A262959 -id:A262959 - cp's OEIS frontend

A262956 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a square or a square minus 1.

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

Offset: 1

Zhi-Wei Sun, Oct 05 2015

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, for any positive integer n, either n or n + 1 can be written as the sum of a fourth power, a square and a positive triangular number.

We also guess that a(n) = 1 only for n = 1, 89, 244, 464, 5243, 14343.

			a(1) = 1 since 1 = 0^4 + 1*2/2 + 0^2.
a(89) = 1 since 89 = 2^4 + 4*5/2 + 8^2 - 1.
a(244) = 1 since 244 = 2^4 + 2*3/2 + 15^2.
a(464) = 1 since 464 = 2^4 + 22*23/2 + 14^2 - 1.
a(5243) = 1 since 5243 = 0^4 + 50*51/2 + 63^2 - 1.
a(14343) = 1 since 14343 = 2^4 + 163*164/2 + 31^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A000217, A000290, A000583, A262941, A262944, A262945, A262954, A262955, A262959.

SQ[n_]:=IntegerQ[Sqrt[n]]||IntegerQ[Sqrt[n+1]]
Do[r=0;Do[If[SQ[n-x^4-y(y+1)/2],r=r+1],{x,0,n^(1/4)},{y,1,(Sqrt[8(n-x^4)+1]-1)/2}];Print[n," ",r];Continue,{n,1,100}]

A306227 Number of ways to write n as w + x^4 + y(y+1)/2 + z(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 3, 4, 4, 2, 3, 3, 4, 5, 3, 3, 4, 4, 3, 4, 3, 4, 5, 4, 2, 2, 3, 5, 7, 4, 3, 3, 2, 3, 4, 4, 4, 5, 5, 2, 3, 4, 4, 5, 2, 4, 4, 4, 4, 4, 3, 3, 5, 3, 2, 4, 5, 6, 5, 2, 2, 4, 5, 4, 4, 2, 3, 4, 4, 2, 3, 6, 7, 8, 4, 5, 4, 3, 5, 5, 3, 4, 7, 7, 6, 6, 4, 7, 6, 4, 5

Offset: 1

Zhi-Wei Sun, Jan 30 2019

nonn

Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer can be written as the sum of two fourth powers one of which is 0 or 1, and two distinct triangular numbers.
We have verified a(n) > 0 for all n = 1..10^6. The conjecture implies that the set S = {x^4 + y*(y+1)/2: x,y = 0,1,2,...} is an additive basis of order two (i.e., the sumset S + S coincides with {0,1,2,...}).
See also A306225 for a similar conjecture.

			a(1) = 1 with 1 = 0 + 0^4 + 0*1/2 + 1*2/2.
a(2) = 1 with 2 = 0 + 1^4 + 0*1/2 + 1*2/2.
a(14) = 1 with 14 = 0 + 1^4 + 2*3/2 + 4*5/2.
a(3774) = 1 with 3774 = 1 + 5^4 + 52*53/2 + 59*60/2.
a(7035) = 1 with 7035 = 0 + 3^4 + 48*49/2 + 107*108/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000217, A000583, A262813, A262941, A262954, A262956, A262959, A270566, A306225.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[r=0;Do[If[TQ[n-x-y^4-z(z+1)/2],r=r+1],{x,0,Min[1,(n-1)/2]},{y,x,(n-1-x)^(1/4)},{z,0,(Sqrt[4(n-1-x-y^4)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

A115159 Numbers that are not the sum of a triangular number, a square and a fourth power.

Original entry on oeis.org

34, 63, 89, 99, 139, 164, 174, 193, 204, 245, 314, 399, 424, 454, 464, 489, 504, 524, 549, 714, 1049, 1149, 1174, 1439, 1504, 1539, 1639, 1799, 1814, 1919, 2164, 2239, 2313, 2374, 2414, 2439, 2764, 2789, 3079, 3319, 3414, 3669, 3774, 3814, 4019, 4114

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 718 such numbers up to 2*10^9, the last one in this range being 99570649.

It is known that each natural number can be written as the sum of two squares and a triangular number. I believe that the sequence only has 718 terms as found by _Giovanni Resta and listed in the b-file. - Zhi-Wei Sun, Apr 15 2020

Zhi-Wei Sun, Table of n, a(n) for n = 1..718
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A022552, A014158, A115160, A115161, A115162, A115163, A262959, A334086, A334113.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[n-x^4-y^2],Goto[aa]],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]}];tab=Append[tab,n];Label[aa],{n,0,4114}];Print[tab] (From Zhi-Wei Sun)

Definition corrected by Giovanni Resta, Aug 17 2011

cp's OEIS Frontend

A262956 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a square or a square minus 1.

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A306227 Number of ways to write n as w + x^4 + y(y+1)/2 + z(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.

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A115159 Numbers that are not the sum of a triangular number, a square and a fourth power.

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cp's OEIS Frontend

A262956 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a square or a square minus 1.

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Comments

Examples

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Mathematica

A306227 Number of ways to write n as w + x^4 + y*(y+1)/2 + z*(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

A115159 Numbers that are not the sum of a triangular number, a square and a fourth power.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A306227 Number of ways to write n as w + x^4 + y(y+1)/2 + z(z+1)/2, where w is 0 or 1, and x, y, z are nonnegative integers with x >= w and y < z.