A055394 -id:A055394 - cp's OEIS frontend

A078375 Smallest prime factor of numbers which can be written as sum of a positive square and a positive cube.

2, 5, 3, 2, 2, 17, 2, 2, 2, 31, 3, 2, 37, 43, 2, 2, 2, 3, 3, 5, 2, 2, 73, 2, 2, 2, 89, 7, 2, 101, 2, 113, 2, 2, 127, 2, 3, 2, 3, 5, 2, 2, 2, 7, 2, 2, 3, 2, 3, 5, 3, 2, 197, 2, 2, 2, 7, 2, 223, 3, 2, 2, 233, 241, 2, 2, 257, 2, 2, 5, 269, 2, 283, 17, 2, 2, 3, 2, 2, 3, 5, 2, 337, 2, 347, 2, 3, 2

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020639, A055394, A078376.

FactorInteger[#][[1, 1]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A020639(A055394(n)).

A078376 Greatest prime factor of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 3, 5, 3, 17, 3, 13, 7, 31, 11, 3, 37, 43, 11, 5, 13, 19, 7, 13, 17, 3, 73, 19, 5, 41, 89, 13, 5, 101, 3, 113, 61, 7, 127, 2, 43, 67, 47, 29, 37, 5, 19, 23, 41, 17, 19, 29, 59, 37, 7, 7, 197, 17, 103, 13, 31, 11, 223, 5, 113, 29, 233, 241, 41, 7, 257, 13, 11, 53, 269, 7

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530, A055394, A078375.

FactorInteger[#][[-1, 1]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A006530(A055394(n)).

A078380 Maximum exponent in prime factorization of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A055394, A078377, A078379.

Max[FactorInteger[#][[;; , 2]]] & /@ Select[Range[300], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A051903(A055394(n)).

A078381 Number of divisors of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 2, 3, 4, 6, 2, 8, 4, 6, 2, 4, 9, 2, 2, 6, 6, 6, 4, 6, 4, 6, 12, 2, 6, 10, 4, 2, 4, 9, 2, 12, 2, 4, 12, 2, 8, 4, 4, 4, 4, 6, 12, 8, 4, 6, 8, 6, 8, 4, 4, 8, 9, 2, 12, 4, 10, 4, 12, 2, 9, 4, 8, 2, 2, 8, 18, 2, 12, 16, 4, 2, 16, 2, 3, 8, 12, 8, 6, 14, 4, 6, 6, 2, 8, 2, 12, 8, 12, 2, 2, 24, 4, 10, 6, 2

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A055394, A078377, A078378, A078382.

DivisorSigma[0, Select[Range[200], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &]] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A000005(A055394(n)).

A078387 Moebius's mu of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

-1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 1, 0, -1, -1, 0, 0, 0, 1, 0, 1, 0, 0, -1, 0, 0, 1, -1, 1, 0, -1, 0, -1, 1, 0, -1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 1, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, -1, -1, 0, -1, 0, 0, 1, -1, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, -1, 0, 1, 0, 0, -1, 1, -1

Offset: 1

Reinhard Zumkeller, Nov 25 2002

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008683, A055394, A078386.

MoebiusMu[Select[Range[200], Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &]] (* Amiram Eldar, Mar 27 2025 *)

a(n) = A008683(A055394(n)).

A078390 Composite numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

9, 10, 12, 24, 26, 28, 33, 36, 44, 50, 52, 57, 63, 65, 68, 72, 76, 80, 82, 91, 100, 108, 122, 126, 128, 129, 134, 141, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 204, 206, 208, 217, 220, 225, 226, 232, 246, 252, 260, 264, 265, 280, 289

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

			A002808(74) = 100 = 6^2 + 4^3, therefore 100 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A002808 and A055394.

Cf. A066649.

