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 1-10 of 22 results. Next

A096677 A060254 indexed by A000040.

Original entry on oeis.org

7, 8, 10, 11, 13, 14, 16, 19, 20, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 40, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 73, 74, 75, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 93, 94, 95, 97, 98, 99, 101, 102
Offset: 1

Views

Author

Ray Chandler, Jul 10 2004

Keywords

Crossrefs

Cf. A060254, A096784, A096785, A096786, A096787, A096788.

Formula

a(n) = k such that A000040(k) = A060254(n).

A096787 Primes of form 4n+3 that are the sum of two consecutive composite numbers.

Original entry on oeis.org

19, 31, 43, 67, 71, 79, 103, 127, 131, 139, 151, 163, 191, 199, 211, 223, 239, 251, 271, 283, 307, 311, 331, 367, 379, 419, 431, 439, 443, 463, 487, 491, 499, 523, 547, 571, 599, 607, 619, 631, 643, 647, 659, 683, 691, 727, 739, 743, 751, 787, 811, 823, 827
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Links

Crossrefs

Subsequence of A060254. See A096788 for values 2n+1. See A096676 for n values.
Cf. A060254, A096784, A096785, A096786, A096788, A096676, A096679.

Programs

  • Mathematica
    2Select[ Range[ 450], PrimeQ[ # ] == PrimeQ[ # + 1] == False && PrimeQ[2# + 1, GaussianIntegers -> True] == True &] + 1 (* Robert G. Wilson v, Jul 11 2004 *)
Select[Total/@Partition[Select[Range[500],CompositeQ],2,1],PrimeQ[#] && IntegerQ[ (#-3)/4]&] (* Harvey P. Dale, Mar 06 2019 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=440;n=4;while(nKlaus Brockhaus, Jul 10 2004

Formula

Equals 1+2*A096788.

Extensions

Corrected and extended by Klaus Brockhaus and Ray Chandler, Jul 10 2004

A096785 Primes of form 4k+1 which are the sum of two consecutive composite numbers.

Original entry on oeis.org

17, 29, 41, 53, 89, 97, 101, 109, 113, 137, 149, 173, 181, 197, 229, 233, 241, 257, 269, 281, 293, 317, 337, 349, 353, 373, 389, 401, 409, 433, 449, 461, 509, 521, 557, 569, 577, 593, 601, 617, 641, 653, 677, 701, 709, 761, 769, 773, 797, 809, 821, 829, 853
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Crossrefs

Subsequence of A060254. See A096786 for values 2n. See A096675 for n values.
Cf. A060254, A096784, A096786, A096787, A096788, A096675, A096678.

Programs

  • Mathematica
    Do[If[PrimeQ[2*n+1]&&Equal[Mod[s, 4], 1]&&!PrimeQ[n]&&!PrimeQ[n+1], Print[2*n+1]], {n, 1, 1000}] (* Labos Elemer *)
2Select[ Range[450], PrimeQ[ # ] == PrimeQ[ # + 1] == PrimeQ[2# + 1, GaussianIntegers -> True] == False && PrimeQ[2# + 1] == True &] + 1 (* Robert G. Wilson v, Jul 11 2004 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=440;n=4;while(nKlaus Brockhaus, Jul 11 2004

Formula

Equals 1 + 2*A096786.

Extensions

Corrected and extended by Klaus Brockhaus, Rick L. Shepherd and Ray Chandler, Jul 10 2004

A096784 Numbers n such that both n and n+1 are composite numbers that sum up to a prime.

Original entry on oeis.org

8, 9, 14, 15, 20, 21, 26, 33, 35, 39, 44, 48, 50, 51, 54, 56, 63, 65, 68, 69, 74, 75, 81, 86, 90, 95, 98, 99, 105, 111, 114, 116, 119, 120, 125, 128, 134, 135, 140, 141, 146, 153, 155, 158, 165, 168, 174, 176, 183, 186, 189, 194, 200, 204, 209, 215, 216, 219, 221
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Crossrefs

See A060254 for the primes 2n+1.
Cf. A096785, A096786, A096787, A096788.

Programs

  • Magma
    [n: n in [0..250]|not IsPrime(n) and not IsPrime(n+1) and IsPrime(2*n+1)] // Vincenzo Librandi, Dec 18 2010
  • Mathematica
    Select[ Range[ 225], PrimeQ[ # ] == PrimeQ[ # + 1] == False && PrimeQ[2# + 1] == True &] (* Robert G. Wilson v, Jul 11 2004 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=230;n=4;while(nKlaus Brockhaus, Jul 11 2004

Formula

Equals (A060254 -1)/2.

