A111925 -id:A111925 - cp's OEIS frontend

A111909 Numbers that cannot be represented as a^4 + b^2 with a, b > 0.

1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23, 24, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 84, 86, 87, 88

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

			3 cannot be represented as a^4 + b^2 and thus is in this sequence while 10 = 1^4 + 3^2 is not.

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

Cf. A111925 (complement), A055394, A022549. A022544 is a subsequence.

list(lim)=lim\=1; my(v=List(),u=vectorsmall(lim),m2); for(m=1,sqrtint(lim-1), m2=m^2; for(n=1,sqrtnint(lim-m2,4), u[m2+n^4]=1)); for(i=1,#u, if(!u[i], listput(v,i))); Set(v) \\ Charles R Greathouse IV, Sep 01 2015

a(n) = n + O(n^(3/4)). - Charles R Greathouse IV, Sep 01 2015

A363711 Number of ways to write n as sum of a positive square and a positive fourth power.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 1

Seiichi Manyama, Jun 17 2023

Cf. A010052, A063725, A078359, A363713.

Cf. A111925, A363714.

G.f.: (Sum_{k>=1} x^(k^2)) * (Sum_{k>=1} x^(k^4)).

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

Original entry on oeis.org

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

A273908 Integers z such that x^2 + y^4 = z^6 where x, y, z > 0, is soluble.

Original entry on oeis.org

5, 15, 20, 34, 39, 41, 45, 55, 60, 65, 80, 85, 111, 125, 135, 136, 145, 150, 156, 164, 175, 180, 194, 219, 220, 240, 245, 255, 260, 265, 299, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 495, 500, 505, 514, 525, 540, 544

Offset: 1

Altug Alkan, Jun 03 2016

nonn

A271576 is a subsequence.
Terms that are not in A271576 are 55, 220, 299, ...
Sequence is infinite since if k is a term then also t^2*k is a term, for every t>0. - Giovanni Resta, Jun 04 2016

			5 is a term because 75^2 + 10^4 = 5^6.

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

Cf. A111925, A266212, A271576.

q[n_] := {} != Select[Range[n^(1/4)]^4, n > # && IntegerQ@ Sqrt[n - #] &]; Select[ Range[100], q[#^6] &] (* Giovanni Resta, Jun 04 2016 *)

A363714 Numbers that are the sum of a positive square and a positive fourth power in more than one way.

Original entry on oeis.org

17, 65, 82, 97, 145, 257, 272, 305, 337, 377, 442, 481, 545, 626, 641, 706, 785, 881, 1025, 1040, 1105, 1297, 1312, 1345, 1377, 1525, 1552, 1780, 1850, 1921, 1937, 2225, 2320, 2385, 2402, 2417, 2465, 2482, 2501, 2657, 2977, 3026, 3505, 3697, 3737, 3845, 4097, 4100, 4105, 4112, 4177, 4225, 4321

Offset: 1

Seiichi Manyama, Jun 17 2023

nonn

Numbers k such that A363711(k) > 1.

			a(1) = 17 = 1^2 + 2^4 = 4^2 + 1^4.

Cf. A111925, A363711.

A274015 Least number of the form x^2 + y^4 in exactly n ways where x and y are nonzero integers.

Original entry on oeis.org

2, 17, 3026, 141457, 4740625, 113260225, 205117028945, 3234825286225

Offset: 1

Altug Alkan, Jun 06 2016

			a(1) = 2 = 1^2 + 1^4.
a(2) = 17 = 1^2 + 2^4 = 4^2 + 1^4.
a(3) = 3026 = 25^2 + 7^4 = 49^2 + 5^4 = 55^2 + 1^4.
a(4) = 141457 = 191^2 + 18^4 = 321^2 + 14^4 = 336^2 + 13^4 = 376^2 + 3^4.
a(5) = 4740625 = 2177^2 + 6^4 = 2175^2 + 10^4 = 1800^2 + 35^4 = 800^2 + 45^4 = 513^2 + 46^4.
a(6) = 113260225 = 10640^2 + 15^4 = 10593^2 + 32^4 = 10368^2 + 49^4 = 10015^2 + 60^4 = 7967^2 + 84^4 = 5640^2 + 95^4.

Cf. A111925.

do(lim,stride=10^7)=lim\=1;my(v,t,r,top);forstep(n=0,lim-1,stride,top=min(n+stride,lim);v=vectorsmall(top-n);for(y=1,sqrtnint(top-1,4),t=y^4;for(x=if(n>t,sqrtint(n-t)+1,1),sqrtint(top-t),v[t+x^2-n]++));for(i=1,#v,if(v[i]>r,r=v[i];print(r" "i+n)))) \\ Charles R Greathouse IV, Jun 07 2016

a(5)-a(6) from Charles R Greathouse IV, Jun 07 2016

a(7)-a(8) from Giovanni Resta, Jun 07 2016

A274026 Perfect powers m^k such that m^k = a^2 + b^4 = c^3 + d^5 for some positive integers a, b, c, d.

Original entry on oeis.org

13140625, 36859543552, 49762009476, 87169610025, 3324163986441, 2988330556640625, 10155995666841600, 28920784535654400, 34328125000000000, 65388757868609536, 101445409544601600, 275625000000000000, 428123439576907776

Offset: 1

Altug Alkan, Jun 07 2016

nonn

Intersection of A001597, A100293, A111925.
3625^2 = 13140625 is the least number with this property.
Sequence is infinite because if m^k = a^2 + b^4 = c^3 + d^5 is a term, then (m*t^60)^k = (a*t^(30*k))^2 + (b*t^(15*k))^4 = (c*t^(10*k))^3 + (d*t^(12*k))^5 is also a term for every t>1. - Giovanni Resta, Jun 07 2016

			13140625 is a term because 13140625 = 3625^2 = 2625^2 + 50^4 = 150^3 + 25^5.

