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.

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A118723 Chen primes whose digital root is also a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 47, 59, 83, 101, 113, 131, 137, 149, 167, 191, 227, 239, 257, 263, 281, 293, 311, 317, 347, 353, 389, 401, 419, 443, 461, 479, 491, 509, 563, 569, 587, 599, 617, 641, 653, 659, 677, 743, 761, 797, 821, 839, 857, 887, 911, 941, 947
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

Keywords

Examples

			167 is in the sequence because (1) it is a Chen prime and (2) the digital root 5 is also a Chen prime.

Crossrefs

Cf. A010888, A109611.

Programs

  • Mathematica
    digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#] &, n];
chenPrimeQ[x_] := PrimeQ[x] && Plus@@Last/@FactorInteger[x+2] <= 2;
Select[Range[1000], chenPrimeQ[#] && chenPrimeQ[digitalRoot[#]]&] (* Vladimir Joseph Stephan Orlovsky, Apr 30 2011, with a function by Eric W. Weisstein *)
  • PARI
    isA118723(n)=[0,0,1,1,0,1,0,1,0][n%9+1]&isprime(n)&bigomega(n+2)<3 \\ Charles R Greathouse IV, Apr 30 2011

A118724 Chen primes for which the multiplicative digital root is also a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 53, 71, 113, 131, 137, 157, 211, 311, 317, 359, 389, 431, 557, 571, 751, 827, 839, 953, 983, 1117, 1151, 1297, 1367, 1511, 1553, 1621, 1637, 1759, 2111, 2179, 2213, 2269, 2339, 2393, 2699, 2719, 2969, 2971, 3167, 3221, 3329, 3511
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

Keywords

Examples

			157 is in the sequence because (1) it is a Chen prime and (2) the multiplicative digital root 5 is also a Chen prime.

Crossrefs

Cf. A109611.

Programs

  • Mathematica
    mdr[n_] := NestWhile[Times @@ IntegerDigits@# &, n, # > 9 &]; chenQ[n_] := PrimeQ[n] && Plus @@ Last /@ FactorInteger[n + 2] < 3; Select[ Prime@ Range@500, chenQ@ mdr@# && chenQ@# &] (* Robert G. Wilson v *)

Extensions

More terms from Robert G. Wilson v, May 22 2006

A118773 Smaller of two consecutive Chen primes with the same multiplicative digital root.

Original entry on oeis.org

101, 107, 167, 179, 197, 251, 293, 401, 443, 491, 503, 509, 521, 563, 577, 587, 617, 631, 653, 809, 1009, 1019, 1031, 1039, 1049, 1061, 1091, 1097, 1283, 1327, 1381, 1409, 1427, 1439, 1451, 1471, 1511, 1559, 1567, 1583, 1601, 1607, 1621, 1787, 1871
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), May 22 2006

Keywords

Crossrefs

Cf. A109611.

A118774 Larger of two consecutive Chen primes with the same multiplicative digital root.

Original entry on oeis.org

107, 109, 179, 181, 199, 257, 307, 409, 449, 499, 509, 521, 541, 569, 587, 599, 631, 641, 659, 811, 1019, 1031, 1039, 1049, 1061, 1091, 1097, 1109, 1289, 1361, 1399, 1427, 1439, 1451, 1459, 1487, 1553, 1567, 1583, 1601, 1607, 1619, 1637, 1801, 1877
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), May 22 2006

Keywords

Crossrefs

Cf. A109611.

A118775 Sum of two consecutive Chen primes.

Original entry on oeis.org

5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 88, 100, 112, 126, 138, 154, 172, 190, 208, 216, 222, 240, 258, 268, 276, 288, 306, 324, 346, 360, 372, 388, 396, 410, 438, 460, 472, 490, 508, 520, 532, 550, 574, 600, 618, 628, 654, 684, 700, 712, 738, 768, 790
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), May 22 2006

Keywords

Crossrefs

Cf. A109611.

Programs

A124050 Difference between (first Chen prime > 10^n) and 10^n.

Original entry on oeis.org

1, 1, 1, 9, 7, 19, 37, 19, 7, 7, 19, 19, 61, 687, 97, 91, 79, 13, 79, 97, 151, 217, 427, 253, 667, 13, 127, 427, 457, 577, 1069, 349, 1147, 1267, 2527, 2833, 709, 871, 259, 361, 1651, 391, 2689, 649, 31, 3007, 1657, 2257, 3757, 5977, 1441, 2779, 5749, 367, 31
Offset: 0

Views

Author

Zak Seidov, Nov 03 2006

Keywords

Comments

A033873(n) <= a(n) <= A124001(n) and a(n) = A033873(n) for n = 0, 1, 2, 3, 4, 7, 8, 9, 10, 25, 44, 54.

Examples

			a(0) = 1 because 2 is prime, 2 + 2 = 4 semiprime and 2 - 10^0 = 1,
a(1) = 1 because 11 and 13 are twin primes and 11 - 10^1 = 1,
a(2) = 1 because 101 and 103 are twin primes and 101 - 10^2 = 1,
a(3) = 9 because 1009 is prime, 1011 = 3*337 semiprime and 1009 - 10^3 = 9,
a(4) = 7 because 10007 and 10009 are twin primes and 10007 - 10^4 = 7,
a(5) = 19 because 100019 is prime, 100021 = 29*3449 semiprime and 100019 - 10^5 = 19, etc.

Crossrefs

Cf. A033873, A109611, A124001.

A172102 Primes in A118482.

