The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A210722 Number of ways to write n = (2-(n mod 2))p+q+2^k with p, q-1, q+1 all prime, and p-1, p+1, q all practical. 2
 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 4, 2, 4, 2, 4, 3, 4, 4, 3, 4, 5, 3, 8, 5, 8, 4, 5, 4, 5, 3, 5, 4, 4, 3, 9, 2, 12, 4, 9, 5, 7, 6, 7, 5, 5, 7, 10, 5, 13, 6, 10, 6, 8, 6, 7, 5, 1, 7, 7, 1, 10, 5, 8, 4, 9, 7, 8, 6, 3, 10, 6, 6, 10, 7, 7, 9, 11, 7, 10, 10, 5, 10, 7, 5, 10, 7, 4, 8, 8, 5, 11, 5, 8, 10, 7, 5, 12, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS Conjecture: a(n)>0 except for n = 1,...,8, 10, 520, 689, 740. Zhi-Wei Sun also guessed that any integer n>6 different from 407 can be written as p+q+F_k, where p is a prime with p-1 and p+1 practical, q is a practical number with q-1 and q+1 prime, and F_k (k>=0) is a Fibonacci number. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..20000 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. EXAMPLE a(1832)=1 since 1832=2*881+6+2^6 with 5, 7, 881 all prime and 6, 880, 882 all practical. a(11969)=1 since 11969=127+11778+2^6 with 127, 11777, 11779 all prime and 126, 128, 11778 all practical. 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) pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True a[n_]:=a[n]=Sum[If[pp[j]==True&&pq[n-2^k-(2-Mod[n, 2])Prime[j]]==True, 1, 0], {k, 0, Log[2, n]}, {j, 1, PrimePi[(n-2^k)/(2-Mod[n, 2])]}] Do[Print[n, " ", a[n]], {n, 1, 100}] CROSSREFS Cf. A005153, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528, A210531, A210533, A210681. Sequence in context: A048225 A155481 A075148 * A162341 A066728 A279161 Adjacent sequences: A210719 A210720 A210721 * A210723 A210724 A210725 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jan 29 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 14 23:20 EDT 2024. Contains 373401 sequences. (Running on oeis4.)