A024614 -id:A024614 - cp's OEIS frontend

A349987 Numbers that can be represented in more than one way as p^2+p*q+q^2 with p and q primes, p<=q.

147, 903, 1911, 3667, 3913, 4627, 5187, 8103, 10137, 11613, 12999, 13117, 13467, 14313, 16023, 16887, 18723, 19047, 19747, 20397, 22197, 23107, 24307, 25833, 28227, 30457, 30847, 31827, 32403, 37947, 38703, 39819, 45163, 46543, 50407, 57603, 58813, 61383, 63147, 68367, 68403, 70707, 71337, 74973

Offset: 1

J. M. Bergot and Robert Israel, Jan 09 2022

nonn

			a(3) = 1911 is a term because 1911 = 5^2+5*41+41^2 = 19^2+19*31+31^2 where 5, 41, 19 and 31 are primes.

Robert Israel, Table of n, a(n) for n = 1..2500

Subsequence of A024614.

Cf. A349986.

N:= 10^6: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
nP:= nops(P):
S:= {}: T:= {}:
for i from 1 to nP do
  for j from 1 to i do
    x:= P[i]^2 + P[i]*P[j]+P[j]^2;
    if x > N then break fi;
    if member(x,S) then T:= T union {x} fi;
    S:= S union {x};
od od:
sort(convert(T,list));

Do[If[Total@Boole[And@@@PrimeQ[{p,q}/.Solve[p^2+p*q+q^2==k&&p>1&&p<=q,{p,q},Integers]]]>1,Print@k],{k,10^6}] (* Giorgos Kalogeropoulos, Jan 09 2022 *)

A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

Original entry on oeis.org

12, 19, 27, 39, 49, 67, 75, 79, 109, 147, 163, 199, 201, 217, 247, 259, 309, 327, 349, 363, 399, 403, 427, 433, 457, 481, 507, 543, 579, 597, 607, 669, 679, 691, 739, 777, 867, 903, 937, 973, 997, 1011, 1027, 1063, 1083, 1093, 1141, 1209, 1227, 1281, 1327, 1387, 1423, 1447, 1489, 1533, 1579, 1587

Offset: 1

J. M. Bergot and Robert Israel, Jan 09 2022

nonn

The only square in this sequence is 49.

			a(3) = 27 is a term because 27 = 3^2+3*3+3^2.
a(4) = 39 is a term because 39 = 2^2+2*5+5^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Largest square written as p^2+pq+q^2 where p, q are primes?

Contains A079705, A244146, A349987.

Subsequence of A024614.

N:= 10^4: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
nP:= nops(P):
S:= {}:
for i from 1 to nP do
  for j from 1 to i do
    x:= P[i]^2 + P[i]*P[j]+P[j]^2;
    if x > N then break fi;
    S:= S union {x};
od od:
sort(convert(S,list));

A300381 Numbers of the form x^3+x*y+y^3 where x,y>0.

Original entry on oeis.org

3, 11, 20, 31, 41, 63, 69, 80, 103, 131, 143, 144, 167, 209, 223, 236, 261, 275, 304, 351, 365, 371, 391, 435, 468, 503, 521, 536, 563, 601, 608, 677, 735, 739, 755, 776, 783, 829, 899, 911, 999, 1011, 1028, 1057, 1088, 1104, 1135, 1175, 1276, 1313, 1343, 1361, 1391, 1413, 1439, 1511

Offset: 1

R. J. Mathar, Mar 04 2018

nonn

Contains all terms larger than 1 from A071568 (contributed by y=1).

Cf. A003136, A266970 (prime subsequence), A024614.

A299505 Numbers of the form x^4 + yx^3 + y^2x^2 + y^3*x + y^4, where x and y are positive integers.

Original entry on oeis.org

5, 31, 80, 121, 211, 341, 405, 496, 781, 1031, 1280, 1441, 1555, 1936, 2101, 2511, 2801, 3125, 3355, 3376, 4141, 4651, 4681, 5261, 5456, 6480, 6505, 6841, 7381, 7936, 8431, 9031, 9801, 9881, 11111, 11605, 12005, 12496, 13981, 14251, 15961, 16105, 16496, 17091, 17891

Offset: 1

Peter Luschny, Mar 02 2018

nonn

Cf. A024614, A002649 (subsequence of primes).

function isA299505(n)
    n % 5 >= 2 && return false
    n == 5 && return true
    K = Int(floor(5.383*log(n)^1.161))
    M = Int(floor(2*sqrt(n/3)))
    for k in 3:K
        for y in 2:M, x in 1:y
            n == x^4+y*x^3+y^2*x^2+y^3*x+y^4 && return true
    end end
    return false
end
A299505list(upto) = [n for n in 1:upto if isA299505(n)]
println(A299505list(18000))

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.

A331121 a(n) is the smallest positive integer k for which tau(k) does not divide sigma(n).

Original entry on oeis.org

2, 2, 4, 2, 6, 16, 4, 2, 2, 6, 16, 4, 4, 16, 16, 2, 6, 2, 4, 6, 4, 16, 16, 24, 2, 6, 4, 4, 6, 16, 4, 2, 16, 6, 16, 2, 4, 24, 4, 6, 6, 16, 4, 16, 6, 16, 16, 4, 2, 2, 16, 4, 6, 36, 16, 36, 4, 6, 24, 16, 4, 16, 4, 2, 16, 16

Offset: 1

Lechoslaw Ratajczak, Jan 10 2020

nonn

Consecutive t satisfying the equation a(t) = 2 are consecutive elements of A028982 (squares and twice squares).

Conjecture: consecutive u satisfying the equation a(u) = 4 are consecutive elements of a sequence defined as follows: (A024614 \ A088535) \ A074384. The conjecture was checked for 10^6 consecutive integers.

			a(10) = 6 because sigma(10) = 18 is divisible by (tau(1) = 1), (tau(2) = 2), (tau(3) = 2), (tau(4) = 3), (tau(5) = 2), and is not divisible by (tau(6) = 4).

Cf. A000005, A000203, A024614, A028982, A074384, A088535.

Array[Block[{k = 1}, While[Mod[DivisorSigma[1, #], DivisorSigma[0, k]] == 0, k++]; k] &, 66] (* Michael De Vlieger, Jan 31 2020 *)

a(n):=(for k:1 while mod(divsum(n), length(divisors(k))) = 0 do z:k, z+1) $ makelist(a(n), n, 1, 100, 1);

a(n) = my(k=1, sn=sigma(n)); while ((sn % numdiv(k)) == 0, k++); k; \\ Michel Marcus, Jan 10 2020

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A349987 Numbers that can be represented in more than one way as p^2+p*q+q^2 with p and q primes, p<=q.

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A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

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A300381 Numbers of the form x^3+x*y+y^3 where x,y>0.

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A299505 Numbers of the form x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, where x and y are positive integers.

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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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A331121 a(n) is the smallest positive integer k for which tau(k) does not divide sigma(n).

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A299505 Numbers of the form x^4 + yx^3 + y^2x^2 + y^3*x + y^4, where x and y are positive integers.