A104242 -id:A104242 - cp's OEIS frontend

A104301 Primes which are the reverse concatenation of two consecutive square numbers.

41, 6449, 196169, 576529, 11561089, 14441369, 27042601, 38443721, 51845041, 60845929, 67246561, 84648281, 1081610609, 2073620449, 2190421609, 2822427889, 3240032041, 4000039601, 4326442849, 5017649729, 5290052441, 6250062001, 7507674529, 8294482369, 103684103041

Offset: 1

Shyam Sunder Gupta, Apr 17 2005

			The first term is 41 which is a prime and is the reverse concatenation of 1 and 4 which are two consecutive square numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

These are the primes in A246972.

Cf. A104242, A104302, A104303, A104304.

cat[s_] := FromDigits[Flatten[IntegerDigits[s]]]; Select[cat /@ Reverse /@ Partition[Range[350]^2, 2, 1], PrimeQ] (* Amiram Eldar, Jul 15 2025 *)

from sympy import isprime
A104301_list = []
for n in range(1, 2000):
    x = int(str((n+1)**2)+str(n**2))
    if isprime(x):
        A104301_list.append(x) # Chai Wah Wu, Sep 13 2014

A246973 n^2 concatenated with (n+1)^2.

Original entry on oeis.org

1, 14, 49, 916, 1625, 2536, 3649, 4964, 6481, 81100, 100121, 121144, 144169, 169196, 196225, 225256, 256289, 289324, 324361, 361400, 400441, 441484, 484529, 529576, 576625, 625676, 676729, 729784, 784841, 841900, 900961, 9611024, 10241089, 10891156, 11561225, 12251296, 12961369, 13691444

Offset: 0

N. J. A. Sloane, Sep 13 2014

			a(2) = 49 because 2^2 = 4 and 3^2 = 9.
a(3) = 916 because 3^2 = 9 and 4^2 = 16.
a(4) = 1625 because 4^2 = 16 and 5^2 = 25.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

For primes see A104242.

Cf. A235497.

[1] cat [Seqint(Intseq(n^2+2*n+1) cat Intseq(n^2)): n in [1..50]]; // Vincenzo Librandi, Sep 13 2014

a:= n-> parse(cat(n^2, (n+1)^2)):
seq(a(n), n=0..40);  # Alois P. Heinz, May 27 2018

Table[FromDigits[Join[IntegerDigits[n^2], IntegerDigits[(n + 1)^2]]], {n, 0, 39}] (* Alonso del Arte, Sep 13 2014 *)

a(n) = eval(Str(n^2,(n+1)^2)) \\ Michel Marcus, Sep 13 2014 and M. F. Hasler, May 27 2018

A246973(n)=n^2*10^logint(10*(n+1)^2,10)+(n+1)^2 \\ Over 4 x faster than using eval(Str(...)). - M. F. Hasler, May 27 2018

A090738 Integers n such that the concatenation of n^2 and (n+1)^2 is prime.

Original entry on oeis.org

8, 12, 18, 20, 28, 40, 48, 82, 88, 92, 96, 98, 112, 128, 132, 140, 142, 218, 232, 238, 240, 246, 252, 272, 286, 288, 330, 332, 346, 356, 360, 376, 380, 450, 458, 460, 462, 466, 488, 500, 518, 532, 538, 550, 588, 590, 596, 602, 610, 612, 616, 630, 640, 646, 648

Offset: 1

Alex Kontorovich (alexk(AT)math.columbia.edu), Jan 19 2004

I conjecture this sequence to be infinite. Searching through the first 200000 values, I found 7000 primes, of which over 400 were "twins", i.e. both n^2*(n+1)^2 and (n+2)^2*(n+3)^2 were prime, where "*" denotes concatenation. I conjecture there to be an infinitude of such twins and the obvious generalizations.

The symmetric problem, i.e., finding two consecutive primes whose concatenation is a square, is somehow harder. Probably the smallest such primes are p = 411828016678198512725064549221 and its successor p+20, whose concatenation is equal to 641738277398347583345401533579^2. - Giovanni Resta, Jul 23 2015

			The first term, n=8 corresponds to the prime 6481, which is the concatenation of 8^2=64 and 9^2=81. The second term, n=12 corresponds to the prime 144169.

Zak Seidov, Table of n, a(n) for n = 1..10000

See A104242 for the corresponding primes.

For[i=1, i<200000, i=i+1, n=2i;e=IntegerPart[2 Log[10, n+1]]+1;x=10^e n^2 + (n+1)^2;y={n, x}; If[ PrimeQ[x], Save["primes.txt", y]]]
Select[Range@ 648, PrimeQ@ FromDigits[IntegerDigits[#^2] ~Join~ IntegerDigits[(# + 1)^2]] &] (* Michael De Vlieger, Jul 23 2015 *)
Position[FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[ Range[ 700]^2, 2,1],?PrimeQ]//Flatten (* _Harvey P. Dale, Dec 23 2018 *)

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A104301 Primes which are the reverse concatenation of two consecutive square numbers.

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A246973 n^2 concatenated with (n+1)^2.

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A090738 Integers n such that the concatenation of n^2 and (n+1)^2 is prime.

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Mathematica