A031924 -id:A031924 - cp's OEIS frontend

A052234 Primes p from A031924 such that A052180(primepi(p)) = 17.

727, 1033, 1747, 2837, 4093, 5387, 5897, 6337, 7121, 7867, 8887, 9467, 10723, 11437, 13751, 15077, 15313, 15791, 16097, 16333, 17047, 17117, 17321, 19597, 20177, 21401, 22147, 23167, 28541, 28573, 30307, 31327, 33641, 41017, 41597

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

A052235 Primes p from A031924 such that A052180(primepi(p)) = 19.

Original entry on oeis.org

433, 587, 1117, 2411, 4007, 4993, 5107, 5147, 5563, 6703, 6857, 6971, 7541, 10847, 12973, 14951, 18787, 21221, 24373, 24527, 27947, 29201, 30341, 30643, 30757, 36913, 37483, 38321, 39877, 40487, 42767, 43451, 45007, 46301, 47287, 48883, 49037

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925.

Reap[For[p = 23, p < 50000, p = q, q = NextPrime[p]; If[q == p + 6 && Max[ FactorInteger[#][[1, 1]]& /@ Range[p+1, q-1]] == 19, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Jan 29 2018 *)

A052236 Primes p from A031924 such that A052180(primepi(p)) = 29.

Original entry on oeis.org

1187, 1361, 2287, 3797, 4723, 5711, 7451, 10061, 10987, 12497, 17021, 18587, 20327, 22067, 25603, 26417, 32563, 41263, 41381, 43991, 50833, 53617, 55997, 60521, 64871, 71713, 77977, 81457, 84011, 87317, 87547, 89983, 90971, 98801

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

A052237 Primes p from A031924 such that A052180(primepi(p)) = 31.

Original entry on oeis.org

1453, 3313, 5981, 6911, 8707, 15467, 22721, 25447, 25633, 26627, 29167, 30097, 31957, 42187, 42373, 42437, 46093, 48017, 48947, 49627, 51673, 52667, 58061, 59113, 62897, 68477, 74923, 78643, 78707, 105613, 106357, 107351, 108217

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

f[p_] := Max[FactorInteger[#][[1, 1]] & /@ Range[ p+1, NextPrime[p] - 1]]; Select[Prime@ Range@ 10300, NextPrime[#] == # + 6 && f[#] == 31 &] (* Giovanni Resta, May 30 2018 *)

A052238 Primes p from A031924 such that A052180(p) = 23.

Original entry on oeis.org

941, 1217, 1907, 3607, 4391, 6047, 6367, 8117, 8713, 9127, 9221, 10093, 10601, 11981, 12577, 14741, 19571, 19753, 23203, 23893, 24677, 25367, 28723, 29921, 36131, 36313, 39857, 41143, 42937, 51907, 52183, 52691, 54667, 55633, 58211

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

A052247 Maximal value of smallest prime divisors of the 5 composite numbers between A031924(n) and A031925(n).

Original entry on oeis.org

5, 5, 7, 5, 5, 7, 5, 7, 5, 7, 13, 5, 5, 11, 7, 5, 5, 5, 5, 7, 13, 5, 19, 5, 5, 5, 13, 5, 5, 19, 5, 5, 13, 11, 5, 7, 17, 11, 5, 23, 13, 7, 11, 5, 5, 17, 11, 7, 5, 19, 7, 7, 29, 23, 5, 5, 5, 5, 5, 29, 37, 5, 31, 5, 7, 5, 7, 5, 5, 11, 5, 17, 7, 13, 5, 11, 23, 5, 11, 5, 5, 5, 47, 5, 29, 5, 5, 13, 5

Offset: 1

Labos Elemer, Feb 01 2000

nonn

			The 5 composites between A031924(1) = 23 and A031925(1) = 29 are 24, 25, 26, 27, 28; their smallest prime divisors are 2 5 2 3 2; the maximal value is 5; so a(1) = 5.

A031924, A031925, A052158.

A078862 Smallest primes from A031924, each belonging to those different residue class of mod 210 which are listed in A078861. Arranged according to possible least positive residues mod 210.

Original entry on oeis.org

1901, 433, 647, 23, 31, 3607, 251, 47, 53, 61, 1117, 73, 83, 727, 941, 733, 947, 331, 131, 557, 353, 151, 157, 373, 167, 173, 601, 607, 3761, 1033

Offset: 1

Labos Elemer, Dec 13 2002

			Several terms are equal to corresponding ones in A078861, while others are larger like: 1033=4.210+193, where r=193 is in A078861.

Cf. A031924, A008364, A078861, A078859-A078864.

f[x_] := Mod[Prime[x], 210] d[x_] := Prime[x+1]-Prime[x] t=Table[0, {210}]; Do[s=f[n]; If[Equal[d[n], 6]&&s<211&&t[[s]]==0, t[[s]]=Prime[n]], {n, 1, 1000}]; t

A023201 Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)

Original entry on oeis.org

5, 7, 11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 131, 151, 157, 167, 173, 191, 193, 223, 227, 233, 251, 257, 263, 271, 277, 307, 311, 331, 347, 353, 367, 373, 383, 433, 443, 457, 461, 503, 541, 557, 563, 571, 587, 593, 601, 607, 613, 641, 647

Offset: 1

David W. Wilson

T. D. Noe, Table of n, a(n) for n = 1..10000
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].
Wikipedia, Sexy prime.

A031924 (primes starting a gap of 6) and A007529 together give this (A023201).

