A036468 -id:A036468 - cp's OEIS frontend

A099468 Numbers n such that there are no primes < 2n in the sequence m(0)=n, m(k+1)=m(k)+4k.

1, 21, 45, 51, 81, 213, 249, 315, 477, 525, 681, 891, 1143, 1221, 1851, 1965, 2415, 5133

Offset: 1

T. D. Noe, Oct 17 2004

nonn

No others < 10^8. Note that 3 divides all these n > 1. This sequence is conjectured to be complete. Related to a question posed in A036468 by Zhang Ming-Zhi. Let r=2s+1 be an odd number. If n = (s+1)^2+s^2, then the sequence m(0)=n, m(k+1)=m(k)+4k for k=0,1,...s calculates the s+1 distinct sums of two squares (r-i)^2+i^2.

			45 is here because 45, 49, 57, 69 and 85 are all composite.

Cf. A036468 (number of ways to represent 2n+1 as a+b with a^2+b^2 prime).

lst={}; Do[n=m; found=False; k=0; While[n=n+4k; !found && n<2m, found=PrimeQ[n]; k++ ]; If[ !found, AppendTo[lst, m]], {m, 1, 10000, 2}]; lst

A102751 Numbers k such that 1 + (k-1)^2 and ((k-1)/2)^2 + ((k+1)/2)^2 = (1/2)*(k^2+1) are primes.

Original entry on oeis.org

3, 5, 11, 15, 25, 85, 95, 121, 131, 171, 181, 205, 231, 261, 271, 315, 441, 445, 471, 545, 571, 595, 715, 751, 781, 861, 921, 951, 1011, 1055, 1081, 1095, 1125, 1151, 1185, 1315, 1411, 1421, 1495, 1615, 1661, 1701, 2035, 2051, 2055, 2065, 2175, 2261, 2315

Offset: 1

Robin Garcia, Feb 09 2005

nonn

Conjectured to be infinite.

			11 is a term because 10^2+1=101 and 5^2+6^2=(1/2)*(11^2+1)=61 are primes.

G. H. Hardy and W. M. Wright, Unsolved Problems Concerning Primes, Section 2.8 and Appendix 3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, p. 19.
P. Ribenboim, The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, pp. 206-208, 1996.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A036468, A002496, A005574.

a:=proc(n) if isprime(1+(n-1)^2)=true and type((n^2+1)/2,integer)=true and isprime((n^2+1)/2)=true then n else fi end: seq(a(n),n=1..3000); # Emeric Deutsch, May 31 2005

Select[Range[2,2500], PrimeQ[1+(#-1)^2]&&PrimeQ[(1/2)*(#^2+1)]&] (* James C. McMahon, Jan 10 2024 *)

More terms from Emeric Deutsch, May 31 2005

A127079 Number of ways to represent prime(n) as a+b with a >= b > 0 and a^2+b^2 prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 6, 5, 8, 9, 7, 7, 9, 9, 11, 11, 9, 11, 13, 15, 14, 14, 18, 16, 17, 16, 20, 18, 22, 18, 21, 23, 21, 24, 24, 22, 24, 22, 28, 30, 23, 27, 24, 29, 30, 30, 28, 29, 24, 28, 30, 34, 33, 36, 35, 31, 37, 32, 36, 37, 41, 42, 42, 42, 43, 42, 38, 34, 43, 38, 45, 44

Offset: 1

J. M. Bergot, Mar 24 2007

nonn

Essentially A036468 restricted to the primes.

a(n) <= floor(prime(n)/2).

			prime(5) = 11 can be represented as 10+1, 9+2, 8+3, 7+4 and 6+5. Among 10^2+1^2 = 101, 9^2+2^2 = 85, 8^2+3^2 = 73, 7^2+4^2 = 65 and 6^2+5^2 = 61 are three primes, hence a(5) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A036468.

f:= proc(n) local a;
add(charfcn[{true}](isprime(a^2 + (n-a)^2)), a=1..n/2)
end proc:
map(f, [seq(ithprime(i),i=1..100)]); # Robert Israel, Jun 03 2019

Reap[Do[p = Prime[n]; c = 0; Do[b = p - a; If[PrimeQ[a^2 + b^2], c++], {a, 1, p/2}]; Sow[c], {n, 1, 75}]][[2, 1]] (* Jean-François Alcover, Aug 19 2020 *)

{for(n=1, 75, p=prime(n); c=0; for(a=1, p\2, b=p-a; if(isprime(a^2+b^2), c++)); print1(c, ","))} /* Klaus Brockhaus, Mar 26 2007 */

Edited and extended by Klaus Brockhaus, Mar 26 2007

A192056 a(n) = |{0

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 2, 1, 3, 1, 2, 2, 3, 3, 2, 1, 3, 4, 2, 6, 2, 1, 8, 3, 3, 6, 2, 1, 3, 3, 1, 5, 7, 5, 4, 4, 3, 3, 6, 3, 3, 6, 3, 5, 3, 7, 5, 7, 6, 4, 5, 1, 8, 8, 2, 4, 6, 1, 5, 2, 4, 9, 8, 3, 6, 7, 3, 5, 5, 5, 3, 3, 5, 9, 4, 13, 6, 5, 9, 7, 7, 3, 10, 9, 8, 9, 7, 4, 7, 13, 5, 7, 10, 4, 4, 11, 4, 5

Offset: 1

Zhi-Wei Sun, Dec 30 2012

nonn

Conjecture: a(n)>0 for all n>2. Moreover, if n>2 is not among 12, 18, 105, 522, then there is 0

k^2+(n-k)^2-3(n-1 mod 2) is prime and also JacobiSymbol[k,p]=1 for any prime divisor p of n+3(n-1 mod 2).

