A071810 -id:A071810 - cp's OEIS frontend

A108018 a(n) = the number of primes representable as the sum of some subset of the set of first n primes.

1, 3, 4, 5, 9, 12, 16, 19, 25, 31, 37, 43, 51, 59, 66, 75, 84, 95, 103, 115, 127, 137, 150, 162, 177, 191, 205, 218, 233, 250, 267, 282, 299, 319, 338, 359, 376, 399, 421, 440, 461, 481, 508, 531, 556, 578, 602, 629, 653, 683, 707, 737, 765, 793, 824, 853, 883

Offset: 1

Max Alekseyev, Sep 08 2006

nonn

a(n) = length of n-th row in A256015; A066028(n) = A256015(n,a(n)). - Reinhard Zumkeller, Jun 01 2015

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A071810 (same count but with multiplicities).
Cf. A010051, A000040.
Cf. A066028, A256015.

import Data.List (subsequences, nub)
a108018 = sum . map a010051' . nub . map sum .
          tail . subsequences . flip take a000040_list
-- Reinhard Zumkeller, Dec 16 2013

(* This program is not suitable to compute a large number of terms. *)
a[n_] := a[n] = Select[Total /@ Subsets[Table[Prime[i], {i, 1, n}]], PrimeQ] // Union // Length;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Mar 12 2019 *)

More terms from Alois P. Heinz, Oct 24 2015

A262765 a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime.

Original entry on oeis.org

1, 2, 3, 7, 14, 28, 57, 113, 227, 437, 834, 1616, 3143, 6144, 12036, 23467, 45713, 89375, 175722, 346193, 681828, 1344815, 2657630, 5253615, 10374965, 20471599, 40401901, 79871358, 158182869, 313402574, 620776183, 1228390053, 2430853614, 4813878134, 9550070608

Offset: 2

Bob Selcoe, Sep 30 2015

nonn

From Bob Selcoe, Oct 02 2015: (Start)
Conjectures:
i. a(n) ~ 2*a(n-1);
ii. a(n) <= 2*a(n-1)+1, a(n) < 2*a(n-1) n>=11;
iii. As n increases to infinity, a(n)/a(n-1) generally increases toward approximately 2, though the limiting ratio must be < 2.
(End)
From Alois P. Heinz, Oct 02 2015: (Start)
a(333) = 2*a(332)+d where d = 608...358 is a 95-digit positive integer.
It is not true that "a(n)/a(n-1) generally increases"; see plot below.
(End)
From Bob Selcoe, Oct 20 2015: (Start)
The plot does, in fact, suggest that a(n)/a(n-1) "generally increases" (i.e., generally a(z*n)/a(z*n-1) > a(n)/a(n-1), when z is sufficiently large). In other words, the peaks and the troughs tend to increase, with peaks tending to be higher than next trough. This behavior is consistent with the observation about "general increase" in conjecture iii, as n increases to infinity. However, the plot does not show that generally a(n)/a(n-1) > a(n+1)/a(n), or that there can't be a maximum value of a(n)/a(n-1) after which all other values decrease (two completely different issues).
Conjecture ii is clearly false since a(n)/(n-1) is slightly > 2 for a few terms 11 <= n <= 400 (n = {333..340}), therefore weakening the observation in conjecture iii that the limiting ratio must be < 2. (End)

			a(5)=7; prime(5)=11: 2+3=5, 2+5=7; 2+11=13; 2+3+5+7=17; 3+5+11=19; 2+3+7+11=23; 5+7+11=23.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Alois P. Heinz, Plot of a(n+1)/a(n)

Cf. A000040 (prime numbers), A007504, A071810.

s:= proc(n) option remember; `if`(n=0, 0, s(n-1)+ithprime(n)) end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1, 0, b(n, i-1, t) +(p->
      `if`(p>n, 0, b(n-p, i-1, max(0, t-1))))(ithprime(i))))
    end:
a:= n-> add(`if`(isprime(k), b(k, n, 2), 0), k=5..s(n)):
seq(a(n), n=2..36);  # Alois P. Heinz, Oct 01 2015

