A266968 -id:A266968

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A271026

Number of ordered ways to write n as x^7 + y^4 + z^3 + w*(3w+1)/2, where x, y, z are nonnegative integers, and w is an integer.

+10
4

1, 4, 7, 7, 4, 2, 3, 4, 5, 6, 5, 3, 2, 4, 5, 4, 6, 7, 5, 3, 2, 3, 4, 6, 8, 5, 3, 5, 7, 8, 6, 5, 5, 3, 3, 5, 6, 4, 2, 4, 5, 4, 5, 7, 6, 3, 2, 1, 2, 4, 5, 5, 5, 5, 3, 2, 2, 3, 5, 6, 4, 1, 1, 2, 3, 6, 7, 6, 5, 4, 4, 5, 5, 3, 2, 2, 2, 3, 7, 9, 6 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	COMMENTS	`Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 47, 61, 62, 112, 175, 448, 573, 714, 1073, 1175, 1839, 2167, 8043, 13844.` `(ii) Any natural number can be written as 3x^6 + y^4 + z^3 + w(3w+1)/2, where x, y, z are nonnegative integers and w is an integer.` `(iii) For every a = 3, 4, 5, 9, 12, any natural number can be written as ax^5 + y^4 + z^3 + w(3w+1)/2, where x, y, z are nonnegative integers and w is an integer. Also, any natural number can be written as x^5 + 2y^4 + 2z^3 + w(3w+1)/2 (or 3x^5 + 2y^4 + z^3 + w(3w+1)/2), where x, y, z are nonnegative integers and w is an integer.` `We have verified that a(n) > 0 for n up to 2*10^6.` `See also A266968 for a related conjecture.`
	LINKS	`Zhi-Wei Sun, Table of n, a(n) for n = 0..10000` `Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.` `Z.-W. Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), 1367-1396.`
	EXAMPLE	`a(47) = 1 since 47 = 1^7 + 2^4 + 2^3 + (-4)(3(-4)+1)/2.` `a(61) = 1 since 61 = 1^7 + 1^4 + 2^3 + (-6)(3(-6)+1)/2.` `a(62) = 1 since 62 = 0^7 + 0^4 + 3^3 + (-5)(3(-5)+1)/2.` `a(112) = 1 since 112 = 1^7 + 3^4 + 2^3 + (-4)(3(-4)+1)/2.` `a(175) = 1 since 175 = 1^7 + 3^4 + 1^3 + (-8)(3(-8)+1)/2.` `a(448) = 1 since 448 = 2^7 + 4^4 + 4^3 + 0(30+1)/2.` `a(573) = 1 since 573 = 1^7 + 4^4 + 6^3 + 8(38+1)/2.` `a(714) = 1 since 714 = 2^7 + 4^4 + 0^3 + (-15)(3(-15)+1)/2.` `a(1073) = 1 since 1073 = 0^7 + 2^4 + 10^3 + 6(36+1)/2.` `a(1175) = 1 since 1175 = 0^7 + 5^4 + 5^3 + (-17)(3(-17)+1)/2.` `a(1839) = 1 since 1839 = 1^7 + 4^4 + 5^3 + 31(331+1)/2.` `a(2167) = 1 since 2167 = 1^7 + 5^4 + 11^3 + (-12)(3(-12)+1)/2.` `a(8043) = 1 since 8043 = 1^7 + 2^4 + 20^3 + 4(34+1)/2.` `a(13844) = 1 since 13844 = 3^7 + 2^4 + 21^3 + (-40)(3(-40)+1)/2.`
	MATHEMATICA	`pQ[n_]:=pQ[n]=IntegerQ[Sqrt[24n+1]]` `Do[r=0; Do[If[pQ[n-x^7-y^4-z^3], r=r+1], {x, 0, n^(1/7)}, {y, 0, (n-x^7)^(1/4)}, {z, 0, (n-x^7-y^4)^(1/3)}]; Print[n, " ", r]; Continue, {n, 0, 80}]`
	CROSSREFS	`Cf. A000326, A000578, A000583, A000584, A001015, A001318, A262813, A262815, A262816, A262827, A266968, A270469, A270488, A270516, A270533, A270559, A270566, A270920.`
	KEYWORD	`nonn`
	AUTHOR	`Zhi-Wei Sun, Mar 29 2016`
	STATUS	`approved`

A271106

Number of ordered ways to write n as x^6 + 3*y^3 + z^3 + w*(w+1)/2, where x and y are nonnegative integers, and z and w are positive integers.

