A073682 -id:A073682 - cp's OEIS frontend

A253664 Partition the sequence A073682 into groups so that the sum of each group is prime, then a(n) is the sum of terms in n-th group.

8893, 14699, 365587, 57097, 364183, 247369, 2225221, 251003, 2112923, 4197343, 174019, 7407013, 5711477, 2210773, 11243371, 884669, 6686107, 10585117, 26803823, 530063, 4427051, 3682759, 19362887, 4756019, 9065123, 4953593, 27207703, 14042257, 5587723, 13678729, 16995289, 15014777, 34884601, 21561341

Offset: 1

Zak Seidov, Jan 07 2015

nonn

This is to A073682 as A073682 is to A000040. It is of interest to reiterate the map "Partition of sequence s into groups with prime sums".

			5,23,101,109,263,211,251,757,1367,941,2053,1901,911 gives a(1)=8893.
2347,1861,1187,1249,1303,2273,1433,1493,1553 gives a(2)=14699.

Zak Seidov, Table of n, a(n) for n = 1..100

Cf. A000040, A073682.

A077277 Duplicate of A073682.

Original entry on oeis.org

5, 23, 101, 109, 263, 211, 251, 757, 1367, 941, 2053, 1901, 911, 2347, 1861, 1187

Offset: 1

dead

A073684 Sum of next a(n) successive primes is prime.

Original entry on oeis.org

2, 3, 5, 3, 5, 3, 3, 7, 9, 5, 9, 7, 3, 7, 5, 3, 3, 3, 5, 3, 3, 3, 5, 5, 57, 25, 49, 3, 9, 5, 11, 3, 5, 5, 5, 5, 17, 25, 3, 3, 5, 3, 7, 9, 5, 3, 3, 3, 15, 3, 3, 3, 3, 3, 3, 3, 15, 3, 5, 33, 5, 3, 3, 9, 7, 3, 33, 3, 3, 5, 3, 15, 3, 5, 9, 7, 13, 5, 11, 3, 3, 11

Offset: 1

Amarnath Murthy, Aug 11 2002

nonn

Group the primes such that the sum of each group is a prime. Each group from the second onwards should contain at least 3 primes: (2, 3), (5, 7, 11), (13, 17, 19, 23, 29), (31, 37, 41), (43, 47, 53, 59, 61), ... Sequence gives number of terms in each group.

			a(1)=2 because sum of first two primes 2+3 is prime; a(2)=3 because sum of next three primes 5+7+11 is prime; a(3)=5 because sum of next five primes 13+17+19+23+29 is prime.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A073682(n) is the sum of terms in n-th group, A073683(n) is the first term in n-th group, A077279(n) is the last term in n-th group.

f[l_List] := Block[{n = Length[Flatten[l]], k = 3, r},While[r = Table[Prime[i], {i, n + 1, n + k}]; ! PrimeQ[Plus @@r], k += 2];Append[l, r]];Length /@ Nest[f, {{2, 3}}, 100] (* Ray Chandler, May 11 2007 *)
cnt = 0; Table[s = Prime[cnt+1] + Prime[cnt+2]; len = 2; While[! PrimeQ[s], len++; s = s + Prime[cnt+len]]; cnt = cnt + len; len, {n, 100}] (* T. D. Noe, Feb 06 2012 *)

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s, i, p = 0, 1, 2
    while True:
        while not(isprime(s:=s+p)) or i < 2:
            i, p = i+1, nextprime(p)
        yield i
        s, i, p = 0, 1, nextprime(p)
print(list(islice(agen(), 82))) # Michael S. Branicky, May 23 2025

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 10 2003

Extended by Ray Chandler, May 02 2007

A073683 Group the primes such that the sum of each group is a prime. Each group from the second onwards should contain at least 3 primes: (2, 3), (5, 7, 11), (13, 17, 19, 23, 29), (31, 37, 41), (43, 47, 53, 59, 61), ... This is the sequence of the leading element in each group.

Original entry on oeis.org

2, 5, 13, 31, 43, 67, 79, 97, 131, 179, 199, 257, 293, 313, 359, 389, 409, 431, 443, 467, 491, 509, 541, 571, 601, 991, 1163, 1523, 1549, 1607, 1627, 1723, 1747, 1787, 1831, 1873, 1907, 2039, 2243, 2269, 2287, 2333, 2347, 2389, 2459, 2521, 2543, 2557, 2593

Offset: 1

Amarnath Murthy, Aug 11 2002

nonn

First prime of n-th group of successive primes in A073684.

			Partition the sequence of primes into groups so that the sum of the terms in each group is prime: {2, 3}, {5, 7, 11}, {13, 17, 19, 23, 29}, {31, 37, 41}, {43, 47, 53, 59, 61}, {67, 71, 73}, {79, 83, 89}, {97, 101, 103, 107, 109, 113, 127}, {131, 137, 139, 149, 151, 157, 163, 167, 173}, {179, 181, 191, 193, 197},..; A073684(n) is the number of terms in n-th group; A073682(n) is the sum of terms in n-th group; a(n) is the first term in n-th group; A077279(n) is the last term in n-th group.

Cf. A073682, A073684, A077279.

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s, i, p = 0, 1, 2
    while True:
        pp = p
        while not(isprime(s:=s+p)) or i < 2:
            i, p = i+1, nextprime(p)
        yield pp
        s, i, p = 0, 1, nextprime(p)
print(list(islice(agen(), 49))) # Michael S. Branicky, May 23 2025

More terms from Zak Seidov, Nov 02 2002

A077279 Last prime of n-th group of successive primes in A073684.

Original entry on oeis.org

3, 11, 29, 41, 61, 73, 89, 127, 173, 197, 251, 283, 311, 353, 383, 401, 421, 439, 463, 487, 503, 523, 569, 599, 983, 1153, 1511, 1543, 1601, 1621, 1721, 1741, 1783, 1823, 1871, 1901, 2029, 2239, 2267, 2281, 2311, 2341, 2383, 2447, 2503, 2539, 2551, 2591

Offset: 1

Zak Seidov, Nov 02 2002

nonn

Partition the sequence of primes into groups so that the sum of the terms in each group is prime: {2, 3}, {5, 7, 11}, {13, 17, 19, 23, 29}, {31, 37, 41}, {43, 47, 53, 59, 61}, {67, 71, 73}, {79, 83, 89}, {97, 101, 103, 107, 109, 113, 127}, {131, 137, 139, 149, 151, 157, 163, 167, 173}, {179, 181, 191, 193, 197},..; A073684(n) is the number of terms in n-th group; A073682(n) is the sum of terms in n-th group; A073683(n) is the first term in n-th group; A077279(n) is the last term in n-th group.

Cf. A073684, A092285, A073683.

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s, i, p = 0, 1, 2
    while True:
        while not(isprime(s:=s+p)) or i < 2:
            i, p = i+1, nextprime(p)
        yield p
        s, i, p = 0, 1, nextprime(p)
print(list(islice(agen(), 48))) # Michael S. Branicky, May 23 2025

cp's OEIS Frontend

A253664 Partition the sequence A073682 into groups so that the sum of each group is prime, then a(n) is the sum of terms in n-th group.

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A077277 Duplicate of A073682.

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A073684 Sum of next a(n) successive primes is prime.

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A073683 Group the primes such that the sum of each group is a prime. Each group from the second onwards should contain at least 3 primes: (2, 3), (5, 7, 11), (13, 17, 19, 23, 29), (31, 37, 41), (43, 47, 53, 59, 61), ... This is the sequence of the leading element in each group.

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Python

Extensions

A077279 Last prime of n-th group of successive primes in A073684.

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Python