A001015 -id:A001015 - cp's OEIS frontend

A240676 Number of digits in the decimal expansion of n^7.

1, 1, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Michelle Tu, Apr 28 2014

			a(1) = 1, because 1^7 = 1.
a(2) = 4, because 2^7 = 1024.

a(n)=max(#digits(n^7),1) \\ Charles R Greathouse IV, May 12 2014

a(n) = A055642(n^7) = A055642(A001015(n)).

A291829 Numbers k such that k^3 is the sum of two positive 7th powers.

Original entry on oeis.org

32, 4096, 69984, 524288, 2500000, 8957952, 26353376, 67108864, 153055008, 320000000, 623589472, 1146617856, 2007952544, 3373232128, 5467500000, 8589934592, 13130837536, 19591041024, 28603895648, 35723051649, 40960000000, 57634833312, 79819452416, 108954414304

Offset: 1

XU Pingya, Sep 03 2017

nonn

When a^7 + b^7 = m, (m^2*a)^7 + (m^2*b)^7 = m^15 is a cube.

When k in this sequence, k*(n^7) (n = 2, 3, ... ) is also in this sequence.

			32^3 = 4^7 + 4^7, so 32 is in the sequence.
35723051649^3 = 16641^7 + 33282^7, so 35723051649 is in the sequence.

Cf. A000578, A001015, A000404, A009003, A050801.

lst={};Do[If[IntegerQ[(n^3-a^7)^(1/7)],AppendTo[lst,n]],{n,1.467*10^11},{a,(n^3/2)^(1/7)}]; lst

A291830 Numbers k such that k^4 is sum of two positive 7th powers.

Original entry on oeis.org

4, 512, 8748, 16641, 65536, 312500, 1119744, 2130048, 3294172, 4787344, 5359225, 8388608, 19131876, 36393867, 40000000, 77948684, 143327232, 250994068, 268468225, 272646144, 344882041, 421654016, 612780032, 683437500, 685980800, 1073741824, 1300078125

Offset: 1

XU Pingya, Sep 03 2017

nonn

When a^7 + b^7 = m, (ma)^7 + (mb)^7 = m^8 is 4th power.

When k in this sequence, k*(n^7) (n = 2, 3, ... ) is also in this sequence.

			4^4 = 2^7 + 2^7, so 4 is in the sequence.
16641^4 = 129^7 + 358^7, so 16641 is in the sequence.

Cf. A000583, A001015, A000404, A009003, A050801.

lst={};Do[If[IntegerQ[(n^4-a^7)^(1/7)],AppendTo[lst,n]],{n,1.4*10^9},{a,(n^4/2)^(1/7)}]; lst

A291831 Numbers k such that k^5 is sum of two positive 7th powers.

Original entry on oeis.org

8, 1024, 17496, 131072, 625000, 2146689, 2239488, 6588344, 16777216, 38263752, 80000000, 155897368, 74776192, 86654464, 501988136, 843308032, 1366875000, 2147483648, 3282709384, 4694808843, 4897760256, 7150973912, 10240000000, 10474708672, 12406605875

Offset: 1

XU Pingya, Sep 03 2017

nonn

When a^7 + b^7 = m, (m^2*a)^7 + (m^2*b)^7 = m^15 is 5th power.

When k in this sequence, k*(n^7) (n = 2, 3, ... ) is also in this sequence.

			8^5 = 4^7 + 4^7, so 8 is in the sequence.
2146689^5 = 16641^7 + 33282^7, so 2146689 is in the sequence.

Cf. A000584, A001015, A000404, A009003, A050801.

lst={};Do[If[IntegerQ[(n^5-a^7)^(1/7)],AppendTo[lst,n]],{n,1.3*10^10},{a,(n^5/2)^(1/7)}]; lst

A291832 Numbers k such that k^6 is sum of two positive 7th powers.

Original entry on oeis.org

64, 8192, 139968, 1048576, 5000000, 17915904, 52706752, 134217728, 306110016, 640000000, 1247178944, 2293235712, 4015905088, 6746464256, 10935000000, 17179869184, 26261675072, 39182082048, 57207791296, 81920000000, 115269666624, 159638904832, 217908828608

Offset: 1

XU Pingya, Sep 03 2017

nonn

When a^7 + b^7 = m, (m^5*a)^7 + (m^5*b)^7 = m^36 is 6th power.