+10
4

0, 0, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 3, 1, 3, 3, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 4, 3, 3, 4, 5, 3, 2, 4, 4, 3, 2, 4, 3, 2, 2, 1, 2, 3, 4, 3, 2, 1, 1, 2, 4, 4, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 1, 5, 5, 5, 3, 4 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,6`
	COMMENTS	`Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 6, 9, 13, 16, 20, 21, 24, 25, 44, 50, 51, 65, 84, 189, 290, 484, 616, 664, 680, 917, 1501, 1639, 3013.` `Based on our computation, we also formulate the following general conjecture.` General Conjecture: Let T(w) = w(w+1)/2. We have {P(x,y,z,w): x,y,z,w = 0,1,2,...} = {0,1,2,...} for any of the following polynomials P(x,y,z,w): x^3+y^3+cz^3+T(w) (c = 2,3,4,6), x^3+y^3+cz^3+2T(w) (c = 2,3), x^3+by^3+3z^3+3T(w) (b = 1,2), x^3+2y^3+3z^3+w(5w-1)/2, x^3+2y^3+3z^3+w(5w-3)/2, x^3+2y^3+cz^3+T(w) (c = 2,3,4,5,6,7,12,20,21,34,35,40), x^3+2y^3+cz^3+2T(w) (c = 3,4,5,6,11), x^3+2y^3+cz^3+w^2 (c = 3,4,5,6), x^3+2y^3+4z^3+w(3w-1)/2, x^3+2y^3+4z^3+w(3w+1)/2, x^3+2y^3+4z^3+w(2w-1), x^3+2y^3+6z^3+w(3w-1)/2, x^3+3y^3+cz^3+T(w) (c = 3,4,5,6,10,11,13,15,16,18,20), x^3+3y^3+cz^3+2T(w) (c = 5,6,11), x^3+4y^3+cz^3+T(w) (c = 5,10,12,16), x^3+4y^3+5z^3+2T(w), x^3+5y^3+10z^3+T(w), 2x^3+3y^3+cz^3+T(w) (c = 4,6), 2x^3+4y^3+8z^3+T(w), x^4+y^3+3z^3+w(3w-1)/2, x^4+y^3+cz^3+T(w) (c = 2,3,4,5,7,12,13), x^4+y^3+cz^3+2T(w) (c = 2,3,4,5), x^4+y^3+2z^3+w^2, x^4+y^3+4z^3+2w^2, x^4+2y^3+cz^3+T(w) (c = 4,5,12), x^4+2y^3+3z^3+2T(w), 2x^4+y^3+2z^3+w(3w-1)/2, 2x^4+y^3+cz^3+T(w) (c = 1,2,3,4,5,6,10,11), 2x^4+y^3+cz^3+2T(w) (c = 2,3,4), 2x^4+2y^3+cz^3+T(w) (c = 3,5), 3x^4+y^3+cz^3+T(w) (c = 1,2,3,4,5,11), 3x^4+y^3+2z^3+2T(w), 3x^4+y^3+2z^3+w^2, 3x^4+y^3+2z^3+w(3w-1)/2, 4x^4+y^3+cz^3+T(w) (c = 2,3,4,6), 4x^4+y^3+2z^3+2T(w), 5x^4+y^3+cz^3+T(w) (c = 2,4), ax^4+y^3+2z^3+T(w) (a = 6,20,28,40), 6x^4+y^3+2z^3+2T(w), 6x^4+y^3+2z^3+w^2, ax^4+y^3+3z^3+T(w) (a = 6,8,11), 8x^4+2y^3+4z^3+T(w), x^5+y^3+cz^3+T(w) (c = 2,3,4), x^5+2y^3+cz^3+T(w) (c = 3,6,8), 2x^5+y^3+4z^3+T(w), 3x^5+y^3+2z^3+T(w), 5x^5+y^3+cz^3+T(w) (c = 2,4), x^6+y^3+3z^3+T(w), x^7+y^3+4z^3+T(w), x^4+2y^4+z^3+w^2, x^4+2y^4+2z^3+T(w), x^4+by^4+z^3+T(w) (b = 2,3,4), 2x^4+3y^4+z^3+T(w), ax^5+y^4+z^3+T(w) (a = 1,2), x^5+2y^4+z^3+T(w). `The polynomials listed in the general conjecture should exhaust all those polynomials P(x,y,z,w) = ax^i+by^j+cz^k+w(s*w+/-t)/2 with {P(x,y,z,w): x,y,z,w = 0,1,2,...} = {0,1,2,...}, where a,b,c,s > 0, 0 <= t <= s, s == t (mod 2), i >= j >= k >= 3, a <= b if i = j, and b <= c if j = k.`
	LINKS	`Zhi-Wei Sun, Table of n, a(n) for n = 0..10000` `Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.` `Z.-W. Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), 1367-1396.`
	EXAMPLE	`a(9) = 1 since 9 = 0^6 + 30^6 + 2^3 + 12/2.` `a(24) = 1 since 24 = 1^6 + 30^6 + 2^3 + 56/2.` `a(1501) = 1 since 1501 = 2^6 + 35^3 + 3^3 + 4546/2.` `a(1639) = 1 since 1639 = 0^6 + 36^3 + 1^3 + 4445/2.` `a(3013) = 1 since 3013 = 3^6 + 33^3 + 13^3 + 34/2.`
	MATHEMATICA	`TQ[n_]:=TQ[n]=n>0&&IntegerQ[Sqrt[8n+1]]` `Do[r=0; Do[If[TQ[n-x^6-3*y^3-z^3], r=r+1], {x, 0, n^(1/6)}, {y, 0, ((n-x^6)/3)^(1/3)}, {z, 1, (n-x^6-3y^3)^(1/3)}]; Print[n, " ", r]; Continue, {n, 0, 70}]`
	CROSSREFS	`Cf. A000217, A000290, A000326, A000578, A000583, A000584, A001014, A005449, A262813, A262824, A262827, A262857, A262880, A266968, A270469, A270488, A270516, A270533, A270559, A270566, A270920, A271026.`
	KEYWORD	`nonn`
	AUTHOR	`Zhi-Wei Sun, Mar 30 2016`
	STATUS	`approved`

