A014092 -id:A014092 - cp's OEIS frontend

A303403 Even numbers that are not the sum of two prime-indexed primes.

2, 4, 12, 18, 24, 26, 30, 32, 38, 40, 50, 54, 56, 60, 66, 68, 74, 80, 92, 96, 102, 104, 106, 110, 116, 122, 128, 136, 146, 148, 152, 154, 156, 164, 170, 172, 178, 180, 200, 204, 206, 212, 226, 230, 234, 248, 256, 260, 264, 268, 276, 290, 292, 296, 298, 302

Offset: 1

Amiram Eldar, May 13 2018

nonn

Bayless et al. conjectured that every even number larger than 80612 is the sum of two prime-indexed primes. If the conjecture is true then this sequence is finite with 733 terms.

Similarly, it appears that 322704332 is the largest of the 1578727 even numbers that cannot be written as prime(prime(prime(i))) + prime(prime(prime(j))). - Giovanni Resta, May 31 2018

			20 is not in the sequence since 20 = 17 + 3 = prime(7) + prime(2).  2 and 7 are primes, so 3 and 17 are prime-indexed primes. - _Michael B. Porter_, May 21 2018

Jonathan Bayless, Dominic Klyve, and Tomás Oliveira e Silva, New Bounds and Computations on Prime-Indexed Primes, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A43, 2013.

Cf. A006450, A014092, A166081.

Equals 2*A174682. - Michel Marcus, May 18 2018

pipQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]; s1falsifiziertQ[s_]:= Module[{ip=IntegerPartitions[s, {2}], widerlegt=False}, Do[If[pipQ[ip[[i, 1]] ] ~And~ pipQ [ip[[i, 2]] ], widerlegt = True; Break[]], {i, 1, Length[ip]}]; widerlegt]; Select[Range[2500],EvenQ[#]&& s1falsifiziertQ[ # ]==False&] (* after Michael Taktikos at A014092 *)
(* or *) p = Prime@ Prime@ Range@ PrimePi@ PrimePi@ 302; Select[Range[2, 302, 2], IntegerPartitions[#, {2}, p] == {} &] (* Giovanni Resta, May 31 2018 *)

isok(n) = {if (n % 2, return (0)); forprime(p=2, n/2, if (isprime(primepi(p)) && isprime(n-p) && isprime(primepi(n-p)), return (0));); return (1);} \\ Michel Marcus, May 18 2018

A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.

Original entry on oeis.org

1, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234, 246, 288, 240, 210, 324, 300, 360, 474, 330, 528, 576, 390, 462, 480, 420, 570, 510, 672, 792, 756, 876, 714, 798, 690, 1038, 630, 1008, 930, 780, 960, 870, 924, 900, 1134, 1434, 840, 990, 1302

Offset: 0

Chai Wah Wu, Mar 11 2022

nonn

Conjecture: a(n) != -1 for all n.
If n > 0 and a(n) != -1, then a(n) is even.
a(0) = A014092(1)
a(1) = A067187(1)
a(2) = A067188(1)
a(3) = A067189(1)
a(4) = A067190(1)
a(5) = A067191(1)
a(6) = A066722(1)
a(7) = A352229(1)
a(8) = A352230(1)
a(9) = A352231(1)
a(10) = A352233(1)

Cf. A014092, A067187, A067188, A067189, A067190, A067191, A066722, A352229, A352230, A352231, A352233.

Essentially the same as A023036 and A001172.

f[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; seq[max] :=  Module[{s = Table[0, {max}], n = 1, c = 0, k}, While[c < max, k = f[n]; If[k < max && s[[k + 1]] == 0, c++; s[[k + 1]] = n]; n++]; s]; seq[50] (* Amiram Eldar, Mar 11 2022 *)

from itertools import count
from sympy import nextprime
def A352296(n):
    if n == 0:
        return 1
    pset, plist, pmax = {2}, [2], 4
    for m in count(2):
        if m > pmax:
            plist.append(nextprime(plist[-1]))
            pset.add(plist[-1])
            pmax = plist[-1]+2
        c = 0
        for p in plist:
            if 2*p > m:
                break
            if m - p in pset:
                c += 1
        if c == n:
            return m

