A256652 -id:A256652 - cp's OEIS frontend

A255830 Numbers D such that D^2 = A^4 + B^5 + C^6 for some positive integers A, B, C.

7, 9, 17, 33, 72, 89, 96, 99, 105, 137, 171, 213, 218, 240, 320, 459, 503, 513, 525, 616, 761, 792, 833, 1048, 1127, 1257, 1369, 1395, 1536, 1551, 2025, 2457, 2600, 2610, 3267, 3312, 3600, 3681, 4032, 4100, 4125, 4128, 4201, 4901, 4976, 5001, 5225, 5880, 5975, 6167

Offset: 1

M. F. Hasler, Apr 06 2015

nonn

The sequence has the infinite subsequence (4^n*(2^n+16), n=0,1,2,...), with corresponding (A,B,C) = (2^(n+2),2^(n+1),2^n).

See A256652 for terms whose square has more than one representation of the given form. See A256613 for the subsequence of terms such that A^2 + B^3 + C^4 is a square, cf. A180241. See A256091 for the analog for sums of 3rd, 4th and 5th power.

			(A, B, C) = (1, 4, 2) = 1^4 + 4^5 + 2^6 = 1 + 1024 + 64 = 1089 = 33^2, so 33 is a term.
(A, B, C) = (1, 4, 8) = 1^4 + 4^5 + 8^6 = 1 + 1024 + 262144 = 263169 = 513^2, so 513 is a term.

Cf. A256091, A256613, A180241, A180242, A256652.

is_A255830(D)=my(B,C=0,D2C6); while(1A255830(D)&&print1(D",")) \\ Converted to integer arithmetic by M. F. Hasler, May 01 2015

Inserted a(4)=33, a(18)=513 and removed doublet 1257 by Lars Blomberg, Apr 26 2015

A256603 Numbers D such that D^2 = A^3 + B^4 + C^5 has more than one solution in positive integers (A, B, C).

Original entry on oeis.org

305, 525, 1206, 1257, 1395, 2048, 2213, 3072, 4348, 6400, 16385, 16640, 16704, 20631, 22872, 23256, 30968, 31407, 32769, 62943, 74515, 77713, 77824, 79776, 82565, 84775, 90432, 98739, 117600, 121250, 133696, 163525, 165628, 171576, 198400, 199872, 243225

Offset: 1

M. F. Hasler, Apr 06 2015

nonn

A subsequence of A256091. Sequences A256604 and A256652 are the analog for A180241 and A255830.

			(A, B, C) = (32, 128, 1): 32^3 + 128^4 + 1^5 = 32768 + 268435456 + 1 = 268468225 = 16385^2
(A, B, C) = (1, 128, 8): 1^3 + 128^4 + 8^5 = 1 + 268435456 + 32768 = 268468225 = 16385^2
so 16385 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..185

Cf. A180241, A180242, A255830, A256091, A256604, A256613, A256652.

for(D=1,9999,f=-1;for(C=1,sqrtn(D^2-1,5),for(B=1,sqrtn(D^2-C^5-.5,4),ispower(D^2-C^5-B^4,3)&&f++&print1(D",")+next(3))))

Inserted a(11),a(16) and added a(19)-a(37) by Lars Blomberg, Apr 17 2015

A256604 Numbers D such that D^2 = A^2 + B^3 + C^4 has more than one solution in positive integers (A, B, C).

Original entry on oeis.org

5, 9, 12, 17, 19, 21, 23, 25, 28, 33, 35, 37, 38, 39, 42, 45, 46, 47, 51, 53, 55, 57, 59, 60, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 77, 80, 81, 82, 84, 87, 88, 89, 91, 93, 95, 97, 98, 99, 100, 103, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 123, 124, 127, 128, 129, 131, 132, 133, 134, 135, 136, 139, 141

Offset: 1

M. F. Hasler, Apr 06 2015

nonn

The subsequence of terms of A180241 whose square has more than one representation of the given form. See A256603 and A256652 are the analog for A256091 and A255830.

			(A, B, C) = (4, 8, 1): 4^2 + 8^3 + 1^4 = 16 + 512 + 1 = 529 = 23^2 and
(A, B, C) = (1, 8, 2): 1^2 + 8^3 + 2^4 = 1 + 512 + 16 = 529 = 23^2,
so 23 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10088

Cf. A180241, A180242, A256613, A255830.

for(D=2,199,my(f=-1,B,D2C4);for(C=1,sqrtint(D),D2C4=D^2-C^4; B=0;while(B++^3M. F. Hasler, May 01 2015

Inserted a(7)=23 by Lars Blomberg, Apr 26 2015

cp's OEIS Frontend

A255830 Numbers D such that D^2 = A^4 + B^5 + C^6 for some positive integers A, B, C.

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A256603 Numbers D such that D^2 = A^3 + B^4 + C^5 has more than one solution in positive integers (A, B, C).

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A256604 Numbers D such that D^2 = A^2 + B^3 + C^4 has more than one solution in positive integers (A, B, C).

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