This conjecture has been verified for n up to 2*10^8. It is stronger than Ming-Zhi Zhang's conjecture that any odd integer n>1 can be written as x+y (x,y>0) with x^2+y^2 prime (see A036468).

			a(33)=1 since 4^2+29^2=857 is prime and JacobiSymbol[4,33]=1.
a(24)=1 since 10^2+14^2-3=293 is prime and JacobiSymbol[10,24+3]=1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A036468, A191004, A185150, A220554.

a[n_]:=a[n]=Sum[If[PrimeQ[k^2+(n-k)^2-3Mod[n-1,2]]==True&&JacobiSymbol[k,n+3Mod[n-1,2]]==1,1,0],{k,1,(n-1)/2}]
Do[Print[n," ",a[n]],{n,1,100}]

A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 5, 4, 1, 2, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 4, 2, 4, 3, 6, 3, 2, 4, 2, 5, 6, 4, 5, 4, 5, 5, 8, 7, 4, 7, 7, 6, 7, 4, 6, 7, 5, 6, 3, 11, 7, 1, 5, 3, 5, 6, 6, 10, 4, 13, 12, 9, 4, 9, 10, 5, 8, 3, 6, 7, 5, 4, 8, 13, 6, 3, 5, 5, 11, 6, 13, 4, 9, 10, 8, 12, 11, 8, 7, 10, 8, 7, 8, 8

Offset: 1

Zhi-Wei Sun, Dec 16 2012

nonn

Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.
This has been verified for n up to 2*10^6.
Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.
Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.

			a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.

a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True,1,0],{k,0,2n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A231883 Number of ways to write n = x + y (x, y > 0) with x^2 + (n-2)*y^2 prime.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 4, 1, 5, 2, 5, 1, 4, 4, 3, 1, 7, 2, 3, 3, 6, 7, 3, 2, 6, 2, 9, 3, 8, 3, 10, 3, 5, 8, 8, 4, 7, 5, 13, 4, 12, 6, 7, 6, 8, 10, 14, 4, 17, 9, 9, 6, 9, 5, 8, 5, 9, 7, 12, 10, 11, 7, 11, 8, 12, 4, 13, 3, 22, 6, 16, 7, 14, 8, 10, 4, 14, 4, 17, 9, 16, 6, 12, 11, 14, 4, 21, 4, 21, 8, 18, 3, 11, 14, 23, 7, 22, 5, 23, 8

Offset: 1

Zhi-Wei Sun, Nov 21 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 8, 12, 16. Moreover, if m and n are positive integers with m >= max{2, n-1} and gcd(m, n+1) = 1, then x^2 + n*y^2 is prime for some positive integers x and y with x + y = m, except for the case m = n + 3 = 29.
(ii) Let m and n be integers greater than one with m >= (n-1)/2 and gcd(m, n-1) = 1. Then x + n*y is prime for some positive integers x and y with x + y = m.
(iii) Any integer n > 3 not equal to 12 or 16 can be written as x + y (x, y > 0) with (n-2)*x - y and (n-2)*x^2 + y^2 both prime.

			 a(8) = 1 since 8 = 5 + 3 with 5^2 + (8-2)*3^2 = 79 prime.
a(12) = 1 since 12 = 11 + 1 with 11^2 + (12-2)*1^2 = 131 prime.
a(16) = 1 since 16 = 15 + 1 with 15^2 + (16-2)*1^2 = 239 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A036468, A219842, A219864, A220417, A232174, A232186, A232194.

a[n_]:=Sum[If[PrimeQ[x^2+(n-2)*(n-x)^2],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

A115397 Erroneous version of A218656.

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 1, 5, 4, 4, 4, 5, 4, 7, 6, 5, 2, 9, 2, 5, 5, 2, 4, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Mar 08 2006

dead

			n=6: 2*6+1 = 13 = 2+11 = 5+8 = 6+7 and
prime(1717)=14657, prime(636)=4721, prime(516)=3697,
a(6) = #{14657=2^4+11^4, 4721=5^4+8^4, 3697=6^4+7^4} = 3.

Cf. A036468, A005408.

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A099468 Numbers n such that there are no primes < 2n in the sequence m(0)=n, m(k+1)=m(k)+4k.

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A102751 Numbers k such that 1 + (k-1)^2 and ((k-1)/2)^2 + ((k+1)/2)^2 = (1/2)*(k^2+1) are primes.

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A127079 Number of ways to represent prime(n) as a+b with a >= b > 0 and a^2+b^2 prime.

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A192056 a(n) = |{0

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A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

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A231883 Number of ways to write n = x + y (x, y > 0) with x^2 + (n-2)*y^2 prime.

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A115397 Erroneous version of A218656.

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