m^2 + 1 is prime and the next prime of this form is 4(m + A214518(n))^2 + 1.

Original entry on oeis.org

1, 3, 13, 20, 47, 92, 175, 248, 1695, 1768, 22685, 41367, 49532, 178582, 420452, 1940278, 13957468, 20258760

Offset: 1

T. D. Noe, Aug 06 2012

Cf. A121326 (primes of the form 1+4*n^2), A214517, A214518 (record differences).

n = 1; last = 1; t = {1}; t2 = {1}; While[Length[t] < 10, n++; p = 1 + 4 n^2; If[PrimeQ[p], If[n - last > t[[-1]], AppendTo[t, n - last]; AppendTo[t2, last]]; last = n]]; t2

A121825 Duplicate of A049423.

Original entry on oeis.org

3, 7, 19, 67, 103, 199, 487, 787, 1447, 2503, 2707, 3847, 4099, 4903, 5479, 5779, 8467, 8839, 11239, 12547, 14887, 16903, 17959, 19603, 21319, 23719, 24967, 25603, 29587, 31687, 47527, 52903, 58567, 59539, 61507, 65539, 75079, 81799, 88807, 92419

Offset: 1

Jonathan Vos Post, Aug 27 2006

dead

See also A121326 (Primes of the form 4*k^2 + 1); see also A049423 (Primes of the form k^2 + 3). For the primes of the form 4*k^2 + 3, the corresponding values of k are 1, 2, 4, 5, 7, 11, 14, 19, 25, 26, 31, 32, 35, 37, 38, 46, 47, 53, 56, 61, 65, 67, 70, 73.

Cf. A000040, A002496, A049423, A121326.

[ a: n in [0..200] | IsPrime(a) where a is 4*n^2+3 ]; // Vincenzo Librandi, Dec 22 2010

a(n) = 3 + (2*A097697(n))^2. - R. J. Mathar, Aug 07 2008

Terms after 21319 added by R. J. Mathar, Aug 07 2008

3 added by Vincenzo Librandi, Dec 22 2010

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cp's OEIS Frontend

A168026 Noncomposite numbers in the southwestern 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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A214517 Differences between the numbers n such that 4n^2 + 1 is prime.

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A090687 Primes of the form 6*k^2 + 1.

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A123669 Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0; or -1 if no such prime exists.

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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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A121834 Primes p of the form 4*n^2 + 1 such that 4*p^2+1 is also prime.

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A214518 Record differences between the numbers n such that 4*n^2 + 1 is prime.

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A214519 Least number m such that 4*m^2 + 1 is prime and the next prime of this form is 4*(m + A214518(n))^2 + 1.

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A121834 Primes p of the form 4n^2 + 1 such that 4p^2+1 is also prime.

A214519 Least number m such that 4m^2 + 1 is prime and the next prime of this form is 4(m + A214518(n))^2 + 1.