A002372 -id:A002372 - cp's OEIS frontend

A233654 |{prime p < n: n - p = sigma(k) for some k > 0}|, where sigma(k) is the sum of all (positive) divisors of k.

0, 0, 1, 1, 1, 3, 1, 3, 3, 3, 3, 2, 2, 4, 4, 3, 5, 4, 4, 6, 4, 3, 5, 3, 6, 5, 5, 1, 4, 4, 7, 5, 6, 4, 8, 3, 8, 5, 5, 2, 9, 5, 10, 8, 8, 4, 10, 3, 11, 6, 10, 2, 8, 4, 11, 5, 8, 3, 12, 5, 13, 7, 7, 3, 13, 3, 14, 7, 7, 5, 12, 3, 14, 9, 11, 6, 12, 2, 16, 7, 11, 5, 12, 3, 18, 8, 12, 2, 11, 3, 19, 6, 11, 4, 13, 4, 17, 8, 10, 6

Offset: 1

Zhi-Wei Sun, Dec 14 2013

nonn

Conjecture: (i) Let n > 1 be an integer. Then we have a(2*n) > 0. Also, 2*n + 1 can be written as p + sigma(k), where p is a Sophie Germain prime and k is a positive integer.
(ii) Each odd number greater than one can be written as sigma(k^2) + phi(m), where k and m are positive integers, and phi(.) is Euler's totient function.
That a(2*n+1) > 0 for n > 1 is a consequence of Goldbach's conjecture, for, if 2*n = p + q with p and q both prime, then 2*n + 1 = p + sigma(q) = q + sigma(p).

			a(3) = 1 since 3 = 2 + 1 = 2 + sigma(1) with 2 prime.
a(7) = 1 since 7 = 3 + 4 = 3 + sigma(3) with 3 prime.
a(10) = 3 since 10 = 2 + 8 = 2 + sigma(7) with 2 prime, 10 = 3 + 7 = 3 + sigma(4) with 3 prime, and 10 = 7 + 3 = 7 + sigma(2) with 7 prime.
a(13) = 2 since 13 = 5 + 8 = 5 + sigma(7) with 5 prime, and 13 = 7 + 6 = 7 + sigma(5) with 7 prime.
a(28) = 1 since 28 = 13 + 15 = 13 + sigma(8) with 13 prime.
a(36) = 3 since 36 = 5 + 31 = 5 + sigma(16) = 5 + sigma(25) with 5 prime, 36 = 23 + 13 = 23 + sigma(9) with 23 prime, and 36 = 29 + 7 = 29 + sigma(4) with 29 prime.
a(148) = 1 since 148 = 109 + 39 = 109 + sigma(18) with 109 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000203, A002372, A002375, A005384, A232270, A233542, A233544, A233547, A233549.

f[n_]:=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
S[n_]:=Union[Table[f[j],{j,1,n}]]
PQ[n_]:=n>0&&PrimeQ[n]
a[n_]:=Sum[If[PQ[n-Part[S[n],i]],1,0],{i,1,Length[S[n]]}]
Table[a[n],{n,1,100}]

A261627 Number of primes p such that n-(pn'-1) and n+(pn'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 4, 2, 3, 2, 2, 1, 2, 2, 3, 1, 3, 2, 2, 3, 3, 3, 3, 3, 3, 1, 4, 1, 3, 2, 3, 4, 4, 3, 3, 2, 4, 3, 6, 2, 3, 2, 2, 3, 5, 3, 4, 4, 4, 2, 5, 4, 6, 1, 4, 2, 4, 3, 5, 4, 3, 4

Offset: 1

Zhi-Wei Sun, Aug 27 2015

nonn

Conjecture: a(n) > 0 for all n > 6, and a(n) = 1 only for n = 5, 7, 10, 11, 12, 19, 22, 30, 34, 44, 46, 72, 142.
This is stronger than Goldbach's conjecture (A002375) and Lemoine's conjecture (A046927).
I have verified the conjecture for n up to 10^8.
Verified for n up to 10^9. - Mauro Fiorentini, Jul 05 2023
Conjecture verified for n < 1.2 * 10^12. - Jud McCranie, Aug 26 2023

			a(19) = 1 since 13, 19-(13-1) = 7 and 19+(13-1) = 31 are all prime.
a(142) = 1 since 41, 142-(2*41-1) = 61 and 142+(2*41-1) = 223 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Cf. A000040, A002372, A002375, A046927, A219055, A237284, A261628.