Select[Range[300], CompositeQ[#] && Length[Reduce[a^2 + b^3 == # && a > 0 && b > 0, {a, b}, Integers]] > 0 &] (* Amiram Eldar, Mar 27 2025 *)

A078393 Squarefree numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 10, 17, 26, 31, 33, 37, 43, 57, 65, 73, 82, 89, 91, 101, 113, 122, 127, 129, 134, 141, 145, 161, 170, 174, 177, 185, 197, 206, 217, 223, 226, 233, 241, 246, 257, 265, 269, 283, 290, 321, 337, 347, 353, 359, 362, 379, 381, 385, 401, 407, 427, 442, 443

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

			7*13 = 91 = 8^2 + 3^3, therefore 91 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..8398

Intersection of A005117 and A055394.

Cf. A066649.

spspcQ[{a_,b_}]:=AllTrue[{Sqrt[a],Surd[b,3]},IntegerQ]||AllTrue[{Sqrt[ b],Surd[ a,3]},IntegerQ]; Select[Range[500],SquareFreeQ[#] && Length[ Select[IntegerPartitions[#,{2}],spspcQ]]>0&] (* Harvey P. Dale, Jan 13 2019 *)

A122956 Least semiprime composed of a square and a positive cube in n different ways.

Original entry on oeis.org

4, 9, 65, 11665, 27289, 3030569, 6808609, 1632201497, 10553247449, 843404126561, 2101614761177, 62537392166201, 100301302204489

Offset: 0

Jonathan Vos Post & Robert G. Wilson v, Sep 29 2006

a(n) for n>0 must be odd.

			a(0)=4 since it is the first semiprime (2*2) not of the form a^2+b^3.
a(1) = 9 = 1^2 + 2^3 = 3*3.
a(2) = 65 = 1^2 + 4^3 = 8^2 + 1^3 = 5*13.
a(3) = 11665 = 108^2 + 1^3 = 107^2 + 6^3 = 87^2 + 16^3 = 5*2333.
a(4) = 27289 = 165^2 + 4^3 = 129^2 + 22^3 = 108^2 + 25^2 = 17^2 + 30^3 = 29*941.
a(5) = 3030569 = 1671^2 + 62^3 = 1587^2 + 80^3 = 1038^2 + 125^3 = 913^2 + 130^3 = 409^2 + 142^3 = 103*29423.
a(6) = 6808609 = 2609^2 + 12^3 = 2445^2 + 94^3 = 1853^2 + 150^3 = 1647^2 + 160^3 = 1522^2 + 165^3 = 1124^2 + 177^3 = 103*66103.
a(7) = 1632201497 = 38425^2 + 538^3 = 38202^2 + 557^3 = 36741^2 + 656^3 = 26177^2 + 982^3 = 18555^2 + 1088^3 = 13477^2 + 1132^3 = 1292^2 + 1177^3. [From _Donovan Johnson_, Aug 31 2008]
Contribution from _Donovan Johnson_, Mar 01 2010: (Start)
a(8) = 10553247449 = 102729^2 + 2^3 = 102393^2 + 410^3 = 101551^2 + 622^3 = 101371^2 + 652^3 = 80357^2 + 1600^3 = 63768^2 + 1865^3 = 13893^2 + 2180^3 = 4581^2 + 2192^3.
a(9) = 843404126561 = 917123^2 + 1318^3 = 902037^2 + 3098^3 = 866353^2 + 4528^3 = 833585^2 + 5296^3 = 634581^2 + 7610^3 = 521169^2 + 8300^3 = 478831^2 + 8500^3 = 259331^2 + 9190^3 = 23805^2 + 9446^3.
a(10) = 2101614761177 = 1449189^2 + 1136^3 = 1448961^2 + 1286^3 = 1448167^2 + 1642^3 = 1421577^2 + 4322^3 = 1315794^2 + 7181^3 = 1271813^2 + 7852^3 = 1119559^2 + 9466^3 = 1085568^2 + 9737^3 = 668475^2 + 11828^3 = 438431^2 + 12406^3.
a(11) = 62537392166201 = 7908053^2 + 448^3 = 7906101^2 + 3140^3 = 7863087^2 + 8918^3 = 7778399^2 + 12670^3 = 7537351^2 + 17890^3 = 7205845^2 + 21976^3 = 6649899^2 + 26360^3 = 5818649^2 + 30610^3 = 5684351^2 + 31150^3 = 2900985^2 + 37826^3 = 1009845^2 + 39476^3.
a(12) = 100301302204489 = 10013433^2 + 3190^3 = 9966435^2 + 9904^3 = 9922058^2 + 12285^3 = 9879183^2 + 13930^3 = 9821564^2 + 15657^3 = 9740881^2 + 17562^3 = 7540415^2 + 35154^3 = 2704995^2 + 45304^3 = 2667144^2 + 45337^3 = 1300067^2 + 46200^3 = 614915^2 + 46404^3 = 54519^2 + 46462^3.
(End)

