A055394 -id:A055394 - cp's OEIS frontend

A303376 Numbers of the form a^6 + b^7, with integers a, b > 0.

2, 65, 129, 192, 730, 857, 2188, 2251, 2916, 4097, 4224, 6283, 15626, 15753, 16385, 16448, 17113, 17812, 20480, 32009, 46657, 46784, 48843, 63040, 78126, 78189, 78854, 82221, 93750, 117650, 117777, 119836, 124781, 134033, 195774, 262145, 262272, 264331, 278528, 279937

Offset: 1

M. F. Hasler, Apr 22 2018

Although it is easy to produce many terms of this sequence, it is nontrivial to check efficiently whether a very large number is of this form.

			The sequence starts with 1^6 + 1^7, 2^6 + 1^7, 1^6 + 2^7, 2^6 + 2^7, 3^6 + 1^7, 3^6 + 2^7, ...

Cf. A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).

Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).

With[{nn=40}, Take[Union[First[#]^6 + Last[#]^7&/@Tuples[Range[nn], 2]], nn]] (* Vincenzo Librandi, Apr 25 2018 *)

is(n,k=6,m=7)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,n)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
A303376_vec(L=10^5,k=6,m=7,S=List())={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k), listput(S,a^m+b^k)));Set(S)} \\ all terms up to limit L

A329807 Numbers k such that k, k+1, k+2 and k+3 are all sums of a positive square and a positive cube.

Original entry on oeis.org

126, 350, 8125, 12742, 19879, 29240, 42974, 76728, 91329, 109241, 140750, 209222, 254681, 258272, 297423, 482958, 744901, 755169, 918601, 986174, 1026214, 1418606, 1515227, 1521233, 1888216, 2082977, 2216080, 2317257, 3510926, 4180848, 4316417, 4330888, 4836895

Offset: 1

Jianing Song, Nov 21 2019

nonn

It is known that there are infinitely many k such that k, k+1, k+2 are all sums of a positive square and a positive cube (see A055394 and A295787). It is natural to ask if this sequence is infinite. There are 243 members here below 10^9.

There are 2 pairs of consecutive numbers below 10^9: (16597502, 16597503) and (593825496, 593825497). Are there infinitely many k such that k, k+1, k+2, k+3 and k+4 are all sums of a positive square and a positive cube?

			350 is here because 350 = 15^2 + 5^3, 351 = 18^2 + 3^3, 352 = 3^2 + 7^3 and 353 = 17^3 + 4^3.

Jianing Song, Table of n, a(n) for n = 1..243 (All terms <= 10^9)

Cf. A055394, A329808, A295787.

isA329807(n) = is(n)&&is(n+1)&&is(n+2)&&is(n+3) \\ is() is defined in A055394.

A329808 Numbers k such that both k and k+1 are sums of a positive square and a positive cube.

Original entry on oeis.org

9, 36, 43, 72, 100, 126, 127, 128, 170, 196, 225, 232, 264, 289, 320, 350, 351, 352, 359, 368, 407, 424, 441, 442, 485, 486, 511, 512, 539, 576, 632, 656, 700, 703, 737, 784, 792, 810, 841, 848, 849, 872, 908, 953, 968, 1000, 1018, 1169, 1183, 1213, 1225, 1240, 1296

Offset: 1

Jianing Song, Nov 21 2019

nonn

It is quite easy to give a constructive proof that this sequence is infinite. For example, 64*x^3 + 49*x^2 + 14*x + 1 = (7*x+1)^2 + (4*x)^3 and 64*x^3 + 49*x^2 + 14*x + 2 = (x+1)^2 + (4*x+1)^3. Moreover, if 97*x^2 + 2*x + 1 = y^2, then 64*x^3 + 49*x^2 + 14*x = y^2 + (4*x-1)^3. Obviously there are infinitely many solutions to 97*x^2 + 2*x + 1 = y^2, so there are infinitely many k such that k, k+1 and k+2 are all sums of a positive square and a positive cube.

			43 is a term because 43 = 4^2 + 3^3, 44 = 6^2 + 2^3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A055394, A295787, A329807.

isA329808(n) = is(n)&&is(n+1) \\ is() is defined in A055394.

A078385 Euler's totient of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

1, 4, 6, 4, 4, 16, 8, 12, 12, 30, 20, 12, 36, 42, 20, 20, 24, 36, 36, 48, 32, 24, 72, 36, 32, 40, 88, 72, 40, 100, 36, 112, 60, 36, 126, 64, 84, 66, 92, 112, 72, 40, 72, 132, 80, 64, 108, 56, 116, 144, 108, 84, 196, 64, 102, 96, 180, 80, 222, 120, 112, 112, 232, 240, 80

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

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

Cf. A000010, A055394.

