A210205 -id:A210205 - cp's OEIS frontend

A382226 Smallest prime in a sequence of n consecutive primes which add to a perfect cube.

3, 439, 4812191, 41051, 1753, 75869, 24359, 1674289, 17509, 6221, 771653, 29863, 6899, 35353, 1073239, 4001, 18959, 1613741, 1033, 12077759, 172433, 1548149, 364079, 199, 4580399, 373, 3847, 411396253, 41863, 1371031, 11491, 135911, 45707, 308149, 364909, 176537, 2089, 32569961, 13619, 625861

Offset: 2

David Dewan, Mar 19 2025

nonn

a(1) does not exist because no single prime is a perfect cube.

			a(2)=3  :       3 + 5 = 8 = 2^3 = A382227(2).
a(3)=439 :      439 + 443 + 449 = 1331 = 11^3 = A382227(3) = A210205(1).
a(4)=4812191 :   4812191 + 4812193 + 4812209 + 4812239 = 19248832 = 268^3 = A382227(4) = A248587(1).

David Dewan, Table of n, a(n) for n = 2..200

Cf. A382227, A382228, A132955.

A382226 := proc(n)
        local i,ps,fp,lp ;
        fp := 2;
        ps := add(ithprime(j),j=1..n) ;
        lp := ithprime(n);
        for i from 1 do
                if isA000578(ps) then #code in A000578
                        return fp;
                end if;
                lp := nextprime(lp) ;
                ps := ps-fp+lp ;
                fp := nextprime(fp) ;
        end do:
end proc:
for n from 2 do
        print(n,A382226(n)) ;
end do:  # R. J. Mathar, Mar 25 2025

a[n_]:=Do[mid=PrimePi[k^3/n]; toTest=Prime[Range[Max[mid-n, 1], mid+n]];
t=Total/@Partition[toTest, n, 1];pos=Position[t, k^3];If[pos!={}, Return[First[toTest[[First[pos]]]]]], {k, 2 , Infinity} ];a/@Range[2, 10]

a(n) = { min prime(k): [ sum(j=k..k+n-1) prime(j)] in A000578 }.

A227475 Cubes which are sum of three consecutive primes.

Original entry on oeis.org

1331, 103823, 3048625, 11089567, 12008989, 19034163, 30664297, 43986977, 48627125, 59776471, 62570773, 68417929, 130323843, 180362125, 182284263, 186169411, 188132517, 263374721, 288804781, 377933067, 498677257, 510082399, 594823321, 697864103, 716917375

Offset: 1

K. D. Bajpai, Sep 02 2013

nonn

			a(2) = 103823 because prime(3696) + prime(3697) + prime(3698) = 34603 + 34607 + 34613 = 103823 = 47^3.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A076306, A210205, A158796, A226524.

Select[Total[#]&/@Partition[Prime[Range[132*10^5]],3,1],IntegerQ[ Surd[ #,3]]&] (* Harvey P. Dale, May 08 2018 *)

n=0; forstep(j=3, 86231, 2, c=j^3; c3=c/3; f=0; if(denominator(c3)==1, if(isprime(c3), if(precprime(c3-1)+c3+nextprime(c3+1)==c, f=1))); p2=precprime(c3); p1=precprime(p2-1); p3=nextprime(c3); p4=nextprime(p3+1); if(p1+p2+p3==c, f=1); if(p2+p3+p4==c, f=1); if(f==1, n++; write("b227475.txt", n " " c))) /* Donovan Johnson, Sep 02 2013 */

a(n) = (A076306(n))^3. - R. J. Mathar, Sep 02 2013

A248587 The smallest of four consecutive primes whose sum is a perfect cube.

Original entry on oeis.org

4812191, 6353029, 8039333, 8821867, 19876711, 60742631, 85017061, 108879847, 127042367, 138853049, 170367959, 238190951, 259108427, 414949357, 485941193, 512095739, 529218559, 582868471, 623331491, 648485381, 771656657, 1001132351, 1098706507, 1172752457

Offset: 1

K. D. Bajpai, Oct 09 2014

nonn

			a(2) = 6353029 is prime. Next three primes are 6353033, 6353051 and 6353071. Their sum = 6353029 + 6353033 + 6353051 + 6353071 = 25412184 = 294^3.
a(3) = 8039333 is prime. Next three primes are 8039359, 8039363 and 8039377. Their sum = 8039333 + 8039359 + 8039363 + 8039377 = 32157432 = 318^3.

K. D. Bajpai and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..42 from K. D. Bajpai.

Cf. A000040 (primes), A000578 (cubes).

Cf. A061308 (two consecutive primes), A210205 (three consecutive primes).

t = {}; p = 2; q = 3; r = 5; Do[v = NextPrime[r]; If[IntegerQ[(p + q + r + v)^(1/3)], AppendTo[t, p]; Print[p]]; p = q; q = r; r = v, {5*10^8}]; t
Select[Partition[Prime[Range[6*10^7]], 4, 1],IntegerQ[Surd[Total[#], 3]] &] [[All, 1]] (* Harvey P. Dale, Oct 07 2016 *)

lista(nn) = {vp = primes(nn); for (i=1, #vp - 3, if (ispower(vp[i]+vp[i+1]+vp[i+2]+vp[i+3], 3), print1(vp[i], ", ")););} \\ Michel Marcus, Oct 24 2014

from sympy import nextprime, prevprime
A248587_list = []
for i in range(3,10**6):
    n = i**3
    p3 = prevprime(n//4)
    p2, p4 = prevprime(p3), nextprime(p3)
    p1 = prevprime(p2)
    q = p1+p2+p3+p4
    while q <= n:
        if q == n:
            A248587_list.append(p1)
        p1, p2, p3, p4 = p2, p3, p4, nextprime(p4)
        q = p1+p2+p3+p4 # Chai Wah Wu, Dec 31 2015

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A382226 Smallest prime in a sequence of n consecutive primes which add to a perfect cube.

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A227475 Cubes which are sum of three consecutive primes.

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A248587 The smallest of four consecutive primes whose sum is a perfect cube.

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