A271076

Number of ordered ways to write n as u^5 + v^4 + x^3 + 2*y^3 + 3*z^3, where u, v , x, y and z are nonnegative integers with v > 0.

+10
2

1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 4, 4, 4, 4, 3, 1, 1, 3, 4, 4, 5, 4, 2, 2, 2, 5, 4, 3, 5, 2, 1, 1, 2, 5, 4, 6, 5, 2, 3, 2, 4, 5, 4, 3, 3, 3, 2, 2, 4, 5, 4, 5, 5, 1, 2, 3, 3, 5, 2, 5, 5, 3, 3, 3, 3, 3, 4, 4, 1, 1, 2, 3, 5 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 8, 9, 10, 11, 13, 14, 15, 16, 23, 24, 38, 39, 61, 76, 77, 104, 118, 188, 214, 229.` (ii) We have {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(u,v,x,y,z) is among the following polynomials: u^5+v^4+x^3+2y^3+5z^3, u^5+2v^4+x^3+2y^3+3z^3, u^5+3v^4+x^3+2y^3+3z^3, 2u^5+v^4+x^3+y^3+4z^3, 2u^5+v^4+x^3+2y^3+4z^3, 3u^5+v^4+x^3+2y^3+4z^3, 5u^5+v^4+x^3+2y^3+4z^3, u^4+2v^4+x^3+y^3+4z^3, u^4+2v^4+x^3+2y^3+3z^3, u^4+2v^4+x^3+2y^3+4z^3, u^4+2v^4+x^3+2y^3+6z^3, u^4+2v^4+x^3+3y^3+4z^3, u^4+2v^4+x^3+4y^3+5z^3, u^4+2v^4+x^3+4y^3+6z^3, u^4+2v^4+x^3+4y^3+10z^3, u^4+3v^4+x^3+2y^3+3z^3, u^4+3v^4+x^3+2y^3+4z^3, u^4+3v^4+x^3+2y^3+6z^3, u^4+4v^4+x^3+y^3+2z^3, u^4+4v^4+x^3+2y^3+3z^3, u^4+4v^4+x^3+2y^3+4z^3, u^4+5v^4+x^3+2y^3+4z^3, u^4+6v^4+x^3+2y^3+3z^3, u^4+7v^4+x^3+2y^3+3z^3, u^4+9v^4+x^3+2y^3+4z^3, 2u^4+4v^4+x^3+2y^3+3z^3,2u^4+6v^4+x^3+2y^3+4z^3, 3u^4+6v^4+x^3+2y^3+4z^3. `(iii) We have {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(u,v,x,y,z) is among the following polynomials: u^5+v^3+x^3+2y^3+4z^3, u^5+v^3+2x^3+3y^3+cz^3 (c = 6,9), au^5+v^3+2x^3+4y^3+5z^3 (a = 1,2,6), bu^5+v^3+2x^3+4y^3+6z^3 (b = 2,3), u^5+v^3+2x^3+4y^3+dz^3 (d = 9,13).` `The listed polynomials in part (ii), together with u^5+v^4+x^3+2y^3+3z^3, should essentially exhaust all those polynomials P(u,v,x,y,z) = su^k+tv^j+ax^3+by^3+cz^3 with s,t,a,b,c positive integers and k >= j > 3, such that {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...}.` `There are also finitely many (but quite a lot) polynomials P(u,v,x,y,z) of the form mu^4+av^3+bx^3+cy^3+d*z^3 with a,b,c,d and m positive integers such that {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...}.` `See also A267826, A271099 and A271237 for related comments.` `Conjectures (i), (ii) and (iii) verified for n up to 10^11 for all polynomials. - Mauro Fiorentini, Sep 20 2023`
	LINKS	`Zhi-Wei Sun, Table of n, a(n) for n = 1..10000`
	EXAMPLE	`a(16) = 1 since 16 = 0^5 + 2^4 +0^3 + 20^3 + 30^3.` `a(104) = 1 since 104 = 0^5 + 2^4 + 4^3 + 20^3 + 32^3.` `a(188) = 1 since 188 = 2^5 + 1^4 + 3^3 + 24^3 + 30^3.` `a(229) = 1 since 229 = 1^5 + 3^4 + 4^3 + 21^3 + 33^3.`
	MATHEMATICA	`CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]` `Do[r=0; Do[If[CQ[n-u^5-v^4-3z^3-2y^3], r=r+1], {u, 0, (n-1)^(1/5)}, {v, 1, (n-u^5)^(1/4)}, {z, 0, ((n-u^5-v^4)/3)^(1/3)}, {y, 0, ((n-u^5-v^4-3z^3)/2)^(1/3)}]; Print[n, " ", r]; Continue, {n, 1, 80}]`
	CROSSREFS	`Cf. A000578, A000583, A000584, A266968, A267826, A271099, A271237.`
	KEYWORD	`nonn`
	AUTHOR	`Zhi-Wei Sun, Apr 07 2016`
	STATUS	`approved`