Do[r=0;Do[If[PrimeQ[n-(3+(-1)^n)/2*Prime[k]+1]&&PrimeQ[n+(3+(-1)^n)/2*Prime[k]-1],r=r+1],{k,1,PrimePi[2n/(3+(-1)^n)]}];Print[n," ",r];Continue,{n,1,80}]

A210480 Number of primes p

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 3, 3, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 4, 4, 4, 5, 5, 7, 6, 6, 3, 4, 3, 3, 5, 5, 4, 5, 5, 7, 7, 6, 4, 3, 3, 4, 4, 3, 2, 4, 4, 7, 6, 6, 3, 3, 4, 4, 4, 4, 2, 4, 4, 6, 5, 5, 3, 2, 4, 4, 6, 3, 3, 4, 4, 7, 5, 6, 4, 4, 4, 4, 7, 6, 5, 4, 3, 8, 5, 7, 3, 3, 5

Offset: 1

Zhi-Wei Sun, Jan 23 2013

nonn

Conjecture: a(n)>0 for all n>3.
This is stronger than Goldbach's conjecture and the author's conjecture that any odd number greater than one is the sum of a prime and a practical number. Also, it implies that there are infinitely many primes p with p-1 and p+1 both practical.
The author has verified this new conjecture for n up to 10^7.

			a(1846)=1 since 1846=1289+557 with 1289 and 557 both prime, and 1288 and 1290 both practical.
a(15675)=1 since 15675=919+14756 with 919 prime, and 918, 920, 14756 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..60000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002372, A005153, A210479, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True&&(PrimeQ[n-Prime[k]]==True||pr[n-Prime[k]]==True),1,0],{k,1,PrimePi[n-1]}]
Do[Print[n," ",a[n]],{n,1,100}]

A258713 A001172(n)/2: Least k such that 2k is a sum of two odd primes in exactly n ways.

Original entry on oeis.org

0, 3, 5, 11, 17, 24, 30, 39, 42, 45, 57, 72, 60, 84, 90, 117, 123, 144, 120, 105, 162, 150, 180, 237, 165, 264, 288, 195, 231, 240, 210, 285, 255, 336, 396, 378, 438, 357, 399, 345, 519, 315, 504, 465, 390, 480, 435, 462, 450, 567, 717, 420, 495, 651

Offset: 0

N. J. A. Sloane, Jun 14 2015

nonn

Up to a(14) also indices of records in A002375, number of ways to write 2n as sum of two odd primes. - M. F. Hasler, Aug 21 2017

Robert Israel, Table of n, a(n) for n = 0..845

Cf. A136244, A001172.

Cf. A002375, A002372, A045917.

g:= add(x^ithprime(i),i=2..1000):
G:= series((g^2+add(x^(2*ithprime(i)),i=2..1000))/2,x,ithprime(1001)+3):
A[0]:= 0:
for k from 1 to (ithprime(1001)+1)/2 do
  m:= coeff(G,x,2*k);
  if not assigned(A[m]) then A[m]:= k fi;
od:
for m from 1 while assigned(A[m]) do od:
seq(A[i],i=0..m-1); # Robert Israel, Aug 21 2017

With[{s = Array[Count[Select[IntegerPartitions[2 #, 2], Length@ # == 2 &], p_ /; AllTrue[p, And[PrimeQ@ #, OddQ@ #] &]] &, 10^3]}, Table[FirstPosition[s, n][[1]] /. 1 -> 0, {n, 0, 53}]] (* Michael De Vlieger, Aug 21 2017 *)

