A066649 -id:A066649 - cp's OEIS frontend

A123364 Primes of the form a^2 + b^3 (with repetition).

2, 5, 17, 17, 31, 37, 43, 73, 89, 89, 101, 113, 127, 197, 223, 233, 233, 241, 257, 269, 283, 337, 347, 353, 359, 379, 401, 443, 449, 449, 487, 521, 577, 577, 593, 593, 599, 677, 701, 733, 743, 811, 827, 829, 919, 953, 1009, 1019, 1049, 1051, 1097, 1129, 1153

Offset: 1

Zak Seidov, Oct 12 2006

nonn

Primes in A022549, A123291. Cf. A066649 Primes of the form a^2 + b^3 (without repetition), with a, b > 0.

			Each of 17, 89, 233 appears two times because 17=3^2+2^3=4^2+1^3, 89=5^2+4^3=9^2+2^3, 233=13^2+4^3=15^2+2^3;
2089 appears three times because 2089=19^2+12^3=33^2+10^3=45^2+4^3;
65537 appears four times because 65537=122^2+37^3=219^2+26^3=255^2+8^3=256^2+1^3.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A022549, A066649, A123291.

t={}; Do[AppendTo[t, n^2+m^3], {n, 100},{m, 100}]; Take[Select[Sort[t], PrimeQ], 60] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

A066650 Numbers not of the form a^2 + b^3 with a, b > 0.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 27, 29, 30, 32, 34, 35, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 69, 70, 71, 74, 75, 77, 78, 79, 81, 83, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Reinhard Zumkeller, Dec 17 2001

nonn

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

Cf. A066649. Complement of A055394.

q[n_] := Length[Reduce[a^2 + b^3 == n && a > 0 && b > 0, {a, b}, Integers]] == 0; Select[Range[100], q] (* Amiram Eldar, Mar 20 2025 *)

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

A232269 Number of ways to write 2*n + 1 = x + y (x, y > 0) with x^3 + y^2 and x^2 + y^2 both prime.

Original entry on oeis.org

1, 3, 1, 2, 3, 2, 1, 6, 4, 1, 4, 6, 3, 8, 1, 1, 6, 1, 1, 9, 2, 4, 5, 3, 1, 2, 7, 4, 5, 8, 1, 12, 4, 4, 12, 3, 4, 9, 10, 1, 5, 9, 5, 11, 7, 4, 9, 2, 4, 19, 1, 1, 14, 4, 6, 16, 8, 5, 8, 7, 2, 11, 8, 1, 16, 3, 5, 9, 4, 3, 8, 8, 6, 16, 4, 3, 12, 13, 5, 11, 5, 3, 10, 10, 7, 12, 7, 4, 17, 20, 1, 17, 5, 6, 15, 4, 5, 18, 5, 7

Offset: 1

Zhi-Wei Sun, Nov 22 2013

nonn

Conjecture: a(n) > 0 for all n > 0. Also, any odd integer greater than one can be written as x + y (0 < x < y) with x^3 + y^2 prime.
The conjecture implies that there are infinitely many primes of the form x^3 + y^2 (x, y > 0) with x^2 + y^2 also prime.
Note that Ming-Zhi Zhang ever asked (cf. A036468) whether any odd integer greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.

			a(10) = 1 since 2*10 + 1 = 1 + 20 with 1^2 + 20^2 = 1^3 + 20^2 = 401 prime.
a(15) = 1 since 2*15 + 1 = 25 + 6 with 25^2 + 6^2 = 661 and 25^3 + 6^2 = 15661 both prime.
a(40) = 1 since 2*40 + 1 = 55 + 26 with 55^2 + 26^2 = 3701 and 55^3 + 26^2 = 167051 both prime.
a(91) =1 since 2*91 + 1 = 85 + 98 with 85^2 + 98^2 = 16829 and 85^3 + 98^2 = 623729 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A036468, A066649, A220413, A232174, A232186.

a[n_]:=Sum[If[PrimeQ[x^3+(2n+1-x)^2]&&PrimeQ[x^2+(2n+1-x)^2],1,0],{x,1,2n}]
Table[a[n],{n,1,100}]

A235471 Primes whose base-8 representation also is the base-3 representation of a prime.

Original entry on oeis.org

2, 17, 73, 521, 577, 593, 1097, 1153, 4177, 8713, 33353, 33857, 37889, 41617, 65537, 65609, 69697, 70289, 70793, 74897, 262153, 262657, 266369, 331777, 331921, 336529, 336977, 529489, 533129, 533633, 590921, 594953, 598537, 2098241, 2101249, 2102417, 2134529

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
Seems to be a subsequence of A066649 and A123364.
Since the trailing digit of the base 7 expansion must (like all others) be less than 3, this is a subsequence of A045381.

			E.g., 17 = 21_8 and 21_3 = 7 are both prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A231478, A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482, A235615 - A235639. See the LINK for further cross-references.

b8b3pQ[n_]:=Module[{id8=IntegerDigits[n,8]},Max[id8]<3&&PrimeQ[ FromDigits[ id8,3]]]; Select[Prime[Range[160000]],b8b3pQ] (* Harvey P. Dale, Mar 16 2019 *)

is(p,b=3,c=8)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,8,3)&&print1(vector(#d=digits(p,3),i,8^(#d-i))*d~,",")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(.,3,8)

A193423 Primes of the form x^2+y^3 with x, y and y^3 - x^2 >= 0.

Original entry on oeis.org

2, 31, 43, 73, 89, 113, 241, 337, 347, 359, 379, 443, 487, 521, 593, 599, 733, 829, 953, 1009, 1049, 1129, 1213, 1289, 1361, 1367, 1753, 1777, 1907, 2017, 2089, 2213, 2297, 2341, 2393, 2521, 2689, 2753, 2953, 2969, 3221, 3391, 3571, 3631, 3797, 3833, 4051, 4133, 4159, 4177, 4217, 4457, 4721, 4937, 5237, 5813

Offset: 1

Juri-Stepan Gerasimov, Aug 27 2011

nonn

Subsequence of A066649.

			2 is in the sequence because 2 is prime, 2=1^2+1^3 and 1^3-1^2=0;
31 is in the sequence because 31 is prime, 31=2^2+3^3 and 3^3-2^2>0;
43 is in the sequence because 43 is prime, 43=4^2+3^3 and 3^3-4^2>0;

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

list(lim)=my(v=List(),t,B); lim\=1; for(b=1,sqrtn(lim+.5,3),B=b^3; for(a=1,min(lim-B,sqrtint(B)),if(isprime(t=B+a^2),listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Aug 28 2011

Corrected by D. S. McNeil, Aug 27 2011

cp's OEIS Frontend

A123364 Primes of the form a^2 + b^3 (with repetition).

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A066650 Numbers not of the form a^2 + b^3 with a, b > 0.

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

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A232269 Number of ways to write 2*n + 1 = x + y (x, y > 0) with x^3 + y^2 and x^2 + y^2 both prime.

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A235471 Primes whose base-8 representation also is the base-3 representation of a prime.

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