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

a(n) = A000010(A055394(n)).

A078388 Arithmetic derivative of numbers which can be written as sum of a positive square and a positive cube.

Original entry on oeis.org

1, 1, 6, 7, 16, 1, 44, 15, 32, 1, 14, 60, 1, 1, 48, 45, 56, 22, 51, 18, 72, 156, 1, 80, 176, 43, 1, 20, 140, 1, 216, 1, 63, 165, 1, 448, 46, 69, 50, 34, 152, 185, 236, 30, 168, 129, 123, 151, 62, 42, 216, 252, 1, 284, 105, 432, 38, 284, 1, 240, 115, 356, 1, 1, 211, 456

Offset: 1

Reinhard Zumkeller, Nov 25 2002

nonn

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

Cf. A003415, A055394, A078386.

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

a(n) = A003415(A055394(n)). - Amiram Eldar, Mar 27 2025

A151952 Palindromes that are the sum of a positive square and a positive cube.

Original entry on oeis.org

2, 5, 9, 33, 44, 101, 141, 161, 171, 232, 252, 353, 414, 424, 464, 505, 525, 616, 626, 656, 737, 848, 868, 898, 909, 919, 969, 1001, 1441, 1881, 2112, 2222, 2332, 2552, 3113, 3993, 4994, 5225, 6776, 8228, 9009, 9559, 10001, 10401, 10601, 10801, 12721

Offset: 1

Claudio Meller, Jul 11 2009

Intersection of A055394 and A002113.

161= 6^2+5^3 ; 252= 6^2+6^3

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Take[With[{nn=100},Select[Union[Flatten[Table[s^2+c^3,{s,nn},{c,nn}]]],PalindromeQ]],50] (* Harvey P. Dale, Jun 12 2025 *)

A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

Original entry on oeis.org

2, 10, 38, 52, 350, 542, 1102, 1460, 1522, 1732, 2510, 2642, 2768, 3692, 4592, 4658, 4690, 7238, 8180, 8320, 8960, 11392, 13468, 14920, 15908, 16600, 16832, 17878, 18820, 19100, 21532, 22060, 23240, 23842, 23968, 24622, 26428, 26638, 27170

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 04 2010

nonn

p = n^2 + 3, q = n^2 + 3^2 = p+6, r = n^2 + 3^3 = p+18 to be primes.
Trivially n is not a multiple of 3 and necessarily LSD of such n is e = 0, 2 or 8 as k^2+3^2 is a multiple of 5 for k = 4 or 6.
Note n^2 + m^k prime (k = 1, 2, 3) in case of m = 2 is (n^2+2,n^2+2^2,n^2+2^3) = (p,p+2,p+6): i.e., a "near square" prime triple of the first kind.
Case k=2: q is also a Pythagorean prime (A002144)
n = 350: first case where p = 122503 = prime(i), q and r are consecutive primes (i = 122503), sod(p) = sod(i) = 13, a so-called Honaker prime.
p = prime(i), q, r consecutive primes, (n,i): (350,11524) (542,25517) (1460,157987) (3692,887608) (4592,1335102) (4690,1389018).

			2^2 + 3 = 7 = prime(4), 2^2 + 3^2 = 13 = prime(6), 2^2 + 3^3 = 31 = prime(11), 2 is first term.
10^2 + 3 = 103 = prime(27), 10^2 + 3^2 = 109 = prime(29), 10^2 + 3^3 = 127 = prime(31), 10 is 2nd term.
Curiously k=0: 10^2 + 3^0 = 101 = prime(26), k=4: 10^2 + 3^4 = 181 = prime(42), necessarily LSD for such n is e = 0, k= 5: 10^2 + 3^5 = 7^3, k=6: 10^2 + 3^6 = 829 = prime(145), 10^2 + 3^7 = 2287 = prime(340), 10^2 + 3^8 = 6661 = prime(859)
n = 8180, primes for exponents k = 0, 1, 2, 3 and 4: p=66912403=prime(3946899), q=66912409=prime(3946900), r=66912427=prime(3946902), n^2+3^0=66912401=prime(3946898) and n^2+3^4=66912481=prime(3946905).
n = 8960, primes for exponents k = 1, 2, 3, 4, 5 and 6: p=80281603=prime(4684862), q=80281609=prime(4684863), r=80281627=prime(4684865), n^2+3^4=80281681=prime(4684868), n^2+3^5=80281843=prime(4684877), n^2+3^5=80282329=prime(4684904).