Edited by M. F. Hasler, Aug 21 2017

Edited by Robert Israel, Aug 21 2017

A156642 Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 3, 4, 3, 4, 6, 3, 2, 4, 3, 4, 5, 3, 2, 5, 4, 4, 5, 4, 4, 7, 4, 4, 5, 3, 6, 7, 3, 5, 7, 4, 4, 7, 4, 5, 10, 5, 4, 7, 3, 7, 9, 5, 6, 8, 5, 5, 9, 5, 5, 11, 6, 5, 9, 5, 6, 10, 5, 6, 8, 6, 6, 9, 5, 5, 12, 6, 5, 9

Offset: 0

Vladimir Shevelev, Feb 12 2009

nonn

Conjecture. For n >= 1, a(n) > 0. This conjecture does not follow from the validity of the Goldbach binary conjecture because numbers of the form 4n+2, generally speaking, also have decompositions into sums of two primes of the form 4k+1.

			From _Lei Zhou_, Mar 19 2013: (Start)
n=1: 4n+2=6, 6=3+3; this is the only case that matches the definition, so a(1)=1;
n=3: 4n+2=14, 14=3+11=7+7; two instances found, so a(3)=2. (End)

Lei Zhou, Table of n, a(n) for n = 0..10000

Cf. A002145, A002375, A061358, A002372, A045917, A214834, A217696.

Table[m = 4*n + 2; p1 = m + 1; ct = 0; While[p1 = p1 - 4; p2 = m - p1; p1 >= p2, If[PrimeQ[p1] && PrimeQ[p2], ct++]]; ct, {n, 1, 100}] (* Lei Zhou, Mar 19 2013 *)

A228623 Determinant of the n X n matrix with (i,j)-entry (i,j = 0,...,n-1) equal to 1 or 0 according as n + i - j and n - i + j are both prime or not.

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 0, 0, 0, -4, -1, 0, 0, 0, -6, 0, 0, -144, 0, 0, 0, -1, 168, 1024, 420, 0, 0, 0, -1, -9801, 0, 144, 0, 0, 3072, 7056, 0, 0, -42346434, 0, 0, -331776, 0, 0, 36528128, -104976, 96545145, 0, 34665386, -62500, 2826240, 2025, 0, -23174596, 0, 0, 255578880, -4, -3, 990172089

Offset: 1

Zhi-Wei Sun, Aug 27 2013

sign

Conjecture: a(n) is nonzero if n is odd and greater than 120.

This implies Goldbach's conjecture for even numbers of the form 4*k + 2.

			a(1) = 0 since 1 + 0 - 0 = 1 is not a prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A002372, A228591, A228615, A228616, A228557, A228559.

a[n_]:=Det[Table[If[PrimeQ[n+j-i]==True&&PrimeQ[n+i-j]==True,1,0],{i,0,n-1},{j,0,n-1}]]
Table[a[n],{n,1,20}]

A230443 Number of decompositions of 2n into a sum of two primes p2 >= p1 such that the number of runs in binary expansion of p2-p1 is less than or equal to 4.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 2, 5, 3, 4, 6, 3, 4, 6, 2, 5, 6, 5, 4, 7, 3, 5, 7, 5, 4, 9, 3, 4, 6, 3, 5, 8, 3, 6, 7, 5, 5, 10, 4, 5, 8, 3, 3, 10, 2, 6, 7, 6, 3, 8, 7, 7, 10, 6, 5, 12, 3, 7, 10, 5, 5, 10, 1, 6, 10, 7, 4

Offset: 2

Lei Zhou, Oct 18 2013

This is a tightly intensified version of Goldbach conjecture.
It is hypothesized that except for n=1402 and 27242, all other terms for n > 1 are greater than zero. Sequence tested up to 1 million without other zero elements.
The definition of "the number of runs in binary expansion of k" is from A005811.
The first difference of this sequence to A002375 is on a(26).

