A209312 -id:A209312 - cp's OEIS frontend

A211165 Number of ways to write n as the sum of a prime p with p-1 and p+1 both practical, a prime q with q+2 also prime, and a Fibonacci number.

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

Offset: 1

Zhi-Wei Sun, Jan 30 2013

nonn

Conjecture: a(n)>0 for all n>5.
This has been verified for n up to 300000.
Note that for n=406 we cannot represent n in the given way with q+1 practical.

			a(6)=a(7)=1 since 6=3+3+0 and 7=3+3+1 with 3 and 5 both prime, 2 and 4 both practical, 0 and 1 Fibonacci 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, New Goldbach-type conjectures involving primes and practical numbers, a message to Number Theory List, Jan. 29, 2013.
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A005153, A000045, A001359, A210479, A210681, A210722, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528, A210531, A210533.

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
q[n_]:=q[n]=PrimeQ[n]==True&&PrimeQ[n+2]==True
a[n_]:=a[n]=Sum[If[k!=2&&Fibonacci[k]

A261641 Number of practical numbers q such that n+(n mod 2)-q and n-(n mod 2)+q are both practical numbers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Aug 27 2015

nonn

Conjecture: a(n) > 0 except for n = 2. Also, for any integer n > 3, there is a practical number q such that n-(n mod 2)-q and n+(n mod 2)+q are both practical numbers.

This is an analog of the author's conjecture in A261627, and it is stronger than Margenstern's conjecture proved by Melfi in 1996.

			a(15) = 1 since 4, 15-(4-1) = 12 and 15+(4-1) = 18 are all practical.
a(2206) = 1 since 2106, 2206-2106 = 100 and 2206+2106 = 4312 are 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], 2012-2015.

Cf. A005153, A209312, A261627.

f[n_]:=FactorInteger[n]
Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
Do[r=0;Do[If[pr[q]&&pr[n+Mod[n,2]-q]&&pr[n-Mod[n,2]+q],r=r+1],{q,1,n}];Print[n," ",r];Continue,{n,1,80}]

A261653 Number of primes p < n such that n-p-1 and n+p+1 are both prime or both practical.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Aug 28 2015

nonn

Conjecture: a(n) > 0 for all n > 6. Also, for any integer n > 2, there is a prime p < n such that n-(p-1) and n+(p-1) are both prime or both practical.

Note that 1 is the only odd practical number and 2 is the only even prime.

			a(31) = 1 since 11, 31-11-1 = 19 and 31+11+1 = 43 are all prime.
a(38) = 17 since 17 is prime, and 38-17-1 = 20 and 38+17+1 = 56 are 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], 2012-2015.

Cf. A000040, A005153, A208243, A209315, A209320, A209312, A261627, A261641.

f[n_]:=FactorInteger[n]
Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
p[n_]:=Prime[n]
Do[r=0;Do[If[(PrimeQ[n-p[k]-1]&&PrimeQ[n+p[k]+1])||(pr[n-p[k]-1]&&pr[n+p[k]+1]),r=r+1],{k,1,PrimePi[n-1]}];Print[n," ",r];Continue,{n,1,80}]

cp's OEIS Frontend

A211165 Number of ways to write n as the sum of a prime p with p-1 and p+1 both practical, a prime q with q+2 also prime, and a Fibonacci number.

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A261641 Number of practical numbers q such that n+(n mod 2)-q and n-(n mod 2)+q are both practical numbers.

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A261653 Number of primes p < n such that n-p-1 and n+p+1 are both prime or both practical.

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