F. Padberg, Zahlentheorie und Arithmetik, Spektrum Akademie Verlag, Heidelberg-Berlin 1999.
M. du Sautoy, Die Musik der Primzahlen: Auf den Spuren des groessten Raetsels der Mathematik, Deutscher Taschenbuch Verlag, 2006.

C. Hooley On nonary cubic forms, Journal für die reine und angewandte Mathematik 386, pp. 32-98, 1988.

Cf. A002144, A049422, A055394, A086380, A114272.

is(n)=isprime(n^2+3) && isprime(n^2+9) && isprime(n^2+27) \\ Charles R Greathouse IV, Jun 13 2017

More terms from R. J. Mathar, Nov 01 2010

A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

Original entry on oeis.org

10, 12, 200, 268, 340, 418, 488, 530, 838, 840, 1102, 1720, 1830, 2240, 2410, 2768, 3148, 3202, 3318, 3322, 3958, 4162, 4610, 5080, 5672, 5700, 5722, 5870, 6178, 6302, 6480, 7490, 8130, 8262, 8888, 9132, 9602, 9618, 10638

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 31 2010

nonn

Subsequence of A176969.

The least-significant digit of all terms is one of 0, 2 or 8, because for odd digits m^2 + 13^k would be even (not prime), and for digits 4 and 6 the number m^2 + 13^2 is a multiple of 5.

			m=10 is in the sequence because 10^2 + 13 = 113 = prime(30), 10^2 + 13^2 = 269 = prime(57), 10^2 + 13^3 = 2297 = prime(342).
m=8888 is in the sequence because 8888^2 + 13 = 78996557 = prime(4614261), 8888^2 + 13^2 = 78996713 = prime(4614269), 8888^2 + 13^3 = 78998741 = prime(4614379).
m=6480 yields a prime 6480^2 + 13^k even for k=0.
m=7490 yields a prime 7490^2 + 13^k even for k=0 and k=4.

B. Bunch: The Kingdom of Infinite Number: A Field Guide, W. H. Freeman, 2001.
R. Courant, H. Robbins: What Is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1996.
G. H. Hardy, E. M. Wright, E. M., An Introduction to the Theory of Numbers (5th edition), Oxford University Press, 1980.

Cf. A000040, A000290, A055394, A056899, A086380, A113536, A176371, A176969.

keyword:base removed by R. J. Mathar, Jul 13 2010

A273546 Integers n such that n^n is the average of a nonzero square and a positive cube.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 12, 13, 14, 18, 24, 25

Offset: 1

Altug Alkan, May 26 2016

Integers n such that n^n = (x^2 + y^3)/2 where x, y > 0, is soluble.

			13 is a term because 13^13 = (24134045^2 + 28561^3)/2.

Cf. A000312, A055394, A267415.

isA055394(n)=for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
is(n)=isA055394(2*n^n) \\ Charles R Greathouse IV, Jun 08 2016

a(10)-a(12) from Giovanni Resta, May 26 2016

A274053 Least squarefree number that is the sum of a nonzero square and a positive cube in exactly n ways.

Original entry on oeis.org

2, 17, 2089, 27289, 3030569, 6808609, 1632201497

Offset: 1

Altug Alkan, Jun 08 2016

			a(1) = 2 because 2 = 1^2 + 1^3.
a(2) = 17 because 17 = 4^2 + 1^3 = 3^2 + 2^3.
a(3) = 2089 because 2089 = 45^2 + 4^3 = 33^2 + 10^3 = 19^2 + 12^3.
a(4) = 27289 because 27289 = 29*941 = 165^2 + 4^3 = 129^2 + 22^3 = 108^2 + 25^3 = 17^2 + 30^3.

Cf. A005117, A055394, A060835, A122956, A173795.

a(6)-a(7) from Giovanni Resta, Jun 12 2016

cp's OEIS Frontend

A303376 Numbers of the form a^6 + b^7, with integers a, b > 0.

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A329807 Numbers k such that k, k+1, k+2 and k+3 are all sums of a positive square and a positive cube.

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A329808 Numbers k such that both k and k+1 are sums of a positive square and a positive cube.

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A078385 Euler's totient of numbers which can be written as sum of a positive square and a positive cube.

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A078388 Arithmetic derivative of numbers which can be written as sum of a positive square and a positive cube.

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A151952 Palindromes that are the sum of a positive square and a positive cube.

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A177173 Numbers n such that n^2 + 3^k is prime for k = 1, 2, 3.

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A178639 Numbers m such that all three values m^2 + 13^k, k = 1, 2, 3, are prime.

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A273546 Integers n such that n^n is the average of a nonzero square and a positive cube.

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