			n=2, 2n=4, 4=2+2, 2-2=0, A005811(0)=1 < 4, only one decomposition, so a(2)=1;
...
n=5, 2n=10, 10=5+5=3+7, 5-5=0, A005811(0)=1<4, 7-3=4, A005811(4)=2<4, so a(5)=2;
...
n=26, 2n=52, 52=5+47=11+41=23+29.  47-5=42, A005811(42)=6>4 [X]; 41-11=30, A005811(30)=2<4 [v]; 29-23=6, A005811(6)=2<4 [v]; so a(26)=2.

Lei Zhou, Table of n, a(n) for n = 2..10000

Cf. A002372, A005811

Table[ev=2*seed;ct=0;cp1=seed-1;While[cp1=NextPrime[cp1];cp1

A233864 a(n) = |{0 < m < 2n: m = sigma(k) for some k > 0, and 2n - 1 - m and 2*n - 1 + m are both prime}|, where sigma(k) is the sum of all (positive) divisors of k.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 3, 1, 1, 3, 3, 3, 3, 2, 4, 5, 3, 4, 4, 4, 4, 4, 3, 5, 4, 5, 4, 5, 3, 4, 7, 4, 5, 6, 4, 8, 8, 4, 4, 4, 7, 5, 6, 5, 6, 8, 4, 6, 8, 6, 7, 6, 6, 5, 5, 9, 7, 9, 7, 6, 8, 7, 7, 8, 6, 9, 9, 6, 6, 12, 9, 6, 10, 8, 9, 12, 7, 7, 11, 5, 10, 9, 9, 10, 7, 11, 8, 9, 6, 8, 14, 10, 8, 8, 10, 12, 9, 6

Offset: 1

Zhi-Wei Sun, Dec 16 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any even number 2*n > 0, 2*n + sigma(k) is prime for some 0 < k < 2*n.
See also A233793 for a related conjecture.
Clearly part (i) of the conjecture implies Goldbach's conjecture for even numbers 2*(2*n - 1) with n > 3; we have verified part (i) for n up to 10^8. Concerning part (ii), we remark that 1024 is the unique positive integer k < 1134 with 1134 + sigma(k) prime, and that sigma(1024) = 2047 > 1134.

			a(7) = 1 since sigma(5) = 6, and 2*7 - 1 - 6 = 7 and 2*7 - 1 + 6 = 19 are both prime.
a(10) = 1 since sigma(6) = sigma(11) = 12, and 2*10 - 1 - 12 = 7 and 2*10 - 1 + 12 = 31 are both prime.
a(11) = 1 since sigma(7) = 8, and 2*11 - 1 - 8 = 13 and 2*11 - 1 + 8 = 29 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000203, A002372, A002375, A232270, A233544, A233654, A233793.

f[n_]:=Sum[If[Mod[n,d]==0,d,0],{d,1,n}]
S[n_]:=Union[Table[f[j],{j,1,n}]]
PQ[n_]:=n>0&&PrimeQ[n]
a[n_]:=Sum[If[PQ[2n-1-Part[S[2n-1],i]]&&PQ[2n-1+Part[S[2n-1],i]],1,0],{i,1,Length[S[2n-1]]}]
Table[a[n],{n,1,100}]

A234808 a(n) = |{0 < k < n: p = k + phi(n-k) and 2*n - p are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 0, 3, 1, 2, 5, 2, 1, 5, 1, 2, 7, 2, 1, 4, 1, 2, 1, 4, 1, 4, 2, 4, 11, 4, 2, 3, 1, 5, 2, 3, 2, 6, 1, 5, 15, 4, 2, 9, 1, 6, 2, 5, 4, 6, 4, 4, 3, 8, 3, 6, 4, 7, 21, 2, 4, 7, 1, 7, 4, 6, 4, 6, 4, 8, 22, 7, 3, 13, 1, 10, 5, 3, 5, 7, 4, 9, 5, 10, 5, 8, 7, 7, 6, 8, 5, 6, 3, 8, 6, 7, 4, 8, 4

Offset: 1

Zhi-Wei Sun, Dec 30 2013

nonn

Conjecture: a(n) > 0 except for n = 1, 8.