Length@ Select[Total /@ ReplaceAll[Subsets[Prime@ Range@ #], {} -> Nothing], PrimeQ] & /@ Range[2, 21] (* _Michael De Vlieger, Oct 01 2015 *)

a(n) = A071810(n) - n. - Alois P. Heinz, Oct 23 2015

a(10)-a(21) from Michael De Vlieger, Oct 01 2015

a(22)-a(36) from Alois P. Heinz, Oct 01 2015

A339485 Number of subsets of the first n primes whose elements have a prime average.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 17, 30, 51, 88, 149, 264, 439, 746, 1261, 2234, 4211, 7996, 14899, 28048, 54037, 106442, 208625, 398588, 735365, 1331590, 2421573, 4481896, 8504953, 16497150, 32595915, 64614636, 127968263, 252470776, 495388085, 962475122, 1847742473, 3504948056

Offset: 1

Ilya Gutkovskiy, Dec 06 2020

nonn

			a(5) = 9 subsets: {2}, {3}, {5}, {7}, {11}, {3, 7}, {3, 11}, {3, 5, 7} and {3, 7, 11}.

Alois P. Heinz, Table of n, a(n) for n = 1..150
Eric W. Weisstein's World of Mathematics, Arithmetic Mean

Cf. A000040, A051293, A071810, A127542, A309160.

b:= proc(n, s, c) option remember; `if`(n=0,
      `if`(c>0 and denom(s)=1 and isprime(s), 1, 0),
       b(n-1, s, c)+b(n-1, (s*c+ithprime(n))/(c+1), c+1))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 08 2020

b[n_, s_, c_] := b[n, s, c] = If[n == 0,
     If[c > 0 && Denominator[s] == 1 && PrimeQ[s], 1, 0],
     b[n-1, s, c] + b[n-1, (s*c + Prime[n])/(c+1), c+1]];
a[n_] := b[n, 0, 0];
Array[a, 40] (* Jean-François Alcover, Jul 09 2021, after Alois P. Heinz *)

from sympy import prime, isprime
from itertools import chain, combinations
def powerset(s): # skip empty set and singletons
    return chain.from_iterable(combinations(s, r) for r in range(2,len(s)+1))
def a(n):
    out = n  # count all singletons
    for s in powerset([prime(i) for i in range(1, n+1)]):
        ss = sum(s)
        if ss%len(s) == 0:
            if isprime(ss//len(s)): out += 1
    return out
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 06 2020

from itertools import combinations
from sympy import prime
def A339485(n):
    c, primeset2 = n, set(prime(i) for i in range(1,n))
    primeset = primeset2 | {prime(n)}
    for l in range(2,n+1):
        for d in combinations(primeset,l):
            a, b = divmod(sum(d),l)
            if b == 0 and a in primeset2:
                c += 1
    return c # Chai Wah Wu, Dec 07 2020

from functools import lru_cache
from sympy import sieve, isprime
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return int(c and s%c == 0 and isprime(s//c))
    return b(n-1, s, c) + b(n-1, s+sieve[n], c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Oct 06 2022

a(10)-a(30) from Michael S. Branicky, Dec 06 2020
a(31)-a(34) from Chai Wah Wu, Dec 07 2020
a(35)-a(36) from Michael S. Branicky, Dec 08 2020
a(37)-a(38) from Chai Wah Wu, Dec 08 2020

A339613 Number of sets of distinct primes whose sum is a prime, the largest element of a set is prime(n).

Original entry on oeis.org

1, 2, 2, 2, 5, 8, 15, 30, 57, 115, 211, 398, 783, 1528, 3002, 5893, 11432, 22247, 43663, 86348, 170472, 335636, 662988, 1312816, 2595986, 5121351, 10096635, 19930303, 39469458, 78311512, 155219706, 307373610, 607613871, 1202463562, 2383024521, 4736192475, 9413441133

Offset: 1

Ilya Gutkovskiy, Dec 09 2020

nonn

			a(6) = 8 sets: {13}, {3, 7, 13}, {5, 11, 13}, {7, 11, 13}, {2, 3, 5, 13}, {2, 3, 11, 13}, {2, 5, 11, 13} and {2, 3, 5, 7, 11, 13}.

