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A209236 List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical. 14
4, 6, 18, 30, 198, 462, 1482, 2550, 3330, 4422, 9042, 11778, 26862, 38610, 47058, 60258, 62130, 65538, 69498, 79902, 96222, 106782, 124542, 143262, 149058, 151902, 184830, 200382, 208962, 225342, 237690, 249858, 251262, 295038, 301182, 312702, 345462, 348462 (list; graph; refs; listen; history; text; internal format)
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).

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.

MAPLE

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

Cf. A005153, A014574, A208243, A208244, A208246, A208249.

Sequence in context: A026689 A138276 A287682 * A182643 A156096 A281861

Adjacent sequences:  A209233 A209234 A209235 * A209237 A209238 A209239

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 13 2013

STATUS

approved

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Last modified June 13 20:01 EDT 2021. Contains 345009 sequences. (Running on oeis4.)