A214723 -id:A214723 - cp's OEIS frontend

A215113 a(n) is the number of different prime divisors of A214723(n).

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 3, 2, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 2, 2, 2, 3, 1, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3

Offset: 1

Vladimir Shevelev, Aug 03 2012

nonn

The records of a(n) are a(1)=1, a(3)=2, a(12)=3, a(132)=4,... Is the sequence unbounded?

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A214723.

nn = 2000; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ ps[[i]]^2 + ps[[j]]^2, {i, Length[ps]}, {j, i, Length[ps]}]]; t =
Select[t, # <= nn &]; PrimeNu[Sort[Transpose[Select[Tally[t], #[[2]] == 1 &]][[1]]]] (* G. C. Greubel, May 16 2017 *)

a(n) = A001221(A214723(n)). - Michel Marcus, Feb 08 2016

A215174 a(n) is the least term of A214723 having n different prime divisors, or 0 if no such term exists.

Original entry on oeis.org

8, 18, 130, 6890, 254930, 3352570, 683351890, 45139676530, 2744959126130

Offset: 1

Vladimir Shevelev, Zak Seidov, Robert G. Wilson v, Aug 05 2012

Conjectures. 1) For every n, a(n)>0; 2) for n>=3, a(n)==0 (mod 10).
If 1) is true, then A215113 is unbounded.
The corresponding positions m=m(n) such that A214723(m)=a(n) are 1,3,12,132,2074,18625,2138668. m(8) exists and the calculations show that it is >= 24866450, although up to now the exact its value is unknown.

Cf. A214723, A215113.

a(9) from Donovan Johnson, Aug 19 2012

A045636 Numbers of the form p^2 + q^2, with p and q primes.

Original entry on oeis.org

8, 13, 18, 29, 34, 50, 53, 58, 74, 98, 125, 130, 146, 170, 173, 178, 194, 218, 242, 290, 293, 298, 314, 338, 365, 370, 386, 410, 458, 482, 530, 533, 538, 554, 578, 650, 698, 722, 818, 845, 850, 866, 890, 962, 965, 970, 986, 1010, 1058, 1082, 1130, 1202, 1250

Offset: 1

Felice Russo

A045698(a(n)) > 0. - Reinhard Zumkeller, Jul 29 2012

All terms greater than 8 are of the form 8k+2 or 8k+5 (A047617). - Giuseppe Melfi, Oct 06 2022

			18 belongs to the sequence because it can be written as 3^2 + 3^2.

A214723 is a subsequence. Complement: A214879.
Cf. A214511 (least number having n orderless representations as p^2 + q^2).
Cf. A047617.

import Data.List (findIndices)
a045636 n = a045636_list !! (n-1)
a045636_list = findIndices (> 0) a045698_list
-- Reinhard Zumkeller, Jul 29 2012

q=13; imax=Prime[q]^2; Select[Union[Flatten[Table[Prime[x]^2+Prime[y]^2, {x,q}, {y,x}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
With[{nn=60},Take[Union[Total/@(Tuples[Prime[Range[nn]],2]^2)],nn]] (* Harvey P. Dale, Jan 04 2014 *)

list(lim)=my(p1=vector(primepi(sqrt(lim-4)),i,prime(i)^2), t, p2=List()); for(i=1,#p1, for(j=i,#p1, t=p1[i]+p1[j];if(t>lim, break, listput(p2,t)))); vecsort(Vec(p2),,8) \\ Charles R Greathouse IV, Jun 21 2012

from sympy import primerange
def aupto(limit):
    primes = list(primerange(2, int((limit-4)**.5)+2))
    nums = [p*p + q*q for i, p in enumerate(primes) for q in primes[i:]]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(1251)) # Michael S. Branicky, Aug 13 2021

A214511 Least number having n orderless representations as p^2 + q^2, where p and q are primes.

Original entry on oeis.org

8, 338, 2210, 10370, 202130, 229970, 197210, 81770, 18423410, 16046810, 12625730, 21899930, 9549410, 370247930, 416392730, 579994610, 338609570, 2155919090, 601741010, 254885930, 10083683090, 4690939370, 29207671610, 30431277890, 22264417370, 23231920010

Offset: 1

T. D. Noe, Jul 26 2012

nonn

A045698(a(n)) = n and A045698(m) < n for m < a(n). - Reinhard Zumkeller, Jul 29 2012

a(53) = 3374376505370. a(52) and terms following a(53) are greater than 4*10^13. - Giovanni Resta, Jul 02 2018

			a(2) = 338 because 338 = 7^2 + 17^2 = 13^2 + 13^2 and 338 is the least number with this property.

