A104301 -id:A104301 - cp's OEIS frontend

A246972 (n+1)^2 concatenated with n^2.

10, 41, 94, 169, 2516, 3625, 4936, 6449, 8164, 10081, 121100, 144121, 169144, 196169, 225196, 256225, 289256, 324289, 361324, 400361, 441400, 484441, 529484, 576529, 625576, 676625, 729676, 784729, 841784, 900841, 961900, 1024961, 10891024, 11561089, 12251156, 12961225, 13691296

Offset: 0

N. J. A. Sloane, Sep 13 2014

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

Cf. A246973, A235497. For primes, see A104301.

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

Table[FromDigits[Join[Flatten[IntegerDigits[{(n + 1)^2, n^2}]]]], {n, 0, 50}] (* Vincenzo Librandi, Sep 13 2014 *)
FromDigits/@(Join[IntegerDigits[#[[2]]],IntegerDigits[#[[1]]]]&/@ Partition[ Range[0,40]^2,2,1]) (* Harvey P. Dale, Apr 16 2015 *)

def A246972(n):
    return int(str((n+1)**2)+str(n**2)) # Chai Wah Wu, Sep 13 2014

A104242 Primes which are the concatenation of two consecutive square numbers.

Original entry on oeis.org

6481, 144169, 324361, 400441, 784841, 16001681, 23042401, 67246889, 77447921, 84648649, 92169409, 96049801, 1254412769, 1638416641, 1742417689, 1960019881, 2016420449, 4752447961, 5382454289, 5664457121, 5760058081, 6051661009

Offset: 1

Shyam Sunder Gupta, Apr 17 2005

The second prime in this sequence, 144169, arises in the theory of modular forms, as observed by Hecke. On page 671 of Hecke (1937), Hecke works out the cusp forms of weight 24 and observes that the Hecke operators have eigenfunctions with Fourier coefficients in the quadratic field of discriminant 144169. Thanks to Jerrold B. Tunnell for this comment. See also the articles by Hida and Zagier. N. J. A. Sloane, Sep 13 2014

			The first term is 6481 which is a prime and is the concatenation of 64 and 81 which are two consecutive square numbers.

E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, Math. Annalen, 114 (1937), 1-28; Werke pp. 644-671. See page 671.

Robert Israel, Table of n, a(n) for n = 1..10000
Haruzo Hida, Arithmetic of Weil numbers and Hecke fields.
Don Zagier, Elliptic Modular Forms and Their Applications

A090738 gives the numbers n such that a(n) = (n^2 concatenated with (n+1)^2) is prime.

These are the primes in A246973. Cf. A104301, A246972.

catn:= proc(a,b) 10^(1+ilog10(b))*a+b end proc:
R:= NULL: count:= 0:
for x from 2 by 2 while count < 100 do
  y:= catn(x^2,(x+1)^2);
  if isprime(y) then count:= count+1; R:= R, y; fi
od:
R; # Robert Israel, May 19 2020

from sympy import isprime
A104242_list = []
for n in range(1,2000):
    x = int(str(n**2)+str((n+1)**2))
    if isprime(x):
        A104242_list.append(x) # Chai Wah Wu, Sep 13 2014

A104302 Primes which are the reverse concatenation of three consecutive square numbers.

Original entry on oeis.org

941, 25169, 169144121, 302529162809, 110251081610609, 166411638416129, 497294928448841, 580815760057121, 930259241691809, 123201122500121801, 139129138384137641, 140625139876139129, 196249195364194481, 214369213444212521, 235225234256233289, 261121260100259081

Offset: 1

Shyam Sunder Gupta, Apr 17 2005

			The first term is 941 which is a prime and is the reverse concatenation of 1, 4 and 9 which are three consecutive square numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A104301, A104303, A104304.

Select[FromDigits[Flatten[IntegerDigits/@#]]&/@(Reverse/@Partition[Range[500]^2,3,1]),PrimeQ] (* Harvey P. Dale, Sep 21 2024 *)

A104303 Primes which are the reverse concatenation of four consecutive square numbers.

Original entry on oeis.org

576529484441, 7056688967246561, 34596342253385633489, 49284488414840047961, 202500201601200704199809, 230400229441228484227529, 260100259081258064257049, 389376388129386884385641, 527076525625524176522729, 553536552049550564549081, 599076597529595984594441

Offset: 1

Shyam Sunder Gupta, Apr 17 2005

			The first term is 576529484441 which is a prime and is the reverse concatenation of 441, 484, 529 and 576 which are four consecutive square numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A104301, A104302, A104304.

Select[FromDigits[Flatten[IntegerDigits/@#]]&/@(Reverse/@Partition[ Range[ 1000]^2,4,1]),PrimeQ] (* Harvey P. Dale, Sep 07 2020 *)

A104304 Primes which are the reverse concatenation of five consecutive square numbers.

Original entry on oeis.org

1876918496182251795617689, 3648136100357213534434969, 297025295936294849293764292681, 466489465124463761462400461041, 16770251674436167184916692641666681, 17556251752976175032917476841745041, 30660013062500305900130555043052009, 31506253147076314352931399843136441

Offset: 1

Shyam Sunder Gupta, Apr 17 2005

			The first term is 1876918496182251795617689 which is a prime and is the reverse concatenation of 17689, 17956, 18225, 18496 and 18769 which are five consecutive square numbers.

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

Cf. A104301, A104302, A104303.

cat[s_] := FromDigits[Flatten[IntegerDigits[s]]]; Select[cat /@ Reverse /@ Partition[Range[3000]^2, 5, 1], PrimeQ] (* Amiram Eldar, Jul 15 2025 *)

More terms from Amiram Eldar, Jul 15 2025

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A246972 (n+1)^2 concatenated with n^2.

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A104242 Primes which are the concatenation of two consecutive square numbers.

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A104302 Primes which are the reverse concatenation of three consecutive square numbers.

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A104303 Primes which are the reverse concatenation of four consecutive square numbers.

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A104304 Primes which are the reverse concatenation of five consecutive square numbers.

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