A034964 -id:A034964 - cp's OEIS frontend

A133540 Sum of fourth powers of five consecutive primes.

17764, 46309, 129749, 259445, 536885, 1229525, 2124485, 3915125, 6610565, 9749525, 13921925, 20888885, 31132085, 42152165, 58884485, 79416485, 99924245, 126756965, 160369445, 202960565, 266078165, 341740325, 415341125, 498962405

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=17764 because 2^4+3^4+5^4+7^4+11^4=17764.

Cf. A034964, A133526, A133531, A133535, A133539, A133541, A133542, A133543.

a = 4; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]

a(n) = A133526(n) + A030514(n+4). - Michel Marcus, Nov 09 2013

A133541 Sum of fifth powers of five consecutive primes.

Original entry on oeis.org

181258, 552519, 1972133, 4445107, 10864643, 31214741, 59472599, 127396699, 240776801, 381348901, 590182759, 979749101, 1625329443, 2354069543, 3557186207, 5132070551, 6786946651, 9149078751, 12243523093, 16477457435

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=181258 because 2^5+3^5+5^5+7^5+11^5=181258.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133542, A133543.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[30]]^5,5,1] (* Harvey P. Dale, Dec 02 2017 *)

a(n) = A133527(n) + A050997(n+4). - Michel Marcus, Nov 09 2013

A133542 Sum of sixth powers of five consecutive primes.

Original entry on oeis.org

1905628, 6732373, 30869213, 77899469, 225817709, 818869469, 1701546341, 4243135181, 8946193541, 15119520701, 25303912709, 46580770157, 86195577389, 132965847509, 217102866629, 334423935221, 463593800381, 664500722261

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=1905628 because 2^6+3^6+5^6+7^6+11^6=1905628.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133541, A133543.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@(Partition[Prime[Range[30]],5,1]^6)  (* Harvey P. Dale, Mar 13 2011 *)

a(n) = A133528(n) + A030516(n+4). - Michel Marcus, Nov 09 2013

A293395 The initial member of 5 consecutive primes whose arithmetic mean is the middle member.

Original entry on oeis.org

71, 271, 337, 431, 631, 661, 769, 1153, 1721, 1789, 2131, 2339, 2381, 2749, 2777, 3313, 3319, 3517, 3919, 4139, 4337, 4729, 4789, 4903, 4937, 4993, 5171, 5303, 5323, 5507, 5849, 5851, 6271, 6323, 6451, 6959, 6983, 7489, 7919, 8221, 8363, 8419, 9349, 9613, 9619

Offset: 1

K. D. Bajpai, Oct 08 2017

nonn

3313 is the smallest term such that 3313 +- 6 are both prime.

			71 is a term because it is the initial member of 5 consecutive primes {71, 73, 79, 83, 89} and (71 + 73 + 79 + 83 + 89)/5 = 79.
271 is a term because it is the initial member of 5 consecutive primes {271, 277, 281, 283, 293} and (271 + 277 + 281 + 283 + 293)/5 = 281.

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

Cf. A006562, A034964, A122535.

A293395:= proc(n)local a, b, c, d, e; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); d:=ithprime(n+3); e:=ithprime(n+4); if (a + b + d + e)/4 = c then RETURN (a); fi; end: seq(A293395(n), n=1..3000);

Select[Prime@ Range[1200], #[[3]] == Mean@ Delete[#, 3] &@ NestList[NextPrime, #, 4] &] (* Michael De Vlieger, Oct 09 2017 *)
Select[Partition[Prime[Range[1200]],5,1],Mean[#]==#[[3]]&][[;;,1]] (* Harvey P. Dale, Jul 31 2025 *)

for(n=1, 1000, a=prime(n); b=prime(n+1); c=prime(n+2); d=prime(n+3); e=prime(n+4); if((a+b+d+e)/4==c, print1(a,", ")));

list(lim)=my(v=List(),p=2,q=3,r=5,s=7); forprime(t=11,lim, if(p+q+s+t==4*r, listput(v,p)); p=q; q=r; r=s; s=t); Vec(v) \\ Charles R Greathouse IV, Oct 09 2017

Definiyion simplified by David A. Corneth, Oct 14 2017

Examples clarified by Harvey P. Dale, Jul 31 2025

A337439 a(n) is the k-th prime, such that abs(prime(k) - Sum_{j=k-2..k+2} prime(j)/5) sets a new record.