Giovanni Resta, Table of n, a(n) for n = 1..51
Index entries for sequences related to sums of squares

Cf. A088919, A045636, A214723.
Cf. A016032 (p and q integers).
Cf. A214512, A214513.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a214511 = (+ 1) . fromJust . (`elemIndex` a045698_list)
-- Reinhard Zumkeller, Jul 29 2012

nn = 10^6; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2, {i, Length[ps]}, {j, i, Length[ps]}]]; t = Select[t, # <= nn &]; t2 = Sort[Tally[t]]; u = Union[Transpose[t2][[2]]]; d = Complement[Range[u[[-1]]], u]; If[d == {}, nLim = u[[-1]], nLim = d[[1]]-1]; t3 = Table[Select[t2, #[[2]] == n &, 1][[1]], {n, nLim}]; Transpose[t3][[1]]

a(21)-a(26) from Donovan Johnson, Jul 29 2012

A045698 Number of ways n can be written as the sum of two squares of primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Felice Russo

a(A214879(n)) = 0; a(A045636(n)) > 0; a(A214723(n)) = 1; a(A214511(n)) = n and a(m) < n for m < A214511(n). - Reinhard Zumkeller, Jul 29 2012

The smallest value of n such that a(n) = 2 is 338. (This helps distinguish it from the characteristic function of A045636.) - Wesley Ivan Hurt, Jun 13 2013

			For example, a(29) = 1 because 29 = 2^2 + 5^2. a(3) = 0 because there is no way to write 3 as sum of two squares of primes.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A001248, A010051, A037213.

a045698 n = length $ filter (\x -> x > 0 && a010051' x == 1) $
map (a037213 . (n -)) $
takeWhile (<= div n 2) a001248_list
-- Reinhard Zumkeller, Jul 29 2012

a(n)=my(s=0,q);forprime(p=2,sqrtint(n\2),if(issquare(n-p^2,&q)&&isprime(q),s++));s \\ Charles R Greathouse IV, Jun 04 2014

More terms from Erich Friedman

A242230 Primes p of the form p^2 + q + 1 where p < q are consecutive primes.

Original entry on oeis.org

61, 4561, 9511, 17299, 19471, 26737, 30109, 37447, 49957, 69439, 94561, 196699, 209311, 259603, 317539, 333517, 352249, 414097, 427069, 459013, 678157, 845491, 886429, 943819, 1027189, 1217719, 1410163, 1472587, 1647379, 2165323, 2200777, 2230549, 2603389

Offset: 1

K. D. Bajpai, May 08 2014

nonn

			a(1) = 61 = 7^2 + 11 + 1: 61 is prime, 7 and 11 are consecutive primes.
a(2) = 4561 = 67^2 + 71 + 1: 4561 is prime, 67 and 71 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..7040

Cf. A242231, A000040, A241945, A045636, A214723, A214511.

with(numtheory): A242230:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)+1); if  isprime(k) then RETURN (k); fi;end: seq(A242230 (),x=1..500);

A242230 = {}; Do[p = Prime[n]^2 + Prime[n + 1] + 1; If[PrimeQ[p], AppendTo[A242230, p]], {n, 500}]; A242230
Select[#[[1]]^2+#[[2]]+1&/@Partition[Prime[Range[300]],2,1],PrimeQ] (* Harvey P. Dale, Mar 28 2016 *)

A242231 Primes p of the form p^2 + q - 1 where p < q are consecutive primes.

Original entry on oeis.org

13, 31, 59, 307, 383, 557, 997, 1409, 1723, 3541, 5113, 5407, 6323, 6977, 8017, 10303, 19469, 52673, 94559, 109897, 151717, 158009, 187927, 193163, 249503, 274069, 326617, 361807, 383791, 419261, 427067, 546863, 573809, 592133, 636017, 684757, 735307, 738743

Offset: 1

K. D. Bajpai, May 08 2014

nonn

			a(1) = 13 = 3^2 + 5 - 1: 13 is prime, 3 and 5 are consecutive primes.
a(2) = 31 = 5^2 + 7 - 1: 31 is prime, 5 and 7 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..6900

Cf. A242230, A000040, A241945, A045636, A214723, A214511.

with(numtheory): A242231:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)-1);if  isprime(k) then RETURN (k); fi;end: seq(A242231 (),x=1..500);

A242231 = {}; Do[p = Prime[n]^2 + Prime[n + 1] - 1; If[PrimeQ[p], AppendTo[A242231, p]], {n, 500}]; A242231
Select[#[[1]]^2+#[[2]]-1&/@Partition[Prime[Range[250]],2,1],PrimeQ] (* Harvey P. Dale, Mar 05 2022 *)

cp's OEIS Frontend

A215113 a(n) is the number of different prime divisors of A214723(n).

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A215174 a(n) is the least term of A214723 having n different prime divisors, or 0 if no such term exists.

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A045636 Numbers of the form p^2 + q^2, with p and q primes.

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A214511 Least number having n orderless representations as p^2 + q^2, where p and q are primes.

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A045698 Number of ways n can be written as the sum of two squares of primes.

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A242230 Primes p of the form p^2 + q + 1 where p < q are consecutive primes.

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A242231 Primes p of the form p^2 + q - 1 where p < q are consecutive primes.

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