Extensions

Corrected and extended by Klaus Brockhaus and Ray Chandler, Jul 10 2004

A096786 Numbers n such that both n and n+1 are composite numbers that sum up to a Pythagorean prime (i.e., of the form 4k+1).

Original entry on oeis.org

8, 14, 20, 26, 44, 48, 50, 54, 56, 68, 74, 86, 90, 98, 114, 116, 120, 128, 134, 140, 146, 158, 168, 174, 176, 186, 194, 200, 204, 216, 224, 230, 254, 260, 278, 284, 288, 296, 300, 308, 320, 326, 338, 350, 354, 380, 384, 386, 398, 404, 410, 414, 426, 428, 440
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Crossrefs

Subsequence (even numbers) of A096784. See A096785 for the associated primes.
Cf. A060254, A096784, A096785, A096787, A096788, A096675.

Programs

  • Mathematica
    Select[ Range[450], PrimeQ[ # ] == PrimeQ[ # + 1] == PrimeQ[2# + 1, GaussianIntegers -> True] == False && PrimeQ[2# + 1] == True &] (* Robert G. Wilson v, Jul 11 2004 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=465;n=4;while(nKlaus Brockhaus, Jul 11 2004

Formula

Equals (A096785 - 1)/2.

Extensions

Corrected and extended by Klaus Brockhaus, Rick L. Shepherd and Ray Chandler, Jul 10 2004

A096788 Numbers m such that both m and m+1 are composite numbers whose sum is a prime of the form 4k+3.

Original entry on oeis.org

9, 15, 21, 33, 35, 39, 51, 63, 65, 69, 75, 81, 95, 99, 105, 111, 119, 125, 135, 141, 153, 155, 165, 183, 189, 209, 215, 219, 221, 231, 243, 245, 249, 261, 273, 285, 299, 303, 309, 315, 321, 323, 329, 341, 345, 363, 369, 371, 375, 393, 405, 411, 413, 429, 441
Offset: 1

Views

Author

Lekraj Beedassy, Jul 09 2004

Keywords

Comments

Odd composite numbers c such that 2*c + 1 is prime. - Alexandre Herrera, Jul 07 2023

Crossrefs

Subsequence (odd numbers) of A096784. See A096787 for the associated primes.
Cf. A060254, A096784, A096785, A096786, A096787, A096676.

Programs

  • Mathematica
    Select[ Range[ 450], PrimeQ[ # ] == PrimeQ[ # + 1] == False && PrimeQ[2# + 1, GaussianIntegers -> True] == True &] (* Robert G. Wilson v, Jul 11 2004 *)
  • PARI
    nextcomposite(k)=if(k<3,4,if(isprime(k),k+1,k));
{m=455;n=4;while(nKlaus Brockhaus, Jul 10 2004

Formula

Equals (A096787 - 1)/2.

Extensions

Corrected and extended by Klaus Brockhaus and Ray Chandler, Jul 10 2000
Incorrect comment about Gaussian primes deleted by N. J. A. Sloane, Mar 02 2011

A079149 Primes p such that either p-1 or p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 2 or bigomega(p+1) <= 2, where bigomega(n) = A001222(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 37, 47, 59, 61, 73, 83, 107, 157, 167, 179, 193, 227, 263, 277, 313, 347, 359, 383, 397, 421, 457, 467, 479, 503, 541, 563, 587, 613, 661, 673, 719, 733, 757, 839, 863, 877, 887, 983, 997, 1019, 1093, 1153, 1187, 1201, 1213, 1237
Offset: 1

Views

Author

Cino Hilliard, Dec 27 2002

Keywords

Comments

There are only 2 primes such that both p-1 and p+1 have at most 2 prime factors - 3 and 5. Proof: If p > 5 then whichever of p-1 and p+1 is divisible by 4 has at least 3 prime factors.
Primes which are not the sum of two consecutive composite numbers. - Juri-Stepan Gerasimov, Nov 15 2009

Crossrefs

Union of A079147 and A079148. Cf. A060254, A079152.

Programs

  • Mathematica
    Select[Prime[Range[500]],MemberQ[PrimeOmega[{#-1,#+1}],2]&] (* Harvey P. Dale, Sep 04 2011 *)
  • PARI
    s(n) = {sr=0; ct=0; forprime(x=2,n, if(bigomega(x-1) < 3 || bigomega(x+1) < 3, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); } \\ Lists primes p<=n such that p+-1 has at most 2 prime factors.