Cf. A046117 (a(n)+6), A087695 (a(n)+3), A098428, A000040, A010051, A006489 (subsequence).

a023201 n = a023201_list !! (n-1)
a023201_list = filter ((== 1) . a010051 . (+ 6)) a000040_list
-- Reinhard Zumkeller, Feb 25 2013

[n: n in [0..40000] | IsPrime(n) and IsPrime(n+6)]; // Vincenzo Librandi, Aug 04 2010

A023201 := proc(n)
    option remember;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+2 by 2 do
            if isprime(a) and isprime(a+6) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 28 2013

Select[Range[10^2], PrimeQ[ # ]&&PrimeQ[ #+6] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Prime[Range[120]],PrimeQ[#+6]&] (* Harvey P. Dale, Mar 20 2018 *)

is(n)=isprime(n+6)&&isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From M. F. Hasler, Jan 02 2020: (Start)
a(n) = A046117(n) - 6 = A087695(n) - 3.
A023201 = { p = A000040(k) | A000040(k+1) = p+6 or A000040(k+2) = p+6 } = A031924 U A007529. (End)

A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 457, 461, 613, 641, 821, 823, 853, 857, 877, 881, 1087, 1091, 1277, 1297, 1301, 1423, 1427, 1447, 1481, 1483, 1487, 1607, 1663, 1693, 1783, 1867, 1871, 1873, 1993, 1997

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Or, prime(m) such that prime(m+2) = prime(m)+6. - Zak Seidov, May 07 2012

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Cf. A031924, A023201, A098414, A098415, A098424, A098416, A098417.

Union of A022004 and A022005.

[NthPrime(n): n in [1..310] | (NthPrime(n)+6) eq NthPrime(n+2)]; // Bruno Berselli, May 07 2012

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(2*i+1, i=1..floor((N+5)/2))]):locs:= select(t -> Primes[t+2]-Primes[t]=6, [$1..nops(Primes)-2]):
Primes[locs]; # Robert Israel, Apr 30 2015

ptrsQ[n_]:=PrimeQ[n+6]&&(PrimeQ[n+2]||PrimeQ[n+4])
Select[Prime[Range[400]],ptrsQ]  (* Harvey P. Dale, Mar 08 2011 *)

p=2;q=3;forprime(r=5,1e4,if(r-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, May 07 2012

a(n) = A098415(n) - 6. - Zak Seidov, Apr 30 2015

A286385 a(n) = A003961(n) - A000203(n).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 3, 12, 12, 3, 1, 17, 3, 9, 11, 50, 1, 36, 3, 21, 23, 3, 5, 75, 18, 9, 85, 43, 1, 33, 5, 180, 17, 3, 29, 134, 3, 9, 29, 99, 1, 69, 3, 33, 97, 15, 5, 281, 64, 54, 23, 55, 5, 255, 19, 177, 35, 3, 1, 147, 5, 15, 171, 602, 35, 51, 3, 45, 49, 87, 1, 480, 5, 9, 121, 67, 47, 87, 3, 381, 504, 3, 5, 271, 25, 9, 35, 171, 7, 291, 75, 93, 57, 15, 41, 963

Offset: 1

Antti Karttunen, May 09 2017

nonn

Are all terms nonnegative? This question is equivalent to the question posed in A285705.
From Antti Karttunen, Aug 05 2020: (Start)
The answer to the above question is yes. Because both A000203 and A003961 are multiplicative sequences, it suffices to prove that for any prime p, and e >= 1, q^e >= sigma(p^e) = ((p^(1+e))-1) / (p-1), where q = A151800(p), i.e., the next larger prime after p. If p is a lesser twin prime, then q = p+2 (and this difference can't be less than 2, apart from case p=2), and it is easy to see that (n+2)^e > ((n^(e+1)) - 1) / (n-1), for all n >= 2, e >= 1.
See comments in A326042.
(End)
This is the inverse Möbius transform of A337549, from which it is even easier to see that all terms are nonnegative. - Antti Karttunen, Sep 22 2020

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).

Cf. A326057 [= gcd(a(n), A252748(n))].

Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] - DivisorSigma[1, #] &, 96] (* Michael De Vlieger, Oct 05 2020 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A286385(n) = (A003961(n) - sigma(n));
for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));

from sympy import factorint, nextprime, divisor_sigma as D
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return 2*a048673(n) - D(n) - 1 # Indranil Ghosh, May 12 2017

(define (A286385 n) (- (A003961 n) (A000203 n)))

a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.
a(n) = A336852(n) - A336851(n). - Antti Karttunen, Aug 05 2020
a(n) = Sum_{d|n} A337549(d). - Antti Karttunen, Sep 22 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) - Pi^2/12 = 1.24152934..., where q(p) = nextprime(p) (A151800). - Amiram Eldar, Dec 21 2023

cp's OEIS Frontend

A052234 Primes p from A031924 such that A052180(primepi(p)) = 17.

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A052235 Primes p from A031924 such that A052180(primepi(p)) = 19.

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A052236 Primes p from A031924 such that A052180(primepi(p)) = 29.

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A052237 Primes p from A031924 such that A052180(primepi(p)) = 31.

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A052238 Primes p from A031924 such that A052180(p) = 23.

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A052247 Maximal value of smallest prime divisors of the 5 composite numbers between A031924(n) and A031925(n).

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A078862 Smallest primes from A031924, each belonging to those different residue class of mod 210 which are listed in A078861. Arranged according to possible least positive residues mod 210.

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Mathematica

A023201 Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)

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A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

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A286385 a(n) = A003961(n) - A000203(n).

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