Original entry on oeis.org

5, 7, 19, 47, 97, 109, 113, 199, 887, 1151, 1277, 1327, 9551, 11777, 14143, 15727, 19609, 25471, 31397, 156007, 360653, 370261, 492113, 1357201, 1357333, 1562051, 2010733, 4652507, 17051887, 20831323, 47326693, 47326913, 122164747, 189695893, 428045741, 436273291, 1453168433

Offset: 1

Hugo Pfoertner, Aug 29 2020

nonn

A337438 are the corresponding values of k.

			List of first terms:
   a(n)  pi(a(n))  average-median
     5,      3,     3/5  = (2 + 3 + 5 + 7 + 11)/5 - 5
     7,      4,     4/5 = (3 + 5 + 7 + 11 + 13)/5 - 7
    19,      8,     6/5 = (13 + 17 + 19 + 23 + 29)/5 - 19
    47,     15,     8/5
    97,     25,   -12/5
   109,     29,    14/5
   113,     30,    22/5
   199,     46,    28/5
   887,    154,    34/5
  1151,    190,   -36/5
  1277,    206,   -38/5
  1327,    217,    12
  9551,   1183,    14

Cf. A034964, A337438, A337488, A337489.

a337439(limp) = {my(p1=0,p2=2,p3=3,p4=5,p5=7,s=p1+p2+p3+p4+p5,d=0);forprime(p=11,limp, s=s-p1+p; my(dd=abs(s/5-p4)); if(dd>d,print1(p4,", ");d=dd); p1=p2;p2=p3;p3=p4;p4=p5;p5=p)};
a337439(500000000)

Name edited by Peter Munn, Aug 01 2025

A362465 a(n) is the least number of 2 or more consecutive signed primes whose sum equals n.

Original entry on oeis.org

3, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 2, 5, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3

Offset: 0

Karl-Heinz Hofmann, Apr 21 2023

nonn

Inspired by a conjecture made by Carlos Rivera in 2000 (see link). Here we remove Rivera's restriction that the primes have to be smaller than n.
For every positive even n, a(n) = 2, provided there are 2 consecutive primes separated by a gap of size n. Polignac's conjecture says: "For any positive even number n, there are infinitely many prime gaps of size n." If so, a(3) is the only 4 in this sequence, as any even number of consecutive odd signed primes has an even sum.
There is also the reversed sequence for negative n with 0 as the symmetry point.
See A362466 for the first occurrences of numbers in this sequence.

			a(1) = 2: -2 + 3 = 1.
a(0) = 3: -2 - 3 + 5 = 0.
a(3) = 4:  2 + 3 + 5 - 7 = 3.
The example below for a(29) gives more detail of the general method employed.
a(29) = 5:  3 - 5 + 7 + 11 + 13 = 29.
Since any even number of consecutive odd signed primes has an even sum, we can show a(29) <> 4.
A test with all triples of consecutive signed primes up to 10^9 gave no solution for 29. The estimated lower bound for the permutation p1 + p2 - p3 is p1 - (p1 + 2)^0.525 and was never surpassed. (See Wikipedia link. "A result, due to Baker, Harman and Pintz in 2001, shows that Theta may be taken to be 0.525".) So the terms are calculated with the assumption that this is true.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..40000
Karl-Heinz Hofmann, Examples for n = 0 to 253
Karl-Heinz Hofmann, Possible solutions for a(n) = 3
Karl-Heinz Hofmann, Possible solutions for a(n) = 5
Thomas R. Nicely, First occurrence prime gaps
Carlos Rivera, Conjecture 21. Rivera's conjecture, The Prime Puzzles and Problems Connection.
Wikipedia, Prime Gap.
Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Volume 179 (2014), Issue 3, pp. 1121-1174.

