A208249 -id:A208249 - cp's OEIS frontend

A209253 Number of ways to write 2n-1 as the sum of a Sophie Germain prime and a practical number.

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

Offset: 1

Zhi-Wei Sun, Jan 14 2013

nonn

Conjecture: a(n)>0 for all n>1.

This has been verified for n up to 5*10^6.

			a(40)=1 since 2*40-1=23+56 with 23 a Sophie Germain prime and 56 a practical number.

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], 2012-2017

Cf. A005384, A005153, A208243, A208244, A208246, A208249.

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

A209236 List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical.

Original entry on oeis.org

4, 6, 18, 30, 198, 462, 1482, 2550, 3330, 4422, 9042, 11778, 26862, 38610, 47058, 60258, 62130, 65538, 69498, 79902, 96222, 106782, 124542, 143262, 149058, 151902, 184830, 200382, 208962, 225342, 237690, 249858, 251262, 295038, 301182, 312702, 345462, 348462

Offset: 1

Zhi-Wei Sun, Jan 13 2013

nonn

Conjecture: a(n) always exists. In other words, there are infinitely many quintuples (m-2, m-1, m, m+1, m+2) with m-1 and m+1 both prime and m-2, m, m+2 all practical.
Note that this sequence is a subsequence of A014574.
Zhi-Wei Sun observed that if m-2, m, m+2 are all practical with m>4 then m is congruent to 2 modulo 4. His PhD student Shan-Shan Du gave the following explanation: If m>4 is a multiple of 4, then m-2 and m+2 are congruent to 2 modulo 4, and one of them is not divisible by 3 and hence not practical (since 4=1+3).
Because all practical numbers greater than 2 are multiples of 4 or 6 (or both), it follows that every term in this sequence after the first is congruent to 6 modulo 12. - Hal M. Switkay, May 03 2022

			a(3)=18 since {17,19} is a twin prime pair and 16, 18, 20 are practical numbers.

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, A014574, A208243, A208244, A208246, A208249.

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); n=0; t = {}; Do[If[PrimeQ[Prime[k]+2] == True && pr[Prime[k]-1] == True && pr[Prime[k]+1] == True && pr[Prime[k]+3] == True, n = n+1; AppendTo[t, Prime[k]+1]], {k, 100}]; t

o=3;forprime(p=5,,(2+o==o=p)||next; is_A005153(p-3) & is_A005153(p-1) & is_A005153(p+1) & print1(p-1,",")) \\ M. F. Hasler, Jan 13 2013

A209312 Number of practical numbers p

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 1, 2, 3, 2, 4, 1, 2, 3, 3, 3, 4, 2, 4, 3, 4, 3, 6, 3, 2, 3, 4, 4, 6, 3, 5, 3, 4, 5, 8, 3, 2, 5, 5, 4, 7, 4, 7, 4, 2, 4, 11, 3, 1, 4, 7, 4, 7, 6, 7, 3, 4, 5, 12, 3, 2, 4, 8, 7, 8, 5, 9, 4, 2, 6, 14, 5, 2, 6, 7, 7, 9, 5, 9, 4, 4, 5, 14, 8, 2, 5, 8, 7, 10, 6, 9, 6, 2, 8, 15, 5, 3, 5, 8

Offset: 1

Zhi-Wei Sun, Jan 19 2013

nonn

Conjecture: a(n)>0 for all n>2.
This has been verified for n up to 10^7.
Except for p=1, all practical numbers are even. Thus, (n-p,n+p) prime is possible only if n is odd, and (n-p,n+p) can be practical only if n is even (except for p=1). - M. F. Hasler, Jan 19 2013

			a(8)=1 since 4, 8-4 and 8+4 are all practical.
a(13)=1 since 6 is practical, and 13-6 and 13+6 are 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, A208243, A208244, A208246, A208249, A209253, A209254.

Cf. A209321: Indices for which a(n)=2.

