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A247048 Composite numbers n such that n+2 is also composite and such that (sopfr(n), sopfr(n+2)) is a twin prime pair. A001414 explains notation 'sopfr(n)'. 1
54, 845, 1083, 37595, 50367, 50898, 101673, 103416, 107909, 112344, 117390, 160020, 172176, 266342, 761926, 839147, 1183488, 1190597, 1629219, 1711950, 1742585, 2247839, 2312190, 2345894, 2372272, 2756502, 3092261, 3679504, 3710280, 4832298, 5267824, 5305023, 5350590, 6720445 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The general case is (b,b+2*k) to give |sopfr(b)-sopfr(b+2*k)|=2*k=p+2*k-p for prime pair (p,p+2*k).  For various k compare all pairs (p,p+2*k)<10^m to determine if the percentage of matching (b,b+2*k)is a maximum for some value of k.  For k=1 and m=6, there are 15 of 8169 twin primes that had matching (b,b+2), or only 0.1836%, about 1 out of 545.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..222

EXAMPLE

sopfr(845)=5+13+13=31 and sopfr(847)=7+11+11=29 and (29,31) are twin primes.

MATHEMATICA

sopfr[n_]:=Total[Flatten[Table[#[[1]], #[[2]]]&/@FactorInteger[n]]]; Select[ Range[ 673*10^4], CompositeQ[#]&&CompositeQ[#+2]&&AllTrue[ {sopfr[ #], sopfr[#+2]}, PrimeQ]&&Abs[sopfr[#]-sopfr[#+2]]==2&] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Apr 16 2018 *)

PROG

(PARI) sopfr(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 1]*f[i, 2]);

is(m, n)=my(p, q); !isprime(m) && !isprime(n) && isprime(p=sopfr(m)) && isprime(q=sopfr(n)) && abs(p-q)==2;

for(n=1, 10^6, if(is(n, n+2), print1(n", ")))

\\ Charles R Greathouse IV, Sep 08 2014

(Python)

from sympy import isprime, factorint

A247048_list = []

for n in range(1, 10**6):

....if not(isprime(n) or isprime(n+2)):

........m = sum(p*e for p, e in factorint(n).items())

........if isprime(m):

............m2 = sum(p*e for p, e in factorint(n+2).items())

............if ((m2 == m+2) or (m == m2+2)) and isprime(m2):

................A247048_list.append(n)

# Chai Wah Wu, Sep 25 2014

CROSSREFS

Cf. A001359, A001414.

Sequence in context: A297865 A298132 A008401 * A280479 A107420 A298069

Adjacent sequences:  A247045 A247046 A247047 * A247049 A247050 A247051

KEYWORD

nonn

AUTHOR

J. M. Bergot, Sep 10 2014

EXTENSIONS

More terms from Charles R Greathouse IV, Sep 10 2014

STATUS

approved

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Last modified March 23 05:36 EDT 2019. Contains 321422 sequences. (Running on oeis4.)