A340771 -id:A340771 - cp's OEIS frontend

A352423 Numbers that are the sum of some number of consecutive prime cubes.

8, 27, 35, 125, 152, 160, 343, 468, 495, 503, 1331, 1674, 1799, 1826, 1834, 2197, 3528, 3871, 3996, 4023, 4031, 4913, 6859, 7110, 8441, 8784, 8909, 8936, 8944, 11772, 12167, 13969, 15300, 15643, 15768, 15795, 15803, 19026, 23939, 24389, 26136, 27467, 27810, 27935, 27962, 27970, 29791

Offset: 1

Michel Marcus, Apr 26 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Cathal O'Sullivan, Jonathan P. Sorenson, and Aryn Stahl, An Algorithm to Find Sums of Consecutive Powers of Primes, arXiv:2204.10930 [math.NT], 2022-2023. See S3 p. 10.

Cf. A030078, A034707, A340771.

lista(nn) = {my(list = List(), ip = primepi(nn), vp = primes(ip)); for(i=1, ip, my(s=vp[i]^3); listput(list, s); for (j=i+1, ip, s += vp[j]^3; if (s >vp[ip]^3, break); listput(list, s); ); ); Vec(vecsort(list, , 8)); }

import heapq
from sympy import prime
from itertools import islice
def agen(): # generator of terms
    p = prime(1)**3; h = [(p, 1, 1)]; nextcount = 2
    while True:
        (v, s, l) = heapq.heappop(h)
        yield v
        if v >= p:
            p += prime(nextcount)**3
            heapq.heappush(h, (p, 1, nextcount))
            nextcount += 1
        v -= prime(s)**3; s += 1; l += 1; v += prime(l)**3
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 47))) # Michael S. Branicky, Apr 26 2022

A352424 Numbers that can be written as sums of squares of consecutive primes in two ways.

Original entry on oeis.org

14720439, 16535628, 34714710, 40741208, 61436388, 603346308, 1172360113, 1368156941, 1574100889, 1924496102, 1989253499, 2021860243, 6774546339, 9770541610, 12230855963, 12311606487, 12540842446, 14513723777, 26423329489, 38648724198, 47638558043, 50195886916, 50811319931, 56449248367

Offset: 1

Michel Marcus, Apr 26 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..991
Cathal O'Sullivan, Jonathan P. Sorenson, and Aryn Stahl, An Algorithm to Find Sums of Consecutive Powers of Primes, arXiv:2204.10930 [math.NT], 2022-2023. See 4.2 Duplicates p. 8-9.
Michael S. Branicky, Python Program

Cf. A001248, A340771.

# see link for a version suitable for producing b-file
from sympy import primerange, integer_nthroot
def aupto(limit):
    adict = dict()
    rootlimit = integer_nthroot(limit, 2)[0]
    for x in primerange(2, rootlimit+1):
        s = x**2
        adict[s] = 1
        for y in primerange(x+1, rootlimit+1):
            s += y**2
            if s <= limit:
                if s not in adict:
                    adict[s] = 1
                else:
                    adict[s] += 1
            else:
                break
    return sorted(s for s in adict if adict[s] == 2)
print(aupto(6*10**10)) # Michael S. Branicky, Apr 26 2022

A372041 Least prime p such that the sum of squares of the 2n + 1 consecutive primes starting with p is prime, or -1 if no such p exists.

Original entry on oeis.org

3, 3, 5, 3, 3, 5, -1, 5, 5, -1, 3, 7, -1, 3, 13, -1, 5, 5, -1, 7, 23, -1, 13, 5, -1, 7, 5, -1, 59, 29, 3, 3, 5, -1, 3, 5, -1, 13, 11, -1, 37, 23, -1, 43, 11, -1, 3, 5, -1, 11, 5, -1, 5, 19, -1, 5, 43, -1, 13, 29, -1, 7, 19, -1, 41, 47, -1, 13, 11, 3, 7, 5, -1, 29, 7, -1, 79, 13, 3, 3

Offset: 1

Michel Lagneau, Apr 17 2024

sign

a(n) = 2 never occurs, since the sum starting at 2 is always even and >= 4, so not prime.
a(n) = 3 iff n is in A370633 (and equivalently iff 2*n+1 is in A071149).
For n == 1 (mod 3), so 2*n+1 is a multiple of 3, a(n) = 3 or -1, since all primes >= 5 are congruent to 1 (mod 6) so the sum starting at 5 or more is a multiple of 3 and so not prime.

			a(6) = 5 because 5 is the smallest of 2*6+1 = 13 consecutive primes whose sum of squares = 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 37^2 + 41^2 + 43^2 + 47^2 = 10453 is prime.
a(7) = -1 because 7 == 1 (mod 3) so its only possibility is that the sum starts at 3, but 3^2 + ... + 53^2 = 13271 is not prime.

Cf. A024450, A089793, A318351, A340771, A370633 (indices of 3's).

a(n) = if ((n % 3) == 1, my(vp = primes(2*n+2)); if (isprime(sum(k=2, #vp, vp[k]^2)), return (3), return(-1));); my(vp = primes(2*n+2)); while(! isprime(sum(k=2, #vp, vp[k]^2)), vp = concat(setminus(vp, Set(vp[1])), nextprime(vp[2*n+2]+1))); vp[2]; \\ Michel Marcus, May 16 2024

A376916 Primes that are the sum of some number of consecutive prime squares.

Original entry on oeis.org

13, 83, 373, 653, 1543, 2393, 3271, 4519, 4723, 5381, 6701, 7591, 8069, 8219, 9439, 10453, 11719, 19541, 20269, 20477, 23599, 24821, 24953, 32939, 35323, 38219, 39631, 41539, 45319, 51031, 53549, 55721, 56179, 56383, 56599, 56909, 65419, 69389, 73331, 74441, 75997, 81299, 87589, 89459, 90199, 93581, 96661, 97847, 98017, 107741, 108827, 109849

Offset: 1

Robert Israel, Oct 09 2024

nonn

Primes in A340771.

			a(3) = 373 is a term because 373 is prime and 373 = 3^2 + 5^2 + 7^2 + 11^2 + 13^2 where 3, 5, 6, 11 and 13 are consecutive primes.

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

Cf. A340771.

N:= 2*10^5: # for terms <= N
PS:= [0, seq(ithprime(i)^2, i=1..numtheory:-pi(floor(sqrt(N))))]:
SPS:= ListTools:-PartialSums(PS):
sort(convert(select(t -> t <= N and isprime(t), {seq(seq(SPS[t]-SPS[s], s=1..t-2), t=2..nops(SPS))}), list);

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A352423 Numbers that are the sum of some number of consecutive prime cubes.

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A352424 Numbers that can be written as sums of squares of consecutive primes in two ways.

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Python

A372041 Least prime p such that the sum of squares of the 2n + 1 consecutive primes starting with p is prime, or -1 if no such p exists.

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PARI

A376916 Primes that are the sum of some number of consecutive prime squares.

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