Clearly, this implies Goldbach's conjecture.

			a(3) = 1 since 2 + phi(1) = 3 and 2*3 - 3 = 3 are both prime.
a(20) = 1 since 11 + phi(9) = 17 and 2*20 - 17 = 23 are both prime.
a(22) = 1 since 1 + phi(21) = 13 and 2*22 - 13 = 31 are both prime.
a(24) = 1 since 9 + phi(15) = 17 and 2*24 - 17 = 31 are both prime.
a(76) = 1 since 67 + phi(9) = 73 and 2*76 - 73 = 79 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A002372, A002375, A233547, A233918, A234200, A234246, A234360, A234470, A234475, A234514, A234567, A234615, A234694

f[n_,k_]:=k+EulerPhi[n-k]
p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[2n-f[n,k]]
a[n_]:=a[n]=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A243485 Sum of all the products formed by multiplying the corresponding smaller and larger parts of the Goldbach partitions of n.

Original entry on oeis.org

0, 0, 0, 4, 6, 9, 10, 15, 14, 46, 0, 35, 22, 82, 26, 94, 0, 142, 34, 142, 38, 263, 0, 357, 46, 371, 0, 302, 0, 591, 58, 334, 62, 780, 0, 980, 0, 578, 74, 821, 0, 1340, 82, 785, 86, 1356, 0, 1987, 94, 1512, 0, 1353, 0, 2677, 106, 1421, 0, 2320, 0, 4242, 118

Offset: 1

Wesley Ivan Hurt, Jun 05 2014

a(n) is even for odd n.
If Goldbach's conjecture is true, a(n) > 0 for all even n > 2.
Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W <= L. For example, a(16) = 94; the two rectangles are 3 X 13 and 5 X 11, and the sum of their areas is 3*13 + 5*11 = 94. - Wesley Ivan Hurt, Oct 28 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A002372, A002375, A045917, A064911, A117929.

with(numtheory): A243485:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=1..floor(n/2)): seq(A243485(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2017

Table[Sum[i*(n - i)*Floor[2/PrimeOmega[i (n - i)]], {i, 2, n/2}], {n,
  50}]

a(n) = Sum_{i=2..n/2} i*(n-i) * A064911(i*(n-i)).

a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017

cp's OEIS Frontend

A233654 |{prime p < n: n - p = sigma(k) for some k > 0}|, where sigma(k) is the sum of all (positive) divisors of k.

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A261627 Number of primes p such that n-(p*n'-1) and n+(p*n'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

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A210480 Number of primes p

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A258713 A001172(n)/2: Least k such that 2k is a sum of two odd primes in exactly n ways.

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A156642 Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

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A228623 Determinant of the n X n matrix with (i,j)-entry (i,j = 0,...,n-1) equal to 1 or 0 according as n + i - j and n - i + j are both prime or not.

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A230443 Number of decompositions of 2n into a sum of two primes p2 >= p1 such that the number of runs in binary expansion of p2-p1 is less than or equal to 4.

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A233864 a(n) = |{0 < m < 2*n: m = sigma(k) for some k > 0, and 2*n - 1 - m and 2*n - 1 + m are both prime}|, where sigma(k) is the sum of all (positive) divisors of k.

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A261627 Number of primes p such that n-(pn'-1) and n+(pn'-1) are both prime, where n' is 1 or 2 according as n is odd or even.

A233864 a(n) = |{0 < m < 2n: m = sigma(k) for some k > 0, and 2n - 1 - m and 2*n - 1 + m are both prime}|, where sigma(k) is the sum of all (positive) divisors of k.