A209312 -id:A209312 - cp's OEIS frontend

A209321 Indices k for which A209312(k) = 2.

4, 5, 6, 7, 9, 11, 14, 19, 26, 38, 46, 62, 70, 74, 86, 94, 118, 134, 194, 206, 278, 286, 566, 598, 778, 842, 934, 1006, 1082, 1214, 1238, 1546, 1574, 1726, 1858, 2018, 2278, 2474, 2774, 3142, 3686

Offset: 1

M. F. Hasler, Jan 19 2013

Many of these seem to be even semiprimes.
Is this sequence infinite, and if so, can someone exhibit an explicitly calculable subsequence?
a(42) > 10^5, if it exists. - Amiram Eldar, May 02 2024

Cf. A209312.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := pracQ[n] = (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), ?(# > 1 &)]) == {}; q[n] := 2 == Sum[If[pracQ[p] && ((PrimeQ[n - p] && PrimeQ[n + p]) || (pracQ[n - p] && pracQ[n + p])), 1, 0], {p, 1, n - 1}]; Select[Range[4000], q] (* Amiram Eldar, May 02 2024 *)

Data corrected by Amiram Eldar, May 02 2024

A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 19 2013

nonn

Conjecture: a(n)>0 for all n>8.
This has been verified for n up to 10^7.
As p+q=2p+(q-p), the conjecture implies Lemoine's conjecture related to A046927.
Zhi-Wei Sun also conjectured that any integer n>2 can be written as p+q, where p is a prime,  one of q and q+1 is prime and another of q and q+1 is practical.

			a(9)=1 since 2*9-1=5+12 with 12 practical, 5 and 12-5 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A046927, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185.

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[PrimeQ[p]==True&&pr[2n-1-p]==True&&PrimeQ[2n-1-2p]==True,1,0],{p,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A209320 Number of ways to write 2n = p+q with p and q both prime, p+1 and q-1 both practical.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 2, 2, 2, 3, 4, 5, 3, 2, 3, 3, 5, 7, 3, 3, 4, 4, 5, 8, 4, 3, 5, 2, 4, 8, 3, 4, 6, 2, 4, 7, 3, 4, 7, 2, 4, 9, 4, 4, 9, 5, 3, 9, 3, 5, 8, 3, 4, 10, 4, 6, 8, 5, 4, 14, 2, 4, 8, 2, 6, 10, 4, 4, 7, 4, 4, 10, 5, 4, 8, 3, 4, 9, 5, 5, 7, 3, 3, 13, 6, 5, 7, 4, 2, 11, 5, 5, 9, 4, 2, 9, 3, 6, 10, 7

Offset: 1

Zhi-Wei Sun, Jan 19 2013

nonn

Conjecture: a(n)>0 for all n>2.
As p+q=(p+1)+(q-1), this unifies Goldbach's conjecture and its analog involving practical numbers.
The conjecture has been verified for n up to 10^7.

			a(8) = 2 since 2*8 = 3+13 = 11+5 with 3, 5, 11, 13 all prime and 3+1, 13-1, 11+1, 5-1 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A002372, A045917, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315.

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[PrimeQ[2n-Prime[k]]==True&&pr[Prime[k]+1]==True&&pr[2n-Prime[k]-1]==True,1,0],{k,1,PrimePi[2n-2]}]
Do[Print[n," ",a[n]],{n,1,100}]

A210445 Least positive integer k with k*n practical.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 1, 2, 2, 6, 1, 6, 2, 2, 1, 12, 1, 12, 1, 2, 3, 12, 1, 4, 3, 2, 1, 12, 1, 16, 1, 2, 6, 4, 1, 18, 6, 2, 1, 20, 1, 20, 2, 2, 6, 24, 1, 4, 2, 4, 2, 24, 1, 4, 1, 4, 6, 24, 1, 24, 8, 2, 1, 4, 1, 30, 3, 4, 2, 30, 1, 30, 9, 2, 3, 4, 1, 36, 1, 2, 10, 36, 1, 4, 10, 4, 1, 36, 1, 4, 3, 6, 12, 4, 1, 42, 2, 2, 1

Offset: 1

Zhi-Wei Sun, Jan 20 2013

Conjecture: a(n) < n for all n>1, and a(n) < n/2 for all n>47.

