A270566 -id:A270566 - cp's OEIS frontend

A334147 Numbers which can be written uniquely as x^4 + y(2y+1) + z(3z+1), where x,y,z are integers with x>=0.

0, 9, 42, 57, 127, 218, 243, 272, 412, 467, 554, 555, 571, 724, 909, 1292, 1385, 1448, 1557, 1604, 1897, 2062, 2410, 3025, 3507, 4328, 5907, 8182, 9018, 14654, 18628, 25479, 25713, 76322, 80488, 152177, 1277405

Offset: 1

Zhi-Wei Sun, Apr 16 2020

nonn

The sequence consists of those n with A334138(n) = 1.
Conjecture: The sequence has no terms greater than a(37) = 1277405.
We have noted that A334138(n) > 1 for all 1277405 < n <= 5*10^6.

			a(10) = 467 with 467 = 0^4 + 15*(2*15+1) + (-1)*(3*(-1)+1).
a(25) = 3507 with 3507 = 6^4 + 33*(2*33+1) + 0*(3*0+1).
a(36) = 152177 with 152177 = 9^4 + (-266)*(2*(-266)+1) + 38*(3*38+1).
a(37) = 1277405 with 1277405 = 22^4 + (-655)*(2*(-655)+1) + (-249)*(3*(-249)+1).

Cf. A000217, A001318, A270566, A334138.

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[12n+1]];
m=0;Do[r=0;Do[If[QQ[n-x^4-y(2y+1)],r=r+1;If[r>1,Goto[aa]]],{x,0,n^(1/4)},{y,-Floor[(Sqrt[8(n-x^4)+1]+1)/4],(Sqrt[8(n-x^4)+1]-1)/4}];If[r==1,m=m+1;Print[m," ",n]];Label[aa],{n,0,152177}]

A343503 Number of ways to write n as x(3x+1)/2 + y(7y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 5, 5, 6, 4, 4, 5, 6, 4, 4, 8, 9, 6, 9, 8, 8, 6, 8, 7, 2, 7, 6, 6, 5, 7, 9, 8, 7, 10, 6, 11, 9, 9, 10, 6, 10, 9, 10, 6, 7, 10, 10, 6, 7, 6, 7, 7, 6, 7, 6, 11, 10, 9, 9, 9, 10, 10, 10, 9, 7, 7, 14, 8, 11, 9, 13, 11, 7, 13, 9, 7, 10, 8, 6, 7, 10, 11, 4, 9, 8, 12, 8, 11, 12, 6, 12, 11, 12, 13, 7, 12, 10, 11, 11, 9

Offset: 1

Zhi-Wei Sun, Apr 17 2021

nonn

The author noted that a(n) > 0 for all n = 1..2*10^7. Giovanni Resta found that 8558169401 is the first value of n with a(n) = 0.

			a(1) = 1 with 1 = 0*(3*0+1)/2 + 0*(7*0+1)/2 + 2^0.
a(25) = 2, and 25 = 1*(3*1+1)/2 + 2*(7*2+1)/2 + 2^3 = (-2)*(3*(-2)+1)/2 + 1*(7*1+1)/2 + 2^4.

Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math. 63 (2020), 501-520.

Cf. A000079, A001318, A270566, A343397, A343411, A343460.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
tab={};Do[r=0;Do[If[PenQ[n-2^k-x(7x+1)/2],r=r+1],{k,0,Log[2,n]},{x,-Floor[(Sqrt[56(n-2^k)+1]+1)/14],(Sqrt[56(n-2^k)+1]-1)/14}];tab=Append[tab,r],{n,1,100}];Print[tab]

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A334147 Numbers which can be written uniquely as x^4 + y(2y+1) + z(3z+1), where x,y,z are integers with x>=0.

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A343503 Number of ways to write n as x(3x+1)/2 + y(7y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.

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cp's OEIS Frontend

A334147 Numbers which can be written uniquely as x^4 + y*(2y+1) + z*(3z+1), where x,y,z are integers with x>=0.

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A343503 Number of ways to write n as x*(3*x+1)/2 + y*(7*y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.

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A334147 Numbers which can be written uniquely as x^4 + y(2y+1) + z(3z+1), where x,y,z are integers with x>=0.

A343503 Number of ways to write n as x(3x+1)/2 + y(7y+1)/2 + 2^k, where x and y are integers, and k is a nonnegative integer.