A035519 -id:A035519 - cp's OEIS frontend

A059799 Primes which when added to their reversals are squares.

2, 29, 47, 83, 263, 461, 20147, 23117, 24107, 63113, 80141, 81131, 300893, 301793, 303593, 308093, 310883, 313583, 324473, 333563, 336263, 342653, 344453, 348053, 350843, 354443, 355343, 356243, 362633, 363533, 364433, 365333, 377123, 378023

Offset: 1

Enoch Haga, Feb 23 2001

Idea from Carlos Rivera's The Prime Puzzles and Problems Connection, Conjecture 23.

			a(4)=263 because 263+362=625 and 625 is a square whose root is 25.

Harvey P. Dale, Table of n, a(n) for n = 1..683 (all such primes up to the 20 millionth prime)

Cf. A035519, A059798.

Cf. A227371 (indices of these primes).

Select[Prime[Range[33000]],IntegerQ[Sqrt[FromDigits[Reverse[ IntegerDigits[#]]]+ #]]&] (* Harvey P. Dale, Aug 18 2012 *)

Add prime to its reverse. If a square, add to sequence.

A059755 Odd rare numbers: odd n such that n-r and n+r are squares, where r is the reverse of n.

Original entry on oeis.org

65, 868591084757, 6979302951885, 6157577986646405, 8052956026592517, 8052956206592517, 8650327689541457, 8650349867341457, 619431353040136925, 619631153042134925, 631688638047992345, 633288858025996145, 633488632647994145, 653488856225994125

Offset: 1

Shyam Sunder Gupta, Feb 11 2001

There are 20 terms up to 10^22. - Shyam Sunder Gupta, Dec 14 2019

			65-56 = 9 and 65 + 56 = 121 are both (perfect) squares.

Shyam Sunder Gupta, "Systematic computations of rare numbers", The Mathematics Education, Vol. XXXII, No. 3, 1998.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..20
Carlos Rivera, Conjecture 23. The Shyam's conjecture about the Rare Numbers, The Prime Puzzles & Problems Connection.
Shyam Sunder Gupta, Rare Numbers
Shyam Sunder Gupta, Rare Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 14, 381-397. {{doi|10.1007/978-981-97-2465-9_14}}

Cf. A035519.

A282116 Numbers k such that k-1/2R(k) and k+1/2R(k) are both positive squares, where R(k) is the digits reverse of k.

Original entry on oeis.org

468, 4842, 27225, 235890, 21030930, 840827745

Offset: 1

Paolo P. Lava, Feb 16 2017

Similar to rare numbers A035519.
No more terms < 10^12. - Lars Blomberg, Jul 12 2017
a(7), if it exists, is larger than 2*10^15. - Giovanni Resta, Jul 14 2017

			(468 - 1/2*864)^(1/2) = (36)^(1/2) = 6  and (468 +1/2*864)^(1/2) = (900)^(1/2) = 30.

Cf. A004086, A035519.

R:=proc(w) local x,y,z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q,k) local n; for n from 1 to q do
if n>k*R(n) then if frac(sqrt(n-k*R(n)))=0 and frac(sqrt(n+k*R(n)))=0
then print(n); fi; fi; od; end: P(10^9,1/2);

a(6) from Lars Blomberg, Jul 12 2017

A332850 Numbers k = a^2 + b^2 such that reversal(k) = a^2 - b^2 for a > b > 0, where reversal is A004086.

Original entry on oeis.org

699796, 4854634, 6752626, 84036010, 931910661, 21584860960, 52554850525, 467170024564, 637843128736, 638730439636, 638734039636, 638943127636, 727830438745, 727834038745, 746710459825, 746754019825, 748943127625, 9894192267061, 401309596403104, 844181015028970

Offset: 1

Metin Sariyar, Feb 26 2020

When b=0, the palindromic numbers m = a^2 + b^2 such that reversal(m) = a^2 - b^2, are A002779 (palindromic squares).

a(19) > 3*10^14, if it exists. - Giovanni Resta, Feb 27 2020

			699796 = 836^2 + 30^2 and 697996 = 836^2 - 30^2.

Cf. A002779, A004086, A035519, A055096, A202386.

Do[If[IntegerReverse[a^2+b^2]==a^2-b^2,Print[{a^2+b^2,a,b}]],{a,1,50000},{b,1,a-1}]

isok(k) = {my(r = fromdigits(Vecrev(digits(k))), s = r+k, d = k-r); d && !(s % 2) && issquare(s/2) && !(d % 2) && issquare(d/2); } \\ Michel Marcus, Feb 27 2020

a(6)-a(18) from Giovanni Resta, Feb 27 2020

a(19)-a(20) from Jinyuan Wang, Apr 10 2025

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A059799 Primes which when added to their reversals are squares.

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A059755 Odd rare numbers: odd n such that n-r and n+r are squares, where r is the reverse of n.

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A282116 Numbers k such that k-1/2*R(k) and k+1/2*R(k) are both positive squares, where R(k) is the digits reverse of k.

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A332850 Numbers k = a^2 + b^2 such that reversal(k) = a^2 - b^2 for a > b > 0, where reversal is A004086.

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A282116 Numbers k such that k-1/2R(k) and k+1/2R(k) are both positive squares, where R(k) is the digits reverse of k.