Cf. A000040, A001223, A034961, A034964, A127334, A127336, A127338.

Cf. A000230, A001632, A362466 (first occurrences).

from sympy import primepi, sieve as prime
import numpy
upto = 50000                   # 5000000 good for 8 GB RAM (3 Minutes)
primepi_of_upto, np, arr = primepi(upto), 1, []
A362465 = numpy.zeros(upto + 1, dtype="i4")
A362465[2:][::2] = 2           # holds if "upto" < 7 * 10^7
for n in range(1,primepi_of_upto + 1): arr.append([prime[n]])
while all(A362465) == 0:
    np += 1
    for k in range(0,primepi_of_upto):
        temp = []
        for i in arr[k]:
            temp.append(i + prime[k+np])
            temp.append(abs(i - prime[k+np]))
        arr[k] = set(temp)
        for n in temp:
            if n <= upto and A362465[n] == 0: A362465[n] = np
print(list(A362465[0:100]))

a(n) = a(-n).

Edited by Peter Munn, Aug 08 2023

A337438 a(n) is the index k of prime(k), such that abs(prime(k) - Sum_{j=k-2..k+2} prime(j)/5) sets a new record.

Original entry on oeis.org

3, 4, 8, 15, 25, 29, 30, 46, 154, 190, 206, 217, 1183, 1410, 1664, 1832, 2225, 2810, 3385, 14358, 30802, 31545, 40933, 104071, 104072, 118506, 149689, 325853, 1094422, 1319945, 2850174, 2850175, 6957876, 10539433, 22749705, 23163299, 72507381, 182837804, 415271758, 486570087

Offset: 1

Hugo Pfoertner, Aug 29 2020

nonn

A337439 are the corresponding primes.

			See A337489.

Cf. A034964, A337439, A337488, A337489.

\\ See A337439, replace p4 by primepi(p4) in call of print1.

Name edited by Peter Munn, Aug 01 2025

A127492 Indices m of primes such that Sum_{k=0..2, k

Original entry on oeis.org

2, 10, 17, 49, 71, 72, 75, 145, 161, 167, 170, 184, 244, 250, 257, 266, 267, 282, 286, 301, 307, 325, 343, 391, 405, 429, 450, 537, 556, 561, 584, 685, 710, 743, 790, 835, 861, 904, 928, 953

Offset: 1

Artur Jasinski, Jan 16 2007

Let p_0 .. p_4 be five consecutive primes, starting with the m-th prime. The index m is in the sequence if the absolute value [x^0] of the polynomial (x-p_0)*[(x-p_1)*(x-p_2) + (x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_1)*[(x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_2)*(x-p_3)*(x-p_4) is two times a prime. The correspondence with A127491: the coefficient [x^2] of the polynomial (x-p_0)*(x-p_1)*..*(x-p_4) is the sum of 10 products of a set of 3 out of the 5 primes. Here the sum is restricted to the 6 products where the two largest of the 3 primes are consecutive. - R. J. Mathar, Apr 23 2023

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489, A127490, A127491.

isA127492 := proc(k)
    local x,j ;
    (x-ithprime(k))* mul( x-ithprime(k+j),j=1..2)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+2))* mul( x-ithprime(k+j),j=3..4) ;
    p := abs(coeff(expand(%/2),x,0)) ;
    if type(p,'integer') then
        isprime(p) ;
    else
        false ;
    end if ;
end proc:
for k from 1 to 900 do
    if isA127492(k) then
        printf("%a,",k) ;
    end if ;
end do: # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2] + Prime[x] Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3] + Prime[x + 1] Prime[x + 3]Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])/2], AppendTo[a, x]], {x, 1, 1000}]; a
prQ[{a_,b_,c_,d_,e_}]:=PrimeQ[(a b c+a c d+a d e+b c d+b d e+c d e)/2]; PrimePi/@Select[ Partition[ Prime[Range[1000]],5,1],prQ][[;;,1]] (* Harvey P. Dale, Apr 21 2023 *)

Definition simplified by R. J. Mathar, Apr 23 2023

Edited by Jon E. Schoenfield, Jul 23 2023

A131686 Sum of squares of five consecutive primes.

