A130555 -id:A130555

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A132214

Numbers that are sums of seventh powers of two distinct primes.

+10
4

2315, 78253, 80312, 823671, 825730, 901668, 19487299, 19489358, 19565296, 20310714, 62748645, 62750704, 62826642, 63572060, 82235688, 410338801, 410340860, 410416798, 411162216, 429825844, 473087190, 893871867, 893873926

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This is to 7th powers as A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as the polynomial x^7 + y^7 factors over Z. Note however that A132215, which is the analog for eighth powers, can be prime, as seen also in A132216.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

{A001015(A000040(i)) + A001015(A000040(j)) for i > j}.

EXAMPLE

a(1) = 2^7 + 3^7 = 2315 = 5 * 463.

MAPLE

P:= select(isprime, [2, seq(i, i=3..100, 2)]): nP:= nops(P):

N:= 2^7 + P[-1]^7:

sort(convert(select(`<=`, {seq(seq(P[i]^7+P[j]^7, j=i+1..nP), i=1..nP-1)}, N), list)); # Robert Israel, Jul 01 2024

MATHEMATICA

Select[Sort[ Flatten[Table[Prime[n]^7 + Prime[k]^7, {n, 15}, {k, n - 1}]]], # <= Prime[15^7] &]

Union[Total/@(Subsets[Prime[Range[10]], {2}]^7)] (* Harvey P. Dale, Jan 03 2012 *)

CROSSREFS

Cf. A000040, A001015, A050997, A120398, A122616, A130873, A130555, A132215, A132216.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Aug 13 2007

STATUS

approved

A132215

Numbers that are sums of eighth powers of two distinct primes.

+10
3

6817, 390881, 397186, 5765057, 5771362, 6155426, 214359137, 214365442, 214749506, 220123682, 815730977, 815737282, 816121346, 821495522, 1030089602, 6975757697, 6975764002, 6976148066, 6981522242, 7190116322, 7791488162

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This is to 8th powers as A132214 is to 7th powers, A130555 is to 6th powers, A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These CAN be prime, as the polynomial x^8 + y^8 is irreducible over Z, as seen in A132216. The first such example is a(11) = A132216(1) = 2^8 + 13^8 = 256 + 815730721 = 815730977, which is prime.

A subset of A003380. - R. J. Mathar, May 11 2008

LINKS

Table of n, a(n) for n=1..21.

FORMULA

{A001016(A000040(i)) + A001016(A000040(j)) for i > j}.

EXAMPLE

a(1) = 2^8 + 3^8 = 256 + 6561 = 6817 = 17 * 401.

MATHEMATICA

Select[Sort[ Flatten[Table[Prime[n]^8 + Prime[k]^8, {n, 15}, {k, n - 1}]]], # <= Prime[15^8] &]

Total/@Subsets[Prime[Range[10]]^8, {2}]//Sort (* Harvey P. Dale, Jun 27 2017 *)

CROSSREFS

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132216.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Aug 13 2007

STATUS

approved

A132216

Primes that are sums of eighth powers of two distinct primes.

+10
3

815730977, 124097929967680577, 6115597639891380737, 144086718355753024097, 524320466699664691937, 3377940044732998170977, 10094089678769799935777, 30706777728209453204417, 58310148000746221725857

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

These primes exist because the polynomial x^8 + y^8 is irreducible over Z. Note that 2^8 + n^8 can be prime for composite n beginning 21, 55, 69, 77, 87, 117.

LINKS

Table of n, a(n) for n=1..9.

FORMULA

Primes in A132215. {A001016(A000040(i)) + A001016(A000040(j)) for i > j and are elements of A000040}.

EXAMPLE

a(1) = 2^8 + 13^8 = 256 + 815730721 = 815730977, which is prime.

a(2) = 2^8 + 137^8 = 124097929967680577, which is prime.

a(3) = 2^8 + 223^8 = 6115597639891380737, which is prime.

a(4) = 2^8 + 331^8 = 144086718355753024097, which is prime.

a(5) = 2^8 + 389^8 = 524320466699664691937, which is prime.

a(6) = 2^8 + 491^8 = 3377940044732998170977, which is prime.

a(7) = 2^8 + 563^8 = 10094089678769799935777, which is prime.

CROSSREFS

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132215.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Aug 13 2007

EXTENSIONS

More terms from Jon E. Schoenfield, Jul 16 2010

STATUS

approved

A132261

Main diagonal of array in A132260.

+10
1

13, 107, 101, 491, 8039, 9349

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..6.

FORMULA

a(n) = A[n,n+2] = n-th prime p such that 2^2^(n+2) + p^2^(n+2) is prime.

EXAMPLE

a(1) = 13 because 13 is the first prime p such that 2^2^3 + p^2^3 is prime.

a(2) = 107 because 107 is the 2nd prime p such that 2^2^4 + p^2^4 is prime.

a(3) = 101 because 101 is the 3rd prime p such that 2^2^5 + p^2^5 is prime.

CROSSREFS

Cf. A000040, A001016, A050997, A120398, A122616, A130873, A130555, A132214, A132215, A132216, A132260.

KEYWORD

more,nonn

AUTHOR

Jonathan Vos Post, Aug 15 2007

STATUS

approved

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