Large values are obtained for prime n: The corresponding subsequence is a(p(n)) = (1, 2, 4, 4, 6, 6, 12, 12, 12, 12, 16, 18, 20, 20, 24, 24, 24, 24, ...), while for composite indices, a(c(n)) = (1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 1, 2, ...). - M. F. Hasler, Jan 21 2013

			a(10)=2 since 2*10=20 is practical but 1*10=10 is not.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Cf. A005153, A210444, A071558, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315, A219320.

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)
Do[Do[If[pr[k*n]==True,Print[n," ",k];Goto[aa]],{k,1,n}];
Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]

A210445(n)={for(k=1,n,is_A005153(k*n)&&return(k))} \\ (Would return 0 if a(n)>n.) - M. F. Hasler, Jan 20 2013

a(n) = 1 iff n is in A005153, therefore a(n) > 1 for all odd n>1. - M. F. Hasler, Jan 21 2013

A210531 Number of nonnegative integers k

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 2, 2, 2, 3, 2, 2, 4, 5, 4, 2, 3, 7, 5, 1, 2, 7, 4, 2, 7, 5, 6, 1, 5, 9, 4, 4, 6, 9, 9, 2, 5, 12, 9, 3, 5, 6, 8, 5, 6, 13, 4, 2, 8, 6, 11, 6, 11, 14, 8, 2, 4, 7, 4, 5, 7, 29, 8, 3, 5, 8, 11, 4, 13, 16, 13, 2, 7, 12, 13, 6, 10, 16, 10, 6, 15, 9, 13, 3, 9, 20, 11, 8, 11, 20, 9, 2, 8, 22, 14, 6, 15, 15

Offset: 1

Zhi-Wei Sun, Jan 28 2013

nonn

Conjecture: a(n)>0 for all n>0. Moreover, if n>0 is different from 74, 138, 166, 542, then n+k^3 is practical for some 0<=k<=sqrt(n)*log(n); if n is not equal to 102, then n+k and n+k^3 are both practical for some k=0,...,n-1.

Zhi-Wei Sun also conjectured that any integer n>1 can be written as x^3+y (x,y>0) with 2x and 4xy both practical.

			a(22)=1 since 22+2^3=30 is practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A185636, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528.

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[n+k^3]==True,1,0],{k,0,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

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}]

A210533 Number of ways to write 2n = x+y (x,y>0) with x-1 and x+1 both prime, and x and x^3+y^3 both practical.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 28 2013

nonn

Conjecture: a(n)>0 for all n>2. Moreover, for each m=2,3,4,... any sufficiently large even integer can be written as x+y (x,y>0) with x-1 and x+1 both prime, and x and x^m+y^m both practical.

			a(17)=1 since 2*17=12+22 with 11 and 13 both prime, and 12 and 12^3+22^3=12376 both practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A210528, A210531, 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[PrimeQ[2k-1]==True&&PrimeQ[2k+1]==True&&pr[2k]==True&&pr[(2k)^3+(2n-2k)^3]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A210681 Number of ways to write 2n = p+q+r (p<=q) with p, q, r-1, r+1 all prime and p-1, p+1, q-1, q+1, r all practical.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 29 2013

nonn

Conjecture: a(n) > 0 for all n > 4.
This conjecture involves two kinds of sandwiches introduced by the author, and it is much stronger than the Goldbach conjecture for odd numbers. We have verified the conjecture for n up to 10^7.
Zhi-Wei Sun also made the following conjectures:
(1) Any even number greater than 10 can be written as the sum of four elements in the set
    S = {prime p: p-1 and p+1 are both practical}.
Also, every n=3,4,5,... can be represented as the sum of a prime in S and two triangular numbers.
(2) Each integer n>7 can be written as p + q + x^2 (or p + q + x(x+1)/2), where p is a prime with p-1 and p+1 both practical, and q is a practical number with q-1 and q+1 both prime.
(3) Every n=3,4,... can be written as the sum of three elements in the set
    T = {x: 6x is practical with 6x-1 and 6x+1 both prime}.
(4) Any integer n>6 can be represented as the sum of two elements of the set S and one element of the set T.
(5) Each odd number greater than 11 can be written in the form 2p+q+r, where p and q belong to S, and r is a practical number with r-1 and r+1 both prime.

			a(5)=1 since 2*5=3+3+4 with 3 and 5 both prime, and 2 and 4 both practical.
a(6)=2 since 2*6=3+3+6=3+5+4 with 3,5,7 all prime and 2,4,6 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56(1996), 205-210.
Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A005153, A068307, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210479, A210528, A210531, A210533, A258838.