When k in this sequence, k*(n^7) (n = 2, 3, ... ) is also in this sequence.

			64^6 = 32^7 + 32^7, so 64 is in the sequence.
(5000000)^6 = (500000)^7 + (500000)^7, so 5000000 is in the sequence.

Cf. A001014, A001015, A000404, A009003, A050801.

lst={};Do[If[IntegerQ[(n^6-a^7)^(1/7)],AppendTo[lst,n]],{n,2.9*10^11},{a,(n^6/2)^(1/7)}]; lst

A324270 a(n) = 137^(7n).

Original entry on oeis.org

13, 10706059, 8816899947037, 7261096233082692091, 5979824975081619492698413, 4924642999453642161875329137259, 4055655269699050826917294183685688637, 3340006507773765415151949203915063077180891, 2750638979431530091290481703239822791770782516813, 2265269477037980585971637173331233381403285546243728459

Offset: 0

Stefano Spezia, Mar 22 2019

x = a(n) and y = A324266(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 7^(14*n+3) = 4*y^7 (see Theorem 2.1 in Chakraborty, Hoque and Sharma).

			For a(0) = 13 and A324266(0) = 2, 13^2 + 7^3 = 512 = 4*2^7.

K. Chakraborty, A. Hoque, R. Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (823543).

Cf. A324266 (2*49^n), A001015 (seventh powers), A000420 (powers of 7), A008595 (multiples of 13).

GAP
```
List([0..20], n->13*823543^n);
```
Magma
```
[13*823543^n: n in [0..20]];
```
Maple
```
a:=n->13*823543^n: seq(a(n), n=0..20);
```
Mathematica
```
13 823543^Range[0, 20]
```
PARI
```
a(n) = 13*823543^n;
```

O.g.f.: 13/(1 - 823543*x).
E.g.f.: 13*exp(823543*x).
a(n) = 823543*a(n-1) for n > 0.
a(n) = 13*823543^n.
a(n) = A008595(A001015((A000420(n)))).

A364654 Numbers which are the sum or difference of two seventh powers.

Original entry on oeis.org

0, 1, 2, 127, 128, 129, 256, 2059, 2186, 2187, 2188, 2315, 4374, 14197, 16256, 16383, 16384, 16385, 16512, 18571, 32768, 61741, 75938, 77997, 78124, 78125, 78126, 78253, 80312, 94509, 156250, 201811, 263552, 277749, 279808, 279935, 279936, 279937, 280064, 282123, 296320

Offset: 1

Geoffrey Caveney, Jul 31 2023

nonn

Don Zagier's conjecture that the polynomial x^7 + 3y^7 is injective on rational numbers is equivalent to the non-existence of any term in this sequence that is exactly 3 times another term in this sequence.

			2059 = 3^7 - 2^7, 2315 = 3^7 + 2^7, 358061 = 6^7 + 5^7, 543607 = 7^7 - 6^7.

Mathematics Stack Exchange, Are all bivariate polynomials of degree < 7 non-injective on rational numbers?
Bjorn Poonen, Multivariable polynomial injections on rational numbers, arXiv:0902.3961 [math.NT], 2009-2010; Acta Arith. 145 (2010), no. 2, 123-127.

Cf. A001015, A247099.

T=thueinit('z^7+1);
is(n) = (n==0) || (#thue(T, n)>0); \\ Michel Marcus, Aug 01 2023

cp's OEIS Frontend

A240676 Number of digits in the decimal expansion of n^7.

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A291829 Numbers k such that k^3 is the sum of two positive 7th powers.

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A291830 Numbers k such that k^4 is sum of two positive 7th powers.

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A291831 Numbers k such that k^5 is sum of two positive 7th powers.

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A291832 Numbers k such that k^6 is sum of two positive 7th powers.

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Mathematica

A324270 a(n) = 13*7^(7*n).

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GAP

Magma

Maple

Mathematica

PARI

Formula

A364654 Numbers which are the sum or difference of two seventh powers.

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PARI

A324270 a(n) = 137^(7n).