A151740 Composites that are the sum of two consecutive composite numbers.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 46, 49, 51, 55, 58, 62, 65, 69, 74, 77, 82, 86, 91, 94, 99, 106, 111, 115, 118, 122, 125, 129, 134, 142, 146, 153, 155, 158, 161, 166, 169, 171, 175, 178, 183, 185, 187, 189, 194, 202, 206, 209, 214, 218, 221, 226, 231, 235, 237, 243, 245
Offset: 1

Views

Author

Claudio Meller, Jun 15 2009

Keywords

Comments

The even terms of this sequence are exactly twice the primes > 3. The odd terms are odd composites c for which the odd integer next to c/2 is not prime. - M. F. Hasler, Jun 16 2009
The English language can be ambiguous! What is meant here is: write down a list of the composite numbers 4,6,8,9,10,12,... Whenever the sum of two adjacent terms is composite, adjoin it to the sequence: 4+6=10, 6+8=14, 10+12=22, ... - N. J. A. Sloane, Nov 26 2019

Links

Crossrefs

Cf. A002808, A151741, A151742, A151743, A151744, A151745, A060254, A060327, A060339, A096785, A096787, A135170. - M. F. Hasler, Jun 16 2009
Cf. A167611 (Essentially the same, except for initial term).

Programs

  • Mathematica
    CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; q=6!;lst2={};Do[If[ !PrimeQ[n],c=CompositeNext[n];a2=n+c;If[ !PrimeQ[a2],AppendTo[lst2,a2]]],{n,q}];lst2 (* Vladimir Joseph Stephan Orlovsky, Jun 17 2009 *)
Module[{c=Select[Range[300],CompositeQ],s2},s2=Total/@Partition[c,2,1];Intersection[c,s2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 27 2019 *)
  • PARI
    isA151740(n)= bittest(n,0) || return(isprime(n/2) && n>6); !isprime(bitor(n\2,1)) && !isprime(n) && n>1 \\ M. F. Hasler, Jun 16 2009
  • Python
    from sympy import isprime, composite
print([totest for k in range(1,92) if not isprime(totest := composite(k) + composite(k+1))]) # Karl-Heinz Hofmann, Feb 06 2024

A151741 Composite which are the sum of three consecutive composite numbers.

Original entry on oeis.org

18, 27, 36, 45, 49, 54, 63, 75, 78, 81, 85, 90, 95, 99, 102, 105, 117, 121, 126, 135, 143, 147, 150, 153, 161, 165, 168, 171, 175, 180, 185, 189, 192, 195, 203, 207, 216, 221, 225, 228, 231, 235, 243, 247, 255, 258, 261, 265, 273, 276, 279, 282, 285, 289, 297
Offset: 1

Views

Author

Claudio Meller, Jun 15 2009

Keywords

Crossrefs

Cf. A151740-A151745, A060254, A060327, A060339, A096785, A096787, A135170.

Programs

  • Mathematica
    Module[{c=Select[Range[300],CompositeQ],s3},s3=Total/@Partition[c,3,1];Intersection[c,s3]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 27 2019 *)
  • PARI
    c1=4;c2=6;for(c3=8,299,isprime(c3) && next;isprime(c1+c2+c3) || print1(c1+c2+c3",");c1=c2;c2=c3) \\ M. F. Hasler, Jun 16 2009

A060328 Primes which are the sum of three consecutive composite numbers.

Original entry on oeis.org

23, 31, 41, 59, 67, 71, 109, 113, 131, 139, 157, 199, 211, 239, 251, 269, 293, 311, 337, 379, 383, 409, 419, 487, 491, 499, 503, 521, 571, 599, 631, 701, 751, 769, 773, 787, 829, 877, 881, 919, 941, 953, 991, 1009, 1013, 1039, 1049, 1061, 1103, 1117, 1151
Offset: 1

Views

Author

Robert G. Wilson v, Mar 30 2001

Keywords

Comments

"Consecutive" necessarily means consecutive in the list of composite numbers as opposed to consecutive in the integers, as the sum of any 3 consecutive integers is a multiple of 3. - Peter Munn, Aug 20 2023

Examples

			a(3) = 41 is equal to 12+14+15.

Crossrefs

Primes that are the sum of other numbers of consecutive composite numbers: A060254 (2), A060329 (4), A060330 (5), A060331 (6), A060332 (7), A060333 (8). See also A037174.
Cf. A034962.
Complement within A166039\{5, 11} of A151741.

Programs

  • Mathematica
    composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); b = {}; Do[ p = composite[ n ] + composite[ n + 1 ] + composite[ n + 2 ]; If[ PrimeQ[ p ], b = Append[ b, p ] ], {n, 1, 1000} ]; b
Showing 1-10 of 22 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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