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)
pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True
a[n_]:=a[n]=Sum[If[pp[j]==True&&pp[k]==True&&pq[2n-Prime[j]-Prime[k]]==True,1,0],{j,1,PrimePi[n-1]},{k,j,PrimePi[2n-Prime[j]]}]
Do[Print[n," ",a[n]],{n,1,100}]

A210444 a(n) = |{0

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 1, 0, 2, 2, 0, 4, 0, 1, 4, 2, 0, 6, 1, 3, 2, 2, 0, 5, 2, 1, 3, 1, 2, 11, 0, 1, 4, 1, 2, 6, 0, 2, 4, 3, 1, 9, 2, 3, 4, 2, 0, 7, 1, 4, 4, 5, 0, 8, 4, 1, 3, 3, 0, 15, 0, 3, 4, 4, 4, 13, 2, 4, 2, 5, 2, 10, 0, 2, 11, 2, 3, 12, 0, 6, 6, 2, 2, 13, 3, 5, 7, 5, 1, 16, 4, 4, 6, 3, 2, 11, 0, 8, 6, 7

Offset: 1

Zhi-Wei Sun, Jan 20 2013

nonn

Conjecture: a(n)>0 for all n>911.

This implies that for each n=2,3,4,... there is a positive integer k

The conjecture has been verified for n up to 10^6.

			a(7) = 1 since 6*7 = 42 is practical, and 41 and 43 are twin primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A071558, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A219185, A219312, A219315, A219320.

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[PrimeQ[k*n-1]==True&&PrimeQ[k*n+1]==True&&pr[k*n]==True,1,0],{k,1,n-1}]
Do[Print[n," ",a[n]],{n,1,100}]

A210722 Number of ways to write n = (2-(n mod 2))p+q+2^k with p, q-1, q+1 all prime, and p-1, p+1, q all practical.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 4, 2, 4, 2, 4, 3, 4, 4, 3, 4, 5, 3, 8, 5, 8, 4, 5, 4, 5, 3, 5, 4, 4, 3, 9, 2, 12, 4, 9, 5, 7, 6, 7, 5, 5, 7, 10, 5, 13, 6, 10, 6, 8, 6, 7, 5, 1, 7, 7, 1, 10, 5, 8, 4, 9, 7, 8, 6, 3, 10, 6, 6, 10, 7, 7, 9, 11, 7, 10, 10, 5, 10, 7, 5, 10, 7, 4, 8, 8, 5, 11, 5, 8, 10, 7, 5, 12, 5

Offset: 1

Zhi-Wei Sun, Jan 29 2013

nonn

Conjecture: a(n)>0 except for n = 1,...,8, 10, 520, 689, 740.

Zhi-Wei Sun also guessed that any integer n>6 different from 407 can be written as p+q+F_k, where p is a prime with p-1 and p+1 practical, q is a practical number with q-1 and q+1 prime, and F_k (k>=0) is a Fibonacci number.

			a(1832)=1 since 1832=2*881+6+2^6 with 5, 7, 881 all prime and 6, 880, 882 all practical.
a(11969)=1 since 11969=127+11778+2^6 with 127, 11777, 11779 all prime and 126, 128, 11778 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
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. A005153, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528, A210531, A210533, A210681.

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)
pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True
a[n_]:=a[n]=Sum[If[pp[j]==True&&pq[n-2^k-(2-Mod[n,2])Prime[j]]==True,1,0],{k,0,Log[2,n]},{j,1,PrimePi[(n-2^k)/(2-Mod[n,2])]}]
Do[Print[n," ",a[n]],{n,1,100}]

Showing 1-10 of 13 results. Next

cp's OEIS Frontend

A209321 Indices k for which A209312(k) = 2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A209315 Number of ways to write 2n-1 = p+q with q practical, p and q-p both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A209320 Number of ways to write 2n = p+q with p and q both prime, p+1 and q-1 both practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A210445 Least positive integer k with k*n practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A210531 Number of nonnegative integers k

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A210480 Number of primes p

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A210533 Number of ways to write 2n = x+y (x,y>0) with x-1 and x+1 both prime, and x and x^3+y^3 both practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A210681 Number of ways to write 2n = p+q+r (p<=q) with p, q, r-1, r+1 all prime and p-1, p+1, q-1, q+1, r all practical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs