A120398 -id:A120398 - cp's OEIS frontend

A130873 Sums of two distinct prime 4th powers.

97, 641, 706, 2417, 2482, 3026, 14657, 14722, 15266, 17042, 28577, 28642, 29186, 30962, 43202, 83537, 83602, 84146, 85922, 98162, 112082, 130337, 130402, 130946, 132722, 144962, 158882, 213842, 279857, 279922, 280466, 282242, 294482, 308402

Offset: 1

Jonathan Vos Post, Jul 24 2007

This is to 4th powers as A120398 is to cubes.

Cf. A030514, A120398.

Select[Sort[ Flatten[Table[Prime[n]^4 + Prime[k]^4, {n, 15}, {k, n - 1}]]], # <= Prime[15^4] &]
Total/@Subsets[Prime[Range[10]]^4,{2}]//Union (* Harvey P. Dale, Oct 20 2024 *)

A138854 Numbers which are the sum of three cubes of distinct primes.

Original entry on oeis.org

160, 378, 476, 495, 1366, 1464, 1483, 1682, 1701, 1799, 2232, 2330, 2349, 2548, 2567, 2665, 3536, 3555, 3653, 3871, 4948, 5046, 5065, 5264, 5283, 5381, 6252, 6271, 6369, 6587, 6894, 6992, 7011, 7118, 7137, 7210, 7229, 7235, 7327, 7453, 8198, 8217, 8315

Offset: 1

M. F. Hasler, Apr 13 2008

nonn

This is a subsequence of A024975. The odd terms of this sequence are A138853, the even terms are 8+{ even terms of A120398 }. Thus primes in this sequence, A137365, are the same as primes in A138853.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..621 from R. J. Mathar)
Index to sequences related to sums of cubes.

Cf. A024975 (a^3+b^3+c^3, a>b>c>0), A138853 (odd terms of this), A120398, A137365 (primes in A138853 / A138854).

isA030078 := proc(n)
    local f ;
    if n < 8 then
        false;
    else
        f := ifactors(n)[2] ;
        if nops(f) = 1 and op(2,op(1,f)) = 3 then
            true;
        else
            false;
        end if;
    end if;
end proc:
isA138854 := proc(n)
    local i,j,p,q,r,rcub ;
    for i from 1 do
        p := ithprime(i) ;
        if p^3+(p+1)^3+(p+2)^3 > n then
            return false;
        end if;
        for j from i+1 do
            q := ithprime(j) ;
            rcub := n-q^3-p^3 ;
            if rcub <= q^3 then
                break;
            fi ;
            if isA030078(rcub) then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 5 do
    if isA138854(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Jun 09 2014

f[upto_]:=Module[{maxp=PrimePi[Floor[Power[upto, (3)^-1]]]}, Select[Union[Total/@(Subsets[Prime[Range[maxp]],{3}]^3)],#<=upto&]]; f[9000]  (* Harvey P. Dale, Mar 21 2011 *)

isA138854(n)={ if( n%2, isA138853(n), isA120398(n-8)) }
for( n=1,10^4, isA138854(n) & print1(n", "))

A138854 = { p(i)^3+p(j)^3+p(k)^3 ; i>j>k>0 } = A138853 union { p(i)^3+p(j)^3+8 ; i>j>1}

A130555 Numbers that are sums of sixth powers of two distinct primes.

Original entry on oeis.org

793, 15689, 16354, 117713, 118378, 133274, 1771625, 1772290, 1787186, 1889210, 4826873, 4827538, 4842434, 4944458, 6598370, 24137633, 24138298, 24153194, 24255218, 25909130, 28964378, 47045945, 47046610, 47061506, 47163530

Offset: 1

Jonathan Vos Post, Aug 09 2007

This is to 6th powers as A130292 is to fifth powers, A130873 is to 4th powers and A120398 is to cubes. These can never be prime, as sixth powers are cubes and the sum of cubes factorizations applies. There are semiprimes for values beginning a(1) = 793, a(2) = 15689 = 29 * 541, a(4) = 117713 = 53 * 2221, a(11) = 4826873 = 173 * 27901.

			a(1) = prime(1)^6 + prime(2)^6 = 2^6 + 3^6 = 64 + 729 = 793 = 13 * 61.

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Cf. A000040, A001014, A050997, A120398, A122616, A130873.

Select[Sort[Flatten[Table[Prime[n]^6 + Prime[k]^6, {n, 15}, {k, n - 1}]]], # <= Prime[15^6] &]
Union[Total/@Subsets[Prime[Range[20]]^6,{2}]] (* Harvey P. Dale, Mar 11 2012 *)

{A001014(A000040(i)) + A001014(A000040(j)) for i > j}.

A132214 Numbers that are sums of seventh powers of two distinct primes.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Aug 13 2007

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.

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

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

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

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

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 *)

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

A138853 Numbers which are the sum of 3 cubes of distinct odd primes.

Original entry on oeis.org

495, 1483, 1701, 1799, 2349, 2567, 2665, 3555, 3653, 3871, 5065, 5283, 5381, 6271, 6369, 6587, 7011, 7137, 7229, 7235, 7327, 7453, 8217, 8315, 8441, 8533, 9083, 9181, 9399, 10387, 11799, 11897, 12115, 12319, 12537, 12635, 13103, 13525, 13623, 13841

Offset: 1

M. F. Hasler, Apr 13 2008

nonn

Dropping the restriction to odd primes would add to this sequence of odd terms the sequence of even terms of the form 8+p(i)^3+p(j)^3 (i>j>1), i.e. 8+{ even terms of A120398 }, cf. A138854.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..560 from R. J. Mathar)
Index to sequences related to sums of cubes.

