A210480 -id:A210480 - cp's OEIS frontend

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

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

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

A211190 Number of ways to write 2n = p+2q+3r with p,q,r terms of A210479.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 03 2013

nonn

Conjecture: a(n)>0 for all n>8. Moreover, for positive integers a<=b<=c, all integers n>=3(a+b+c) with n-a-b-c even can be written as a*p+b*q+c*r with p,q,r terms of A210479, if and only if (a,b,c) is among the following 6 triples: (1,2,3), (1,2,4), (1,2,8), (1,2,9), (1,3,5), (1,3,8).

The author also conjectured that if n>8 is odd, different from 201 and 447, and not congruent to 1 or -1 modulo 12, then n can be written as a sum of three terms of A210479.

			a(10)=1 since 2*10=5+2*3+3*3 with 3 and 5 terms of A210479.

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, 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, A210479, A210480, A210681, A211165.

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)
p[k_]:=p[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
q[n_]:=q[n]=PrimeQ[n]==True&&pr[n-1]==True&&pr[n+1]==True
a[n_]:=a[n]=Sum[If[p[j]==True&&p[k]==True&&q[2n-2Prime[j]-3Prime[k]]==True,1,0],{j,1,PrimePi[n]},{k,1,PrimePi[(2n-2Prime[j])/3]}]
Do[Print[n," ",a[n]],{n,1,100}]

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

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

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A211190 Number of ways to write 2n = p+2q+3r with p,q,r terms of A210479.

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