1,4,9,16,25,36,64 - cp's OEIS frontend

A072862 The smallest k>1 such that k divides sigma(k*n) is equal to 6.

1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041

Offset: 1

Benoit Cloitre, Jul 26 2002

The smallest k>1 such that k divides sigma(k*n) is always 2, 3 or 6. Sequence generates squares but not all the squares (49, 169, 196, 361, 441, 676, 784, ... are the first squares not in the sequence).

This is not the sequence of solutions to sigma(n) == 1 (mod 6), because 2401 is in this sequence but sigma(2401) == 5 (mod 6). See A074384 for more examples. - R. J. Mathar, May 19 2020

{n: A049605(n) = 6}. - R. J. Mathar, May 19 2020

A062295 A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 64, 81, 100, 169, 256, 289, 441, 484, 576, 625, 841, 1089, 1296, 1444, 1936, 2025, 2401, 2601, 3136, 4225, 4356, 4624, 5329, 5476, 5776, 6084, 7569, 9025, 10201, 11449, 11664, 12321, 12996, 13456, 14400, 16129, 17956, 20164, 22201

Offset: 1

Labos Elemer, Jul 02 2001

nonn

			36 is in the sequence since the pairwise sums of {1, 4, 9, 16, 25, 36} are all distinct: 2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 61, 72.
49 is not in the sequence since 1 + 49 = 25 + 25.

Klaus Brockhaus, Table of n, a(n) for n = 1..4944
Eric Weisstein's World of Mathematics, B2-Sequence
Index entries for B_2 sequences

Cf. A000290, A011185, A010672, A025582, A005282, A062292, A133743, A133744.

from itertools import count, islice
def A062295_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in (n**2 for n in count(1)):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2 |= bset2
A062295_list = list(islice(A062295_gen(),30)) # Chai Wah Wu, Sep 05 2023

Edited, corrected and extended by Klaus Brockhaus, Sep 24 2007

A068802 Smaller of two consecutive squares which have no common digits.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 64, 196, 361, 484, 841, 1936, 5929, 8836, 69696, 1999396, 29997529

Offset: 0

Amarnath Murthy, Mar 06 2002

There are no more terms. Sketch of proof: Suppose n^2 and (n+1)^2 have no common digits. Then their first digits differ, so n+1 = ceiling(sqrt(d*10^r)) with 1<=d<=9 and r>=0. In other words,
n+1 = ceiling(sqrt(e*100^s)) with e in {1,2,...,9,10,20,...,90} and s>=0. The cases e=1, 4 and 9 are easy. Otherwise note that about half the digits of (n+1)^2 equal 0, so n^2 has no 0's. We have
(n+1)^2 - n^2 = 2n+1 ~ 10^s*2*sqrt(e). For e=20, this is about 10^s * 8.94427190999916. So for s>=10, the decimal expansion of (n+1)^2 - n^2 has 2 consecutive 9's. (In fact 4 for large s, but 2 is enough.) Since n^2 has no 0's this implies that n^2 and (n+1)^2 have the same digit in the position of the first of the 2 9's. The same idea works for other values of e, but the consecutive 9's occur later.

			29997529 is a term since 29997529 and 30008484 are two consecutive squares with no common digits.

s := X->convert(convert(X,base,10),set); seq(`if`((s(n^2) intersect s((n+1)^2))={},n^2,printf("")),n=1..350000);

For[lastn=-1; r=0, r<500, r++, For[d=1, d<10, d++, n=Ceiling[Sqrt[d*10^r]]-1; If[n>lastn, lastn=n; If[Intersection[IntegerDigits[n^2], IntegerDigits[(n+1)^2]]=={}, Print[n^2]]]]]
First /@ Select[Partition[Range[0, 6000]^2, 2, 1], Intersection @@ IntegerDigits /@ # == {} &] (* Jayanta Basu, Aug 06 2013 *)

Edited by Dean Hickerson and Francois Jooste (phukraut(AT)hotmail.com), Mar 19 2002

A238203 Squares s such that s^2+s+41 is prime.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 64, 100, 144, 169, 196, 225, 324, 400, 441, 484, 529, 576, 625, 841, 900, 961, 1089, 1444, 1521, 1849, 2209, 2601, 2704, 2809, 3025, 3136, 3249, 3364, 3721, 3844, 4096, 4225, 4356, 4489, 5476, 5625, 5776, 6241, 7056, 7921, 8464, 8836, 9025

Offset: 1

K. D. Bajpai, Feb 20 2014

nonn

n^2+n+41: Euler’s prime generating polynomial.

First 6 terms in the sequence are first 6 consecutive squares.

			9 is in the sequence because 9 = 3^2 and 9^2+9+41 = 131 is prime.
36 is in the sequence because 36 = 6^2 and 36^2+36+41 = 1373 is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..4285

Cf. A000040, A097823, A238242.

with(numtheory):KD := proc() local a,b; a:=(n^2);b:=a^2+a+41; if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..500);

Select[Table[k = n^2, {n, 100}], PrimeQ[#^2 + # + 41] &] (* or *) c = 0; Do[k = n^2; If[PrimeQ[k^2 + k + 41], c = c + 1; Print[c, " ", k]], {n, 1, 10000}];
Select[Range[100]^2,PrimeQ[#^2+#+41]&] (* Harvey P. Dale, Dec 13 2021 *)

A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041, 5184, 5625, 6400, 6561, 6724, 6889, 7225, 7569

Offset: 1

Altug Alkan, Mar 02 2018

Or Loeschian numbers (A003136) that are not in A024614.
Squares that are not in this sequence are 49, 169, 196, 361, 441, 676, ...
This is the list of squares not of the form A050931(k)^2. A number n is in this sequence iff n = m^2 with m having no prime factor == 1 (mod 6). - M. F. Hasler, Mar 04 2018

			Loeschian number 25 = 5^2 is a term because 25 = x^2 + x*y + y^2 has no solution for positive integers x, y.

Cf. A000548, A003136, A024614, A050931, A230780.

isA024614:= proc(n) local x,y;
for x from 1 to floor(sqrt(n-1)) do
   if issqr(4*n-3*x^2) then return true fi
od:
false
end proc:
isA024614(0):= false:
remove(isA024614, [seq(i^2,i=0..200)]); # Robert Israel, Mar 02 2018

sol[s_] := Solve[0 < x <= y && s == x^2 + x y + y^2, {x, y}, Integers];
Select[Range[0, 100]^2, sol[#] == {}&] (* Jean-François Alcover, Oct 26 2020 *)

is(n,m)=issquare(n,m)&&!setsearch(Set(factor(m)[,1]%6),1) \\ second part is equivalent to is_A230780(m), this is sufficient to test (e.g., to produce a list) if we know that n = m^2. - M. F. Hasler, Mar 04 2018

a(n) = A230780(n-1)^2 for n > 1.

cp's OEIS Frontend

A072862 The smallest k>1 such that k divides sigma(k*n) is equal to 6.

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A062295 A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.

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A068802 Smaller of two consecutive squares which have no common digits.

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Maple

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A238203 Squares s such that s^2+s+41 is prime.

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Maple

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A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

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Maple

Mathematica

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