Cf. A024975 (a^3+b^3+c^3, a>b>c>0), A138854, A120398.

isA138853(n)= local( c,d); n>494 && forprime( p=floor( sqrtn( n\3+1,3))+1, floor( sqrtn( n-151,3)), d=n-p^3; forprime( q=floor( sqrtn( d\2+1,3))+1, min( p-1, floor( sqrtn( d-26,3))), round( sqrtn( c=d-q^3,3 ))^3==c || next; isprime( round( sqrtn( c,3 ))) && return(1)))
forstep(n=3^3+5^3+7^3,10^5,2, isA138853(n)&print1(n", "))

A138853={ p(i)^3+p(j)^3+p(k)^3 ; i>j>k>1 }

A130292 Numbers that are sums of fifth powers of two distinct primes.

Original entry on oeis.org

275, 3157, 3368, 16839, 17050, 19932, 161083, 161294, 164176, 177858, 371325, 371536, 374418, 388100, 532344, 1419889, 1420100, 1422982, 1436664, 1580908, 1791150, 2476131, 2476342, 2479224, 2492906, 2637150, 2847392, 3895956

Offset: 1

Jonathan Vos Post, Aug 06 2007

This is to 5th powers as A120398 is to cubes and A130873 is to 4th powers.

			a(1) = prime(1)^5 + prime(2)^5 = 2^5 + 3^5 = 32 + 243 = 275.

Cf. A000040, A050997, A120398, A122616, A130873.

Select[Sort[ Flatten[Table[Prime[n]^5 + Prime[k]^5, {n, 15}, {k, n - 1}]]], # <= Prime[15^5] &]

A132215 Numbers that are sums of eighth powers of two distinct primes.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Aug 13 2007

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

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

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

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 *)

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

A132216 Primes that are sums of eighth powers of two distinct primes.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Aug 13 2007

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.

			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.

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

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

More terms from Jon E. Schoenfield, Jul 16 2010

A192926 Sums of two or more distinct prime cubes.

Original entry on oeis.org

35, 133, 152, 160, 351, 370, 378, 468, 476, 495, 503, 1339, 1358, 1366, 1456, 1464, 1483, 1491, 1674, 1682, 1701, 1709, 1799, 1807, 1826, 1834, 2205, 2224, 2232, 2322, 2330, 2349, 2357, 2540, 2548, 2567, 2575, 2665, 2673, 2692, 2700, 3528

Offset: 1

Jonathan Vos Post, Jul 12 2011

This is to cubes of primes as A192336 is to squares of integers.

			2^3 + 3^3 + 7^3 = 8 + 27 + 343 = 378 is in this sequence.

Cf. A120398, A192336.

More terms from Alois P. Heinz, Jul 14 2011

A291339 Primes p such that p^3*q^3 + p^3 + q^3 is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 7, 19, 37, 47, 83, 89, 107, 137, 181, 251, 257, 349, 379, 569, 631, 653, 677, 691, 797, 823, 839, 863, 883, 919, 1009, 1021, 1223, 1229, 1361, 1423, 1571, 1609, 1831, 1873, 1907, 1993, 2053, 2113, 2207, 2239, 2293, 2309, 2579, 2833, 3137, 3319, 3593, 3673

Offset: 1

K. D. Bajpai, Aug 22 2017

nonn

			a(2) = 3 is prime; 5 is the next prime: 3^3*5^3 + 3^3 + 5^3 = 27*125 + 27 + 125 = 3527 that is a prime.
a(3) = 7 is prime; 11 is the next prime: 7^3*11^3 + 7^3 + 11^3 = 343*1331 + 343 + 1331 = 458207 that is a prime.

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

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241.

[p: p in PrimesUpTo (5000) | IsPrime(p^3*q^3+p^3+q^3)];

select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^3+nextprime(p)^3)]), [seq(p, p=1..10^4)]);

Prime@Select[Range[1000], PrimeQ[Prime[#]^3*Prime[# + 1]^3 + Prime[#]^3 + Prime[# + 1]^3] &]
Select[Partition[Prime[Range[600]],2,1],PrimeQ[Times@@(#^3)+Total[#^3]]&][[;;,1]] (* Harvey P. Dale, Apr 28 2025 *)

is(n) = my(q=nextprime(n+1)); ispseudoprime(n^3*q^3+n^3+q^3)
forprime(p=1, 3700, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 22 2017

list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p3+q3), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017

cp's OEIS Frontend

A130873 Sums of two distinct prime 4th powers.

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A138854 Numbers which are the sum of three cubes of distinct primes.

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A130555 Numbers that are sums of sixth powers of two distinct primes.

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A132214 Numbers that are sums of seventh powers of two distinct primes.

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A138853 Numbers which are the sum of 3 cubes of distinct odd primes.

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A130292 Numbers that are sums of fifth powers of two distinct primes.

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A132215 Numbers that are sums of eighth powers of two distinct primes.

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A132216 Primes that are sums of eighth powers of two distinct primes.

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