Original entry on oeis.org

3, 11, 29, 59, 101, 239, 619, 809, 4253, 5323, 5923, 6551, 29131, 37277, 48341, 54413, 58711, 60937, 70537, 101063, 110533, 214993, 224603, 417203, 445069, 466537, 473867, 511391, 519089, 534629, 633449, 686269, 713771, 741913, 770767, 1000537
Offset: 1

Views

Author

Jonathan Vos Post, Jan 25 2010

Keywords

Crossrefs

Cf. A118482.

Programs

  • Maple
    isA001358 := proc(n) return ( numtheory[bigomega](n) = 2 ); end proc:
isA109611 := proc(n) isprime(n) and ( isprime(n+2) or isA001358(n+2) ); end proc:
A109611 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1) ) ; while not isA109611(a) do a := nextprime(a) ; end do ; return a; end if; end proc:
A118482 := proc(n) option remember ; 1+add( A109611(j),j=1..n) ; end proc:
isA172102 := proc(n) if isprime(n) then for j from 1 do if A118482(j) > n then return false; elif A118482(j) = n then return true; end if; end do ; else false ; end if; end proc:
for n from 1 to 10000000 do if isA172102(n) then printf("%d,\n",n) ; end if; end do ;
# R. J. Mathar, Feb 07 2010

Extensions

Extended by R. J. Mathar, Feb 07 2010

A258400 Perfect powers m^k such that m, k and m+k are primes.

Original entry on oeis.org

8, 9, 25, 32, 121, 289, 841, 1681, 2048, 3481, 5041, 10201, 11449, 18769, 22201, 32041, 36481, 38809, 51529, 57121, 72361, 78961, 96721, 120409, 131072, 175561, 185761, 212521, 271441, 323761, 358801, 380689, 410881, 434281, 654481, 674041, 683929, 734449
Offset: 1

Views

Author

Carlos Eduardo Olivieri, May 28 2015

Keywords

Comments

Necessarily either m or k = 2, thus if a(n) is even, it is a power of 2 with odd prime exponent, otherwise (if a(n) is odd), it is a square of odd prime.
For each term m^k, there will be another k^m.
a(3), a(5), a(11) are of the form n! + 1.
Let F(m,k) = m*k, such that m^k = a(n), so A108605 is a subsequence of F. For example a(1) = 2^3 and F(2,3) = A108605(1).

Examples

			a(1) = 8, because 8 = 2^3 and 2+3 = 5.
a(4) = 32, because 32 = 2^5 and 2+5 = 7.
a(5) = 121, because 121 = 11^2 and 11+2 = 13.
a(25) = 131072, because 131072 = 2^17 and 2+17 = 19.

Crossrefs

Subsequence of A001597, A000961.
Cf. A000079, A001248, A108605, A109611.

Programs

  • Mathematica
    SmallestDivisor[n_] := If[n == 1, 1, Divisors[n][[2]]]; perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; ppl = Select[Range[200000], perfectPowerQ]; base[n_] := ppl[[n]]^(1/exp[n]); exp[n_] := SmallestDivisor[GCD @@ FactorInteger[ppl[[n]]][[All, 2]] ]; pp2l = Table[ {base[n], exp[n]}, {n, Length[ppl]}]; p[n_] := pp2l[[n]][[1]]; q[n_] := pp2l[[n]][[2]]; lt = Select[Range[Length[pp2l]], PrimeQ[p[#]] && PrimeQ[q[#]] && PrimeQ[p[#] + q[#]] &]; ppl[[lt]]
Select[Range[10^6], Length[f = FactorInteger@ #] == 1 && PrimeQ@ f[[1, 2]] && PrimeQ@ Total@ f[[1]] &] (* Giovanni Resta, Jun 23 2015 *)

Extensions

a(28)-a(38) from Giovanni Resta, Jun 23 2015

A291531 Number of Chen primes up to 10^n.

Original entry on oeis.org

4, 20, 115, 633, 4234, 29949, 225630, 1762579, 14176573, 116718282, 979244657, 8343503219
Offset: 1

Views

Author

Charles R Greathouse IV, Aug 25 2017

Keywords

Comments

Chen primes are primes p such that p + 2 is either prime or semiprime.

Links

Crossrefs

Cf. A109611.

Programs

  • PARI
    a(n)=my(N=10^n+2,s,p=3); forprime(p=2,sqrtint(N), forprime(q=p,N\p, if(isprime(p*q-2), s++))); forprime(q=5,N, if(q-p==2, s++); p=q); s

Extensions

a(10)-a(12) from Giovanni Resta, Aug 26 2017

A321420 Primes p whose reversal is a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 71, 73, 101, 107, 113, 131, 149, 157, 167, 179, 181, 191, 199, 311, 347, 353, 359, 389, 701, 733, 739, 743, 751, 761, 787, 797, 919, 941, 953, 967, 971, 983, 991, 1009, 1021, 1031, 1061, 1091, 1097, 1103, 1109, 1151, 1153, 1217, 1223
Offset: 1

Views

Author

Paolo Galliani, Nov 09 2018

Keywords

Comments

73 is the smallest non-Chen prime whose reversal is a Chen prime.

Examples

			73 is in the sequence because its reversal is 37 which is a Chen prime (because 37 + 2 = 39 has at most two prime factors).

Crossrefs

Cf. A118725, A109611, A102540.

Programs

  • Mathematica
    cpQ[n_] := Module[{rev = FromDigits[Reverse[IntegerDigits[n]]]}, PrimeQ[rev] && PrimeOmega[rev + 2] < 3]; Select[Prime[Range[400]], cpQ] (* Amiram Eldar, Nov 09 2018 after Harvey P. Dale at A118725 *)
  • PARI
    is(n) = if(isprime(n), rn = fromdigits(Vecrev(digits(n))); return(isprime(rn) && bigomega(rn+2) <= 2), 0) \\ David A. Corneth, Nov 09 2018
Previous Showing 61-70 of 71 results. Next

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