A123388 -id:A123388 - cp's OEIS frontend

A162930 Primes that can be written as a sum of a positive square and a positive cube in more than one way.

17, 89, 233, 449, 577, 593, 1289, 1367, 1601, 1753, 2089, 2521, 3391, 4481, 4721, 5953, 6121, 6427, 7057, 7577, 8081, 9649, 10313, 10657, 10729, 11969, 12329, 13121, 13457, 15137, 15193, 15641, 15661, 16033, 16649, 18523, 21673, 21961, 23201

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 17 2009

nonn

A subset of these, 2089, 4481, 7057, 15193, 15641, etc., allows this representation in more than two ways (See A206606).

			The prime 17 can be written 1^3 + 4^2 as well as 2^3 + 3^2.

David A. Corneth, Table of n, a(n) for n = 1..10808 (first 500 terms from Seiichi Manyama, terms <= 8*10^7)

Cf. A000040, A054402, A123364, A123388, A173795, A206606.

isA162930 := proc(n) if isprime(n) then wa := 0 ; for y from 1 to n/2 do if issqr(n-y^3) then if n -y^3 > 0 then wa := wa+1 ; fi; fi; od: RETURN( wa>1) ; else false; fi; end:
for i from 1 to 2700 do if isA162930 ( ithprime(i)) then printf("%d,",ithprime(i)) ; fi; od: # R. J. Mathar, Jul 21 2009

lst={};Do[Do[AppendTo[lst,n^2+m^3],{n,2*5!}],{m,2*5!}];lst=Sort[lst]; lst2={};Do[If[lst[[n]]==lst[[n+1]]&&PrimeQ[lst[[n]]],AppendTo[lst2, lst[[n]]]],{n,Length[lst]-1}];lst2;

upto(n) = {my(res = List(), v = vector(n), i, j, i2); for(i = 1, sqrtint(n), i2 = i^2; for(j = 1, sqrtnint(n - i^2, 3), v[i2 + j^3]++)); forprime(p = 2, n, if(v[p] > 1, listput(res, p))); kill(v); res} \\ David A. Corneth, Jun 20 2023

A000040 INTERSECT A054402.

Slightly edited by R. J. Mathar, Jul 21 2009

A122054 Least positive number with exactly n partitions into a square and a cube.

Original entry on oeis.org

1, 2, 17, 1737, 1025, 92025, 3375900, 5472225, 35964225, 930860225, 1000837225, 4979585600, 38515961025, 88154795025, 203947076025, 88813460025, 5684061441600, 77806025000000, 64745012358225

Offset: 0

Zak Seidov, Oct 15 2006

The sequence is not monotonic.

Equals 1 followed by A060835.

			a(0)=1 because there's no partition of 1 into a square and a cube;
a(1)=2 because 2 = 1^2 + 1^3, one partition;
a(2)=17 because 17 = 3^2 + 2^3 = 4^2 + 1^3, two partitions;
a(3)=1737 because 1737 = 3^2 + 12^3 = 35^2 + 8^3 = 39^2 + 6^3, three partitions, etc.
Table of partitions:
{m,{a_i,b_i}}
{1,{-}}
{2,{1,1}}
{17,{3,2},{4,1}}
{1737,{3,12},{35,8},{39,6}}
{1025,{5,10},{30,5},{31,4},{32,1}}
{92025,{30,45},{152,41},{213,36},{255,30},{303,6}}
{3375900,{30,150},{1430,110},{1551,99},{1794,54},{1830,30},{1837,11}}
{5472225,{143,176},{990,165},{1935,120},{2035,110},{2251,74},{2277,66},{2321,44}}
{35964225,{165,330},{3167,296},{4035,270},{4191,264},{5610,165},{5885,110},{5973,66},{5997,6}}
{930860225,{13315,910},{23139,734},{26817,596},{29560,385},{30271,244},{30335,220},{30460,145},{30465,140},{30510,5}}
{1000837225,{915,1000},{16921,894},{19665,850},{20960,825},{24735,730},{29465,510},{30221,444},{31608,121},{31635,40},{31636,9}}
{4979585600,{8160,1700},{24512,1636},{40392,1496},{48785,1375},{49640,1360},{64515,935},{65560,880},{66840,800},{69960,440},{70024,424},{70565,55}}
{38515961025,{20330,3365},{41507,3326},{73755,3210},{95084,3089},{115595,2930},{118431,2904},{132255,2760},{153953,2456},{176355,1950},{190305,1320},{195570,645},{196254,69}}

Cf. A060835, A123388.

m=a^2+b^3, a,b>0.

Corrected a(7), added a(8)-a(12) from Lars Blomberg, Feb 15 2016

A122034 Prime numbers with five partitions into a square and cube.

Original entry on oeis.org

3193361, 4251817, 4297609, 17969129, 22618657, 25266041, 30003697, 46918369, 58896881, 61953641, 64166273, 70045993, 70735409, 76033889, 82761857, 95049881, 102087721, 120618073, 127318537, 131584841, 177350489, 184799233

Offset: 1

Zak Seidov, Oct 15 2006

nonn

Cf. A123388 Prime numbers with four partitions into a square and cube.

			{ p {a1,b1}..{a5,b5} }
{{3193361,{1769,40},{1606,85},{1481,100},{1047,128},{285,146}},
{4251817,{2061,16},{2035,48},{1005,148},{618,157},{17,162}},
{4297609,{1981,72},{1799,102},{1772,105},{1603,120},{654,157}},
{17969129,{4239,2},{4002,125},{3945,134},{3924,137},{627,260}},
{22618657,{4751,36},{4643,102},{4629,106},{4132,177},{3183,232}},
{25266041,{4693,148},{4482,173},{4065,206},{3552,233},{2453,268}}}

Cf. A123388.

p=a^2+b^3 for five different pairs {a,b}.

a(7)-a(22) from Donovan Johnson, Dec 14 2009

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A162930 Primes that can be written as a sum of a positive square and a positive cube in more than one way.

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A122054 Least positive number with exactly n partitions into a square and a cube.

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A122034 Prime numbers with five partitions into a square and cube.

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