Original entry on oeis.org

208, 373, 653, 989, 1469, 2189, 2981, 4061, 5381, 6701, 8069, 9917, 12029, 14069, 16709, 19541, 22061, 24821, 27989, 31421, 35789, 40661, 45029, 49589, 53549, 56909, 62837, 69389, 76709, 84149, 93581, 100253, 107741, 115541, 124109, 131837

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=208 because 2^2+3^2+5^2+7^2+11^2=208

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133539, A133540, A133541, A133542, A133543.

a = 2; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]

A283873 Smallest number that is the sum of n successive primes and also the sum of n successive semiprimes, n > 1.

Original entry on oeis.org

24, 749, 48, 311, 690, 251, 2706, 2773, 6504, 1081, 2162, 1753, 11356, 6223, 1392, 2303, 9838, 637, 14510, 1995, 3154, 21459, 72960, 5691, 8140, 1475, 2350, 3647, 1593, 7607, 55074, 2719, 9852, 12143, 106562, 12615, 9036, 19883, 15438, 28369, 8560, 8415, 3831

Offset: 2

Zak Seidov, Mar 17 2017

nonn

The sequence is non-monotone.

			a(2) = 24 = A000040(5) + A000040(6) = 11 + 13 = A001358(4) + A001358(5) = 10 + 14,
a(3) = 749 = A000040(53) + A000040(54) + A000040(55) = 241 + 251 + 257 = A001358(79) + A001358(80) + A001358(81) = 247 + 249 + 253.

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Cf. A000040 Primes, A001358 Semiprimes, A118717 Sum of two consecutive semiprimes.

Sum of k consecutive primes: A001043 k=2, A034961 k=3, A034963 k=4, A034964 k=5, A127333 k=6, A127334 k=7, A127335 k=8, A127336 k=9, A127337 k=10, A127338 k=11, A127339 k=12.

issp:= n-> is(not isprime(n) and numtheory[bigomega](n)=2):
ithsp:= proc(n) option remember; local k; for k from 1+
        `if`(n=1, 1, ithsp(n-1)) while not issp(k) do od; k
        end:
ps:= proc(i, j) option remember;
       ithprime(j)+`if`(i=j, 0, ps(i, j-1))
     end:
ss:= proc(i, j) option remember;
       ithsp(j)+`if`(i=j, 0, ss(i, j-1))
     end:
a:= proc(n) option remember; local i, j, k, l, p, s;
      i, j, k, l, p, s:= 1, n, 1, n, ps(1, n), ss(1, n);
      do if p=s then return p
       elif pAlois P. Heinz, Mar 24 2017

Mathematica

sp=Select[Range[4,100000],2==PrimeOmega[#]&];pr=Prime[Range[PrimePi[Max[sp]]]]; Table[Intersection[(Total/@Partition[pr,k,1]),Total/@Partition[sp,k,1]][[1]],{k,2,100}]

Extensions

More terms from Alois P. Heinz, Mar 24 2017

cp's OEIS Frontend

A133540 Sum of fourth powers of five consecutive primes.

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A133541 Sum of fifth powers of five consecutive primes.

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A133542 Sum of sixth powers of five consecutive primes.

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A293395 The initial member of 5 consecutive primes whose arithmetic mean is the middle member.

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A337439 a(n) is the k-th prime, such that abs(prime(k) - Sum_{j=k-2..k+2} prime(j)/5) sets a new record.

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A362465 a(n) is the least number of 2 or more consecutive signed primes whose sum equals n.

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A337438 a(n) is the index k of prime(k), such that abs(prime(k) - Sum_{j=k-2..k+2} prime(j)/5) sets a new record.

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A127492 Indices m of primes such that Sum_{k=0..2, k

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