Cf. A001597, A100293, A111925.

isA111925(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0
isA100293(n)=for(y=1, sqrtnint(n-1, 5), if(ispower(n-y^5, 3), return(1))); 0
list(lim)=my(v=List(), b4, t); for(e=2,logint(lim\=1,2), for(m=2,sqrtnint(lim,e), t=m^e; if(isA111925(t) && isA100293(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jun 07 2016

a(2)-a(13) from Giovanni Resta, Jun 07 2016

A274033 Numbers k such that k = a^2 + b^4 and n^2 = c^3 + d^5 for some positive integers a, b, c, d.

Original entry on oeis.org

81250, 1062882, 11529602, 12500000, 170061120, 200000000, 2662400000, 5897400777, 7309688832, 12814453125, 34297420960, 37019531250

Offset: 1

Altug Alkan, Jun 07 2016

In other words, values of a^2 + b^4 such that (a^2 + b^4)^2 is of the form c^3 + d^5 where a, b, c, d > 0.
81250 is the least number with this property.
Sequence is infinite: If n = a^2 + b^4 and n^2 = c^3 + d^5, then n*k^60 = (a*k^30)^2 + (b*k^15)^4 and (n*k^60)^2 = (c*k^40)^3 + (d*k^24)^5. So if n is in this sequence, then n*k^60 is in this sequence for all nonzero values of k.

			81250 is a term because 81250 = 175^2 + 15^4 and 81250^2 = 1875^3 + 25^5.

Cf. A100293, A111925.

a(2)-a(6) from Giovanni Resta, Jun 07 2016
a(7) from Chai Wah Wu, Jun 14 2016
a(8)-a(12) from Chai Wah Wu, Jul 07 2016

A274173 Numbers n such that n is not divisible by 5 and n^k is of the form a^2 + b^4 for all k not divisible by 4.

Original entry on oeis.org

609076, 9745216, 28998521, 36924641

Offset: 1

Altug Alkan, Jun 12 2016

Numbers n such that n, n^2 and n^3 is in A111925 and n is not divisible by 5.

Since there cannot be fourth power in A111925, this sequence focuses on the values of k that is not divisible by 4 in definition.

			609076 is a term because 609076 = 710^2 + 18^4, 609076^2 = 581940^2 + 424^4, 609076^3 = 475342920^2 + 676^4 and it is not divisible by 5.

Cf. A111925.

isA111925(n) = for(b=1, sqrtnint(n-1, 4), if(issquare(n-b^4), return(1))); 0;
isok(n) = n%5 != 0 && isA111925(n) && isA111925(n^2) && isA111925(n^3);

A274356 Numbers n such that n^k is of the form (a^2 + b^4)/2 for all k > 0 (a, b > 0).

Original entry on oeis.org

1, 5, 16, 25, 41, 80, 81, 125, 256, 400, 405, 425, 625, 656, 841, 1225, 1280, 1296, 1681, 2000, 2025, 2401, 3125, 3321, 3721, 4096, 6400, 6480, 6561, 6800, 8281, 8381, 10000, 10125, 10496, 12005, 13456, 14161, 14641, 15625, 19600, 20480, 20736, 25625

Offset: 1

Altug Alkan, Jun 18 2016

nonn

Numbers n such that n^k is the average of a nonzero square and a nonzero fourth power for all k > 0.
If n^k = (a^2 + b^4)/2, then n^(k+4) = ((n^2*a)^2 + (n*b)^4)/2. So this sequence lists numbers n such that 2*n, 2*n^2 and 2*n^3 are in A111925.
If n is in this sequence, then n^t is also in this sequence for all t > 1. So in this sequence there are infinitely many prime powers.
There are infinitely many (x, y) pairs in this sequence where x and y are distinct terms of this sequence such that x*y is also in this sequence.

			5 is a term because 5 = (3^2 + 1^4)/2, 5^2 = (7^2 + 1^4)/2, 5^3 = (13^2 + 3^4)/2, 5^4 = ((5^2)^2 + 5^4)/2.

Cf. A111925.

isA111925(n)=for(b=1, sqrtnint(n-1, 4), if(issquare(n-b^4), return(1))); 0;
lista(nn)=for(n=1, nn, if(isA111925(2*n) && isA111925(2*n^2) && isA111925(2*n^3), print1(n, ", ")));

cp's OEIS Frontend

A111909 Numbers that cannot be represented as a^4 + b^2 with a, b > 0.

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A363711 Number of ways to write n as sum of a positive square and a positive fourth power.

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

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A273908 Integers z such that x^2 + y^4 = z^6 where x, y, z > 0, is soluble.

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A363714 Numbers that are the sum of a positive square and a positive fourth power in more than one way.

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A274015 Least number of the form x^2 + y^4 in exactly n ways where x and y are nonzero integers.

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A274026 Perfect powers m^k such that m^k = a^2 + b^4 = c^3 + d^5 for some positive integers a, b, c, d.

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A274033 Numbers k such that k = a^2 + b^4 and n^2 = c^3 + d^5 for some positive integers a, b, c, d.

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A274173 Numbers n such that n is not divisible by 5 and n^k is of the form a^2 + b^4 for all k not divisible by 4.

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