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A209312
Number of practical numbers p<n with n-p and n+p both prime or both practical.
14
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
1,4
COMMENTS
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
LINKS
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.
EXAMPLE
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.
MATHEMATICA
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}]
PROG
(PARI) 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
CROSSREFS
Cf. A209321: Indices for which a(n)=2.
Sequence in context: A347037 A240301 A289641 * A054715 A254668 A306250
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 19 2013
STATUS
approved