A062303 Number of ways writing the n-th prime as a sum of two nonprimes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 6, 7, 8, 9, 9, 11, 13, 14, 15, 16, 17, 18, 19, 21, 24, 25, 26, 26, 27, 27, 33, 34, 36, 37, 40, 41, 42, 44, 45, 47, 49, 50, 53, 54, 54, 55, 59, 64, 65, 66, 66, 68, 69, 72, 74, 76, 78, 79, 80, 81, 82, 85, 91, 92, 93, 93, 99, 101, 105, 106, 106, 108

Offset: 1

Labos Elemer, Jul 05 2001

nonn

			n=10,p(10)=29 has 14 partitions of form a+b=29; 1+28=4+25=8+21=9+20=14+15 are the 5 relevant partitions, so a(10)=5.

Cf. A061358, A062602, A062610, A000040, A014092, A025584.

Table[c = 0; Do[If[i + j == Prime[n] && ! PrimeQ[i] && ! PrimeQ[j], c = c + 1], {i, Prime[n] - 1}, {j, i}]; c, {n, 72}] (* Jayanta Basu, Apr 22 2013 *)
cnpQ[{a_,b_}]:=(!PrimeQ[a]&&CompositeQ[b])||(!PrimeQ[b]&&CompositeQ[a]); Join[{1},Table[Length[Select[IntegerPartitions[Prime[n],{2}],cnpQ]],{n,2,80}]] (* Harvey P. Dale, Sep 30 2018 *)

A062610(A000040(n)) = number of [nonprime+composite] partitions of p(n).

Offset and name corrected by Sean A. Irvine, Mar 25 2023

A178400 Sums of two primes, an array by antidiagonals.

Original entry on oeis.org

4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 16, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 25, 25, 26, 26, 26, 26, 26, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 31

Offset: 1

Clark Kimberling, Dec 21 2010

The joint-rank array of this array is A178431.

			Northwest corner:
4....5....7....9...13...
5....6....8...10...12...
7....8...10...12...16...
9...10...12...14...14...
T(1,1)=2+2=4.

Cf. A000040, A014092.

T(i,j)=p(i)+p(j), where p=(2,3,5,7,...)=A000040.

A269801 Total sum of the divisors of the primes p,q such that n=p+q and p>=q.

Original entry on oeis.org

0, 0, 0, 0, 6, 7, 8, 9, 10, 11, 24, 0, 14, 15, 32, 17, 36, 0, 40, 21, 44, 23, 72, 0, 78, 27, 84, 0, 60, 0, 96, 33, 68, 35, 144, 0, 152, 0, 80, 41, 126, 0, 176, 45, 138, 47, 192, 0, 250, 51, 208, 0, 162, 0, 280, 57, 174, 0, 240, 0, 372, 63, 192, 65, 330, 0

Offset: 0

Wesley Ivan Hurt, Mar 05 2016

			a(5) = 7; Since 5 can be expressed in one way as the sum of the two primes 2 and 3, we add the sum of their divisors separately: sigma(2) + sigma(3) = 3 + 4 = 7.
a(10) = 24; Since 10 can be expressed in two ways as the sum of two primes, we add the sum of the divisors of each prime p and q: 10 = 3+7 = 5+5, so sigma(3) + sigma(7) + sigma(5) + sigma(5) = 4 + 8 + 6 + 6 = 24.

Cf. A000203 (sigma), A010051, A014092, A061358.

with(numtheory): A269801:=n->(n+2)*sum((pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=2..floor(n/2)): seq(A269801(n), n=0..100);

Table[(n+2) Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, 2, Floor[n/2]}], {n, 0, 80}]

a(n) = sum(i=0, n\2, if (isprime(i) && isprime(n-i), sigma(i)+sigma(n-i))); \\ Michel Marcus, Mar 05 2016

a(n) = (n+2) * A061358(n).
a(n) = (n+2) * Sum_{i=2..floor(n/2)} A010051(i) * A010051(n-i).
a(n) = Sum_{i=2..floor(n/2)} (A000203(i) + A000203(n-i)) * A010051(i) * A010051(n-i).

A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 13, 14, 17, 18, 19, 26, 33, 38, 46, 62, 82

Offset: 1

Ilya Gutkovskiy, Jul 20 2017

The sequence is conjectured to be complete.

