A226562
Numbers which are the sum of two squared primes in exactly three ways (ignoring order).
Original entry on oeis.org
2210, 3770, 5330, 6290, 12818, 16490, 18122, 19370, 24050, 24650, 26690, 32810, 33410, 34970, 36530, 39650, 39770, 44642, 45050, 45890, 49010, 50690, 51578, 57770, 59450, 61610, 63050, 66170, 67490, 72410, 73610, 74210, 80330, 85202, 86210, 86330, 88010
Offset: 1
2210 = 19^2 + 43^2 = 23^2 + 41^2 = 29^2 + 37^2;
- Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.
Cf.
A214511 (least number having n representations).
Cf.
A226539 (restricted to sums decomposed in exactly three ways).
-
Prime2PairsSum := s -> select( x -> `if`(andmap(isprime, x), true, false), numtheory:-sum2sqr(s)):
for n from 2 to 10 do
if nops(Prime2PairsSum(n)) = 3 then print(n,Prime2PairsSum(n)) fi
od;
-
Select[Range@20000, Length[Select[ PowersRepresentations[#, 2, 2], And @@ PrimeQ[#] &]] == 3 &] (* Giovanni Resta, Jun 11 2013 *)
A226599
Numbers which are the sum of two squared primes in exactly four ways (ignoring order).
Original entry on oeis.org
10370, 10730, 11570, 12410, 13130, 19610, 22490, 25010, 31610, 38090, 38930, 39338, 39962, 40970, 41810, 55250, 55970, 59330, 59930, 69530, 70850, 73730, 76850, 77090, 89570, 98090, 98930, 103298, 118898, 125450, 126290, 130730, 135218, 139490
Offset: 1
10370 = 13^2 + 101^2 = 31^2 + 97^2 = 59^2 + 83^2 = 71^2 + 73^2.
10730 = 11^2 + 103^2 = 23^2 + 101^2 = 53^2 + 89^2 = 67^2 + 79^2.
- Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.
Cf.
A045636 (sum of two squared primes is a superset).
Cf.
A214511 (least number having n representations).
Cf.
A225104 (numbers having at least three representations is a superset).
-
Prime2PairsSum := s -> select(x ->`if`(andmap(isprime, x), true, false),
numtheory:-sum2sqr(s)):
for n from 2 to 10^6 do
if nops(Prime2PairsSum(n)) = 4 then print(n, Prime2PairsSum(n)) fi;
od;
-
(* Assuming mod(a(n),24) = 2 *) Reap[ For[ k = 2, k <= 2 + 240000, k = k + 24, pr = Select[ PowersRepresentations[k, 2, 2], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]; If[Length[pr] == 4 , Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 14 2013 *)
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