OFFSET
1,1
COMMENTS
Conjecture: a(n) always exists. In other words, there are infinitely many quintuples (m-2, m-1, m, m+1, m+2) with m-1 and m+1 both prime and m-2, m, m+2 all practical.
Note that this sequence is a subsequence of A014574.
Zhi-Wei Sun observed that if m-2, m, m+2 are all practical with m>4 then m is congruent to 2 modulo 4. His PhD student Shan-Shan Du gave the following explanation: If m>4 is a multiple of 4, then m-2 and m+2 are congruent to 2 modulo 4, and one of them is not divisible by 3 and hence not practical (since 4=1+3).
Because all practical numbers greater than 2 are multiples of 4 or 6 (or both), it follows that every term in this sequence after the first is congruent to 6 modulo 12. - Hal M. Switkay, May 03 2022
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(3)=18 since {17,19} is a twin prime pair and 16, 18, 20 are practical numbers.
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); n=0; t = {}; Do[If[PrimeQ[Prime[k]+2] == True && pr[Prime[k]-1] == True && pr[Prime[k]+1] == True && pr[Prime[k]+3] == True, n = n+1; AppendTo[t, Prime[k]+1]], {k, 100}]; t
PROG
(PARI) o=3; forprime(p=5, , (2+o==o=p)||next; is_A005153(p-3) & is_A005153(p-1) & is_A005153(p+1) & print1(p-1, ", ")) \\ M. F. Hasler, Jan 13 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 13 2013
STATUS
approved