Cf. A000469, A014092, A071068, A071331, A098235, A239508, A239509.

nmax = 82; f[x_] := Sum[Boole[SquareFreeQ[k] && PrimeNu[k] != 1] x^k, {k, 1, nmax}]^2; b = Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]; c = Complement[Range[nmax], b][[1 ;; 19]]

A200677 Smallest semiprime such that the sum of the two prime factors equals n, or zero if impossible.

Original entry on oeis.org

0, 0, 0, 4, 6, 9, 10, 15, 14, 21, 0, 35, 22, 33, 26, 39, 0, 65, 34, 51, 38, 57, 0, 95, 46, 69, 0, 115, 0, 161, 58, 87, 62, 93, 0, 155, 0, 217, 74, 111, 0, 185, 82, 123, 86, 129, 0, 215, 94, 141, 0, 235, 0, 329, 106, 159, 0, 265, 0, 371, 118, 177, 122, 183, 0

Offset: 1

Michel Lagneau, Nov 20 2011

nonn

For n > 3, a(n) = 0 if n-2 is an odd composite.
The sequence without zeros is a subsequence of A189553. - Manfred Scheucher, Aug 08 2015
The two prime factors are not necessarily distinct; a(6) = 9, both of whose prime factors are 3s. - Jon E. Schoenfield, Aug 09 2015

			a(10) = 21 because 21 = 3*7 and 3+7 = 10, and there is no semiprime smaller than 21 whose two prime factors sum to 10.

Cf. A001358, A014091, A014092, A135093.

with(numtheory):for n from 1 to 65 do:ii:=0:for k from 1 to 1000 while(ii=0)do:m1:=bigomega(k):x:=factorset(k): m2:=nops(x):if m1=2 and m2=2 and x[1]+x[2]= n or m1=2 and m2=1 and 2*x[1]= n then ii:=1: printf(`%d, `,k):else fi:od:if ii=0 then printf(`%d, `,0):else fi:od:

a(A014091(n)) > 0; a(A014092(n)) = 0. - Michel Marcus, Aug 10 2015

Edited by Jon E. Schoenfield and Manfred Scheucher, Aug 09 2015

A339020 Largest value of (p*q mod n), for primes p and q, where p + q = n and p <= q (or 0 if no such primes exist).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 7, 5, 5, 0, 11, 9, 7, 11, 7, 0, 11, 15, 11, 17, 19, 0, 23, 21, 17, 0, 19, 0, 29, 27, 23, 29, 25, 0, 35, 0, 27, 35, 39, 0, 41, 39, 39, 41, 37, 0, 47, 45, 41, 0, 43, 0, 47, 51, 55, 0, 53, 0, 59, 57, 53, 59, 55, 0, 65, 0, 59, 65, 69, 0, 71, 69, 65, 71, 71, 0, 53

Offset: 1

Wesley Ivan Hurt, Nov 22 2020

nonn

a(m) = 0 for m in A014092.

			a(14) = 7; There are two partitions of 14 into two primes, (3,11) and (7,7). Since (3*11 mod 14) = 5 and (7*7 mod 14) = 7, then 7 is the largest. Therefore, a(14) = 7.

Cf. A014092, A061358, A338768.

Table[If[n == 1, 0, Max[Table[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]) Mod[i (n - i), n], {i, Floor[n/2]}]]], {n, 100}]
Table[Max[Mod[Times@@#,n]&/@Select[IntegerPartitions[n,{2}],AllTrue[#,PrimeQ]&]],{n,80}]/.(-\[Infinity]->0) (* Harvey P. Dale, Mar 24 2024 *)

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A303403 Even numbers that are not the sum of two prime-indexed primes.

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A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.

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A062303 Number of ways writing the n-th prime as a sum of two nonprimes.

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A178400 Sums of two primes, an array by antidiagonals.

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A269801 Total sum of the divisors of the primes p,q such that n=p+q and p>=q.

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A290136 Positive numbers that are not the sum of two nonprime squarefree numbers (A000469).

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A200677 Smallest semiprime such that the sum of the two prime factors equals n, or zero if impossible.

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A339020 Largest value of (p*q mod n), for primes p and q, where p + q = n and p <= q (or 0 if no such primes exist).

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