Cf. A001358, A055394, A066649, A123048.

semiPrimeQ[x_] := Plus @@ Last /@ FactorInteger@x == 2; t = Table[0, {10}]; Do[ If[ semiPrimeQ@n, c = Count[IntegerQ /@ Sqrt[n - Range@Floor[n^(1/3)]^3], True]; If[ t[[c + 1]] == 0, t[[c + 1]] = n; Print[{c, n}] ]], {n, 731000000}]; t

a(7) from Donovan Johnson, Aug 31 2008

a(8)-a(12) from Donovan Johnson, Mar 01 2010

A123048 Semiprimes that are the sum of a positive square and a positive cube.

Original entry on oeis.org

9, 10, 26, 33, 57, 65, 82, 91, 122, 129, 134, 141, 145, 161, 177, 185, 206, 217, 226, 265, 289, 321, 362, 381, 407, 427, 485, 505, 511, 537, 566, 626, 633, 667, 681, 689, 703, 737, 745, 778, 785, 793, 841, 842, 849, 898, 901, 905, 985, 1018, 1041, 1057, 1081

Offset: 1

Jonathan Vos Post, Sep 25 2006

Semiprime analog of A066649, Primes of the form a^2 + b^3 with a, b > 0.

			a(1) = 9 = 2^3 + 1^2 = 3*3.
a(2) = 10 = 3^2 + 1^3 = 2*5.
a(3) = 26 = 5^2 + 1^3 = 2*13.
a(4) = 33 = 5^2 + 2^3 = 3*11.
a(5) = 57 = 7^2 + 2^3 = 3*19.
a(6) = 65 = 1^2 + 4^3 = 8^2 + 1^3 = 5*13.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A001358, A055394, A066649.

Select[ Union[ Plus @@@ Tuples[{Range[4^3]^2, Range[4^2]^3}]], # < 1082 && Plus @@ Last /@ FactorInteger[#] == 2 &] (* Giovanni Resta, Jun 12 2016 *)

A001358 INTERSECTION A055394.

More terms from Robert G. Wilson v, Sep 29 2006

A173795 Smallest prime that is the sum of a square and a positive cube in n different ways.

Original entry on oeis.org

3, 2, 17, 2089, 65537, 3193361, 445341529, 4190216689, 25140740257, 813368268793, 333413867957257, 1057543811051633, 1448734752622601

Offset: 0

Donovan Johnson, Mar 01 2010

nonn

From Kevin T. Acres Sep 22 2012 (Start)
Noam D. Elkies has determined, after an exhaustive search, to 7.5 * 10^15, that 1057543811051633 and 1448734752622601 are the lowest primes such that they are sums of a square and a positive cube in 11 and 12 different ways respectively.
107122676734733201 remains a potential, but unproven, candidate for n = 13 and 14.
107122676734733201 = 18076^3 + 327286985^2 = 56276^3 + 327023625^2 = 83413^3 + 326408198^2 = 128726^3 + 324021045^2 = 180440^3 + 318194601^2 = 319330^3 + 273056899^2 = 339826^3 + 260535965^2 = 344065^3 + 257666476^2 = 385333^3 + 223400642^2 = 403688^3 + 203312727^2 = 415601^3 + 187984920^2 = 447428^3 + 132481143^2 = 457750^3 + 105867851^2 = 460826^3 + 96236115^2
(End)

