The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A209312 Number of practical numbers p
 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 (list; graph; refs; listen; history; text; internal format)
 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 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. 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. A005153, A208243, A208244, A208246, A208249, A209253, A209254. Cf. A209321: Indices for which a(n)=2. Sequence in context: A210452 A240301 A289641 * A054715 A254668 A306250 Adjacent sequences:  A209309 A209310 A209311 * A209313 A209314 A209315 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 19 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 10 21:08 EDT 2021. Contains 343780 sequences. (Running on oeis4.)