cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A063637 Primes p such that p+2 is a semiprime.

Original entry on oeis.org

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 167, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 21 2001

Keywords

Comments

Primes of the form p*q - 2, where p and q are primes.
Union of A049002 and A115093. - T. D. Noe, Mar 01 2006

Examples

			From _K. D. Bajpai_, Sep 06 2014: (Start)
a(3) = 13, which is prime, and 13 + 2 = 15 = 3 * 5, which is a semiprime.
a(4) = 19, which is prime, and 19 + 2 = 21 = 3 * 7, which is a semiprime.
(End)

References

  • J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16 (1973), 157-176.

Links

Crossrefs

Cf. A005383, A001358, A063638.
Cf. A109611 (Chen primes).

Programs

  • Haskell
    a063637 n = a063637_list !!(n-1)
a063637_list = filter ((== 1) . a064911 . (+ 2)) a000040_list
-- Reinhard Zumkeller, Nov 15 2011
  • Maple
    select(t -> isprime(t) and numtheory:-bigomega(t+2)=2, [2, seq(2*i+1,i=1..500)]); # Robert Israel, Sep 07 2014
  • Mathematica
    f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; Select[ Prime[ Range[ 123]], f[ # + 2] == 2 &] (* Robert G. Wilson v, Apr 30 2005 *)
Select[Prime[Range[500]],PrimeOmega[#+2]==2&]  (* K. D. Bajpai, Sep 06 2014 *)
  • PARI
    { n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 2) == 2, write("b063637.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

Formula

a(n) = A062721(n) - 2.
A010051(a(n)) * A064911(a(n) + 2) = 1. - Reinhard Zumkeller, Nov 15 2011

A376830 Numbers k such that tau(k + tau(k)) = tau(k) + tau(tau(k)), where tau = A000005.

Original entry on oeis.org

1, 13, 19, 31, 37, 53, 67, 83, 88, 89, 109, 113, 127, 131, 139, 152, 157, 181, 190, 199, 211, 225, 233, 251, 257, 263, 276, 286, 293, 307, 317, 337, 344, 353, 379, 389, 401, 406, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 536, 541, 557, 563, 571, 577, 587, 612, 631, 642, 647, 653, 658, 677
Offset: 1

Views

Author

Robert Israel, Oct 06 2024

Keywords

Examples

			a(9) = 88 is a term because tau(88) = 8, tau(8) = 4 and tau(88 + 8) = tau(96) = 12 = 8 + 4.

Links

Crossrefs

Cf. A000005, A376831, A376843, A376844, A376848, A376849, A376851. Includes A115093.

Programs

  • Maple
    filter:= proc(k) uses numtheory; local s;
 s:= tau(k);
 tau(k+s) = s + tau(s)
end proc:
select(filter, [$1..1000]);
  • Mathematica
    Select[Range[680],DivisorSigma[0,#+DivisorSigma[0,#]]==DivisorSigma[0,#]+DivisorSigma[0,DivisorSigma[0,#]] &] (* Stefano Spezia, Oct 06 2024 *)

A286256 Compound filter: a(n) = P(A046523(n), A046523(2+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 12, 5, 40, 5, 84, 12, 86, 14, 142, 5, 148, 23, 216, 27, 367, 5, 265, 23, 148, 27, 412, 12, 430, 59, 142, 44, 832, 5, 1860, 23, 698, 61, 826, 27, 856, 23, 412, 27, 1402, 5, 850, 80, 148, 90, 1384, 12, 1759, 40, 265, 27, 607, 23, 1105, 61, 430, 27, 2086, 5, 2140, 80, 2352, 148, 4342, 27, 850, 23, 832, 27, 5080, 5, 2998, 80, 142, 148, 832, 27, 2956, 138, 1426
Offset: 1

Views

Author

Antti Karttunen, May 07 2017

Keywords

Links

Crossrefs

Cf. A000027, A046523, A286255, A286257, A286258.
Cf. A001359 (gives the positions of 5's), A049002 (of 12's), A115093 (of 23's).

Programs

  • PARI
    A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286256(n) = (2 + ((A046523(n)+A046523(2+n))^2) - A046523(n) - 3*A046523(2+n))/2;
for(n=1, 10000, write("b286256.txt", n, " ", A286256(n)));
  • Python
    from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), a046523(n + 2)) # Indranil Ghosh, May 07 2017
  • Scheme
    (define (A286256 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 2 n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 2 n)))) 2)))

Formula

a(n) = (1/2)*(2 + ((A046523(n)+A046523(2+n))^2) - A046523(n) - 3*A046523(2+n)).

A115267 Primes of the form p*q*r-2, where p, q and r are distinct primes.

Original entry on oeis.org

103, 163, 193, 229, 271, 283, 383, 397, 433, 463, 593, 607, 613, 643, 661, 739, 757, 859, 883, 967, 1013, 1021, 1063, 1093, 1103, 1129, 1171, 1237, 1279, 1307, 1433, 1453, 1489, 1493, 1531, 1543, 1549, 1579, 1597, 1613, 1657, 1693, 1741, 1747, 1831
Offset: 1

Views

Author

Zak Seidov, Mar 04 2006

Keywords

Examples

			103 = 3*5*7-2, 163 = 3*5*11-2 etc.

Links

Crossrefs

Cf. A115093 Primes of the form p*q-2, where p and q are distinct primes.
Cf. A000040, A001359, A007304, A115093.

Programs

  • Mathematica
    Select[Prime[Range[400]],Length[FactorInteger[ #+2]]==Plus@@(Last/@FactorInteger[ #+2])==3&]

Formula

{A007304(n)-2} INTERSECTION {A000040}. - Jonathan Vos Post, Mar 06 2006
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