Cf. A000040, A071810 (partial sums), A127542.

from sympy import prime, isprime
from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, s, c):
  if n == 0:
    if isprime(s): return 1
    return 0
  return b(n-1, s, c) + b(n-1, s+prime(n), c+1)
a = lambda n: b(n-1, prime(n), 1)
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Dec 10 2020

a(35)-a(37) from Michael S. Branicky, Dec 09 2020

A364535 a(n) is the number of subsets of the first n primes whose sum is not a prime.

Original entry on oeis.org

1, 1, 1, 3, 9, 20, 44, 93, 191, 390, 787, 1600, 3250, 6563, 13227, 26609, 53484, 107588, 216413, 434894, 872834, 1750938, 3512454, 7043770, 14119562, 28300792, 56733873, 113746102, 228033527, 456999525, 915558925, 1834081043, 3674191081, 7361544506, 14749015536, 29545860199, 59169406092

Offset: 0

Darío Clavijo, Oct 20 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000040, A000295, A071810, A262765.

a(n) = 2^n - A071810(n).

a(n) = A000295(n) - A262765(n) + 1 for n > 2.

A385571 Number of subsets of the first n twin primes (A001097) whose sum is a twin prime.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 22, 37, 64, 119, 227, 450, 884, 1673, 3019, 5113, 8877, 16119, 30250, 57282, 109178, 210547, 412261, 819165, 1641582, 3298602, 6600608, 13150469, 26176431, 52289653, 104981405, 210846036, 420699038, 828442946, 1610436120, 3102364760

Offset: 1

Ilya Gutkovskiy, Jul 03 2025

nonn

			a(5) = 8 subsets: {3}, {5}, {7}, {11}, {13}, {3, 5, 11}, {5, 11, 13} and {7, 11, 13}.

David Radcliffe, Table of n, a(n) for n = 1..501
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001097, A071810.

a[n_] := Module[{t = {} , p = 2}, While[Length[t] < n, p = NextPrime[p]; If[PrimeQ[p - 2] || PrimeQ[p + 2], AppendTo[t, p]]]; Count[Total /@ Subsets[t], ?(PrimeQ[#] && (PrimeQ[# - 2] || PrimeQ[# + 2]) &)]]; Array[a, 20] (* _Amiram Eldar, Jul 03 2025 *)

a(23)-a(25) from Amiram Eldar, Jul 03 2025

a(26)-a(36) from David Radcliffe, Jul 04 2025

A339556 Number of subsets of the first n primes whose elements have a prime root-mean-square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 16, 19, 30, 41, 54, 69, 106, 177, 272, 397, 686, 1299, 2416, 4225, 7196, 11701, 20352, 36305, 70134, 132721, 248722, 473391, 894318, 1674923, 3054022, 5452067, 9626552, 16696543, 29086462, 51830095, 96887612, 192393735, 397875694

Offset: 1

Ilya Gutkovskiy, Dec 08 2020

nonn

			a(7) = 11 subsets: {2}, {3}, {5}, {7}, {11}, {13}, {17}, {7, 17}, {5, 7, 17}, {7, 13, 17} and {5, 7, 11, 17}.

Eric Weisstein's World of Mathematics, Root-Mean-Square

Cf. A000040, A071810, A127542, A163974, A309160, A339454, A339485.

a(10)-a(40) from Alois P. Heinz, Dec 08 2020

cp's OEIS Frontend

A108018 a(n) = the number of primes representable as the sum of some subset of the set of first n primes.

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A262765 a(n) = the number of ways that at least two distinct primes <= prime(n) sum to a prime.

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A339485 Number of subsets of the first n primes whose elements have a prime average.

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A339613 Number of sets of distinct primes whose sum is a prime, the largest element of a set is prime(n).

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Python

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A364535 a(n) is the number of subsets of the first n primes whose sum is not a prime.

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A385571 Number of subsets of the first n twin primes (A001097) whose sum is a twin prime.

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Programs

Mathematica

Extensions

A339556 Number of subsets of the first n primes whose elements have a prime root-mean-square.

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Author

Keywords

Examples

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Extensions