A271365

Number of ordered ways to write n as u^2 + v^3 + x^4 + y^5 + z^6, where u is a positive integer, and v, x, y, z are nonnegative integers.

+10
1

1, 4, 6, 5, 5, 6, 4, 1, 2, 7, 9, 6, 4, 3, 1, 1, 6, 12, 10, 4, 3, 3, 1, 1, 6, 12, 11, 7, 6, 4, 2, 4, 9, 12, 8, 5, 10, 12, 6, 2, 5, 9, 8, 8, 10, 6, 2, 3, 8, 13, 10, 8, 11, 8, 2, 1, 6, 10, 8, 7, 6, 2, 2, 7, 15, 20, 14, 9, 13, 11 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	COMMENTS	`Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 8, 15, 16, 23, 24, 56. Moreover, the only positive integers not represented by u^2+v^3+x^4+y^5 (u > 0 and v,x,y >= 0) are 8, 15, 23, 55, 62, 71, 471, 478, 510, 646, 806, 839, 879, 939, 1023, 1063, 1287, 2127, 5135, 6811, 7499, 9191, 26471.` `Note that 1/2+1/3+1/4+1/5+1/6 = 29/20 < 3/2.`
	LINKS	`Zhi-Wei Sun, Table of n, a(n) for n = 1..10000` `Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.`
	EXAMPLE	`a(1) = 1 since 1 = 1^2 + 0^3 + 0^4 + 0^5 + 0^6.` `a(8) = 1 since 8 = 2^2 + 1^3 + 1^4 + 1^5 + 1^6.` `a(15) = 1 since 15 = 2^2 + 2^3 + 1^4 + 1^5 + 1^6.` `a(16) = 1 since 16 = 4^2 + 0^3 + 0^4 + 0^5 + 0^6.` `a(23) = 1 since 23 = 2^2 + 1^3 + 2^4 + 1^5 + 1^6.` `a(24) = 1 since 24 = 4^2 + 2^3 + 0^4 + 0^5 + 0^6.` `a(56) = 1 since 56 = 4^2 + 2^3 + 0^4 + 2^5 + 0^6.`
	MATHEMATICA	`SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]` `Do[r=0; Do[If[SQ[n-x1^6-x2^5-x3^4-x4^3], r=r+1], {x1, 0, n^(1/6)}, {x2, 0, (n-x1^6)^(1/5)}, {x3, 0, (n-x1^6-x2^5)^(1/4)}, {x4, 0, (n-x1^6-x2^5-x3^4)^(1/3)}]; Print[n, " ", r]; Continue, {n, 1, 70}]`
	CROSSREFS	`Cf. A000290, A000578, A000583, A000584, A001014, A262813, A262824, A262827, A262857, A266968, A270488, A270516, A270533, A270559, A270566, A270920, A271026, A271106, A271325.`
	KEYWORD	`nonn`
	AUTHOR	`Zhi-Wei Sun, Apr 05 2016`
	STATUS	`approved`

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