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

A209312(n)=sum(p=1,n-1, is_A005153(p) && ((is_A005153(n-p) && is_A005153(n+p)) || (isprime(n-p) && isprime(n+p)))) \\ (Could be made more efficient by separating the case of odd and even n.) - M. F. Hasler, Jan 19 2013

A210479 Primes p with p-1 and p+1 both practical: "Sandwich of the first kind".

Original entry on oeis.org

3, 5, 7, 17, 19, 29, 31, 41, 79, 89, 127, 197, 199, 271, 307, 379, 449, 461, 463, 521, 701, 727, 811, 859, 881, 919, 929, 967, 991, 1217, 1231, 1289, 1301, 1409, 1471, 1481, 1483, 1567, 1721, 1889, 1951, 1999, 2129, 2393, 2441, 2549, 2551, 2729, 2753, 2861, 2969, 3041, 3079, 3319, 3329, 3331, 3499, 3739, 3761, 4049

Offset: 1

Zhi-Wei Sun, Jan 23 2013

nonn

When p is a prime with p-1 and p+1 both practical,  {p-1, p, p+1} is a sandwich of the first kind introduced by Zhi-Wei Sun. He conjectured that there are infinitely many such sandwiches. See also A210480 for a strong conjecture involving terms in the current sequence.
No term can be congruent to 1 or -1 modulo 12. In fact, if p>3 and 12|p-1, then neither 3 nor 4 divides p+1, hence p+1 is not practical since 4 is not a sum of some distinct divisors of p+1. Similarly, if 12|p+1 then p-1 is not practical.
Conjecture: The sequence a(n)^(1/n) (n=9,10,...) is strictly decreasing to the limit 1. Also, if {b(n)-1,b(n),b(n)+1} is the n-th sandwich of the second kind, then the sequence b(n)^(1/n) (n=1,2,3,...) is strictly decreasing to the limit 1.
This conjecture is similar to Firoozbakht's conjecture for primes.

			a(1)=3 since 2 and 4 are practical.
a(2)=5 since 4 and 6 are practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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, A208249, A209236, A210480, 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)
n=0
Do[If[pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True,n=n+1;Print[n," ",Prime[k]]],{k,1,100}]

is_A210479(p)={is_A005153(p-1) && is_A005153(p+1) && isprime(p)} \\ M. F. Hasler, Jan 23 2013

A210479(n,print_all=0)={forprime(p=3,, is_A005153(p-1) & is_A005153(p+1) & !(print_all & print1(p",")) & !n-- & return(p))} \\ M. F. Hasler, Jan 23 2013

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}]

A287681 Twin practical numbers: numbers k such that both k and k+2 are practical numbers.

Original entry on oeis.org

2, 4, 6, 16, 18, 28, 30, 40, 54, 64, 78, 88, 126, 160, 196, 198, 208, 270, 304, 306, 340, 378, 390, 414, 448, 460, 462, 510, 520, 544, 558, 700, 702, 726, 798, 810, 858, 868, 880, 918, 928, 966, 990, 1024, 1120, 1216, 1230, 1240, 1288, 1300, 1350, 1408, 1456

Offset: 1

Amiram Eldar, May 29 2017

nonn

Melfi proved that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Giuseppe Melfi, A note on twin practical numbers, Le Matematiche, Vol. 57 No. 1 (2004), pp. 111-117.
Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.
Giuseppe Melfi, Twin practical numbers.

Cf. A005153, A210479, A208249, A209236, A210480, A209236, A287682, A287683.

practicalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
a={}; p1=False; k=2; While[Length[a]<100, p2=practicalQ[k]; If[p1 && p2, a=AppendTo[a,k-2]]; p1 = p2; k+=2];a

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}]

cp's OEIS Frontend

A209253 Number of ways to write 2n-1 as the sum of a Sophie Germain prime and a practical number.

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A209236 List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical.

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A209312 Number of practical numbers p

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A210479 Primes p with p-1 and p+1 both practical: "Sandwich of the first kind".

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

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

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A287681 Twin practical numbers: numbers k such that both k and k+2 are practical numbers.

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A210445 Least positive integer k with k*n practical.

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