			a(0) = 3 (smallest prime not of the form a^2 + b^3).
a(1) = 2 = 1^2 + 1^3.
a(2) = 17 = 4^2 + 1^3 = 3^2 + 2^3.
a(3) = 2089 = 45^2 + 4^3 = 33^2 + 10^3 = 19^2 + 12^3.
a(4) = 65537 = 256^2 + 1^3 = 255^2 + 8^3 = 219^2 + 26^3 = 122^2 + 37^3.
a(5) = 3193361 = 1769^2 + 40^3 = 1606^2 + 85^3 = 1481^2 + 100^3 = 1047^2 + 128^3 = 285^2 + 146^3.
a(6) = 445341529 = 21023^2 + 150^3 = 20955^2 + 184^3 = 20898^2 + 205^3 = 20773^2 + 240^3 = 11195^2 + 684^3 = 2523^2 + 760^3.
a(7) = 4190216689 = 64729^2 + 72^3 = 64005^2 + 454^3 = 61219^2 + 762^3 = 42867^2 + 1330^3 = 36008^2 + 1425^3 = 20915^2 + 1554^3 = 17479^2 + 1572^3.
a(8) = 25140740257 = 155951^2 + 936^3 = 155440^2 + 993^3 = 153739^2 + 1146^3 = 151371^2 + 1306^3 = 126172^2 + 2097^3 = 121809^2 + 2176^3 = 116477^2 + 2262^3 = 38097^2 + 2872^3.
a(9) = 813368268793 = 901707^2 + 664^3 = 900233^2 + 1434^3 = 808084^2 + 5433^3 = 693429^2 + 6928^3 = 610741^2 + 7608^3 = 432210^2 + 8557^3 = 392373^2 + 8704^3 = 379349^2 + 8748^3 = 275817^2 + 9034^3.
a(10) = 333413867957257 = 18202887^2 + 12742^3 = 18190720^2 + 13593^3 = 16205565^2 + 41368^3 = 15621373^2 + 44712^3 = 14905630^2 + 48093^3 = 12187395^2 + 56968^3 = 11330919^2 + 58966^3 = 10486383^2 + 60682^3 = 9216035^2 + 62868^3 = 3854589^2 + 68296^3.
a(11) = 1057543811051633 = 7534^3 + 32513323^2 = 33184^3 + 31953127^2 = 46552^3 + 30929945^2 = 57377^3 + 29472900^2 = 69374^3 + 26901003^2 = 87989^3 + 19399158^2 = 94369^3 + 14735668^2 = 94874^3 + 14267997^2 = 95114^3 + 14038467^2 = 97952^3 + 10850535^2 = 101828^3 + 1302009^2
a(12) = 1448734752622601 = 30668^3 + 37681437^2 = 42326^3 + 37052775^2 = 49498^3 + 36434353^2 = 55000^3 + 35810051^2 = 68585^3 + 33557676^2 = 68890^3 + 33493199^2 = 78020^3 + 31206051^2 = 85838^3 + 28570377^2 = 88258^3 + 27590783^2 = 94820^3 + 24417699^2 = 105368^3 + 16700163^2 = 111901^3 + 6894130^2

Cf. A000040, A000290, A000578, A055394, A066649, A122956.

More terms (n=11 and 12) from Noam D Elkies.

Outdated comments removed by Kevin T. Acres, Sep 22 2012

cp's OEIS Frontend

A078375 Smallest prime factor of numbers which can be written as sum of a positive square and a positive cube.

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A078376 Greatest prime factor of numbers which can be written as sum of a positive square and a positive cube.

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A078380 Maximum exponent in prime factorization of numbers which can be written as sum of a positive square and a positive cube.

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A078381 Number of divisors of numbers which can be written as sum of a positive square and a positive cube.

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A078387 Moebius's mu of numbers which can be written as sum of a positive square and a positive cube.

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A078390 Composite numbers which can be written as sum of a positive square and a positive cube.

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A078393 Squarefree numbers which can be written as sum of a positive square and a positive cube.

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A122956 Least semiprime composed of a square and a positive cube in n different ways.

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A123048 Semiprimes that are the sum of a positive square and a positive cube.

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