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 1-6 of 6 results.

A092064 Prime numbers in A092063.

Original entry on oeis.org

2, 3, 7, 19, 31, 79, 89, 137, 149, 181, 6151
Offset: 1

Views

Author

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 20 2004

Keywords

Comments

6151 corresponds to a probable prime. - Jason Yuen, Aug 25 2024

Crossrefs

Cf. A120271.

Programs

  • PARI
    A120271(n) = numerator(sum(k=1, n, 1/(prime(k)-1)));
for (i=1,500,if(isprime(i) && isprime(A120271(i)),print1(i,",")));

Extensions

a(11) from Jason Yuen, Aug 25 2024

A137692 Primes of the form A128646(k)+1 for some k (listed in A137691), where A128646 = denominators of partial sums of 1/(p(i)-1).

Original entry on oeis.org

2, 3, 5, 11, 13, 61, 18481, 55441, 53413361, 11827018732969441
Offset: 1

Views

Author

M. F. Hasler, Feb 07 2008

Keywords

Comments

The next term is A128646(376)+1, which has 226 decimal digits.

Crossrefs

Cf. A128646, A137689, A137690, A137691, A092063.

Programs

  • PARI
    A137691v = [1,2,3,4,5,6,10,11,12,13,14,18,38,376,377,378,379,380,381,382,383,384,385] /*see there*/; A137692 = vecsort(vector(15,k,A128646(A137691v[k])+1),8) /* ...,8 removes duplicate entries in PARI > 2.4.1 */

A137689 Indices m such that A128646(m)-1 is prime, where A128646 = denominator of partial sums of 1/(p(i)-1).

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 23, 24, 26, 47, 48, 54, 78, 79, 80, 243, 244, 245, 246, 247, 367, 368, 369, 370, 371, 372, 373, 374, 375, 447, 453, 635, 636, 1656, 1657, 1658, 1659, 1660, 18618, 18619, 18620, 18621, 18622, 18623, 18624, 18625, 18626, 18627, 18628, 18629, 18630, 18631, 18632, 18633, 18634, 18635
Offset: 1

Views

Author

M. F. Hasler, Feb 07 2008

Keywords

Comments

Terms corresponding to indices m = a(k) > 1000 are not certified primes but at least probable primes. Is there a simple explanation for the large gaps between a(k)=80, a(k+1)=243 and a(k)=636, a(k+1)=1656?

Crossrefs

Cf. A128646, A137690, A137691, A137692, A092063.

Programs

  • PARI
    print_A137689(i=0/*start checking at i+1*/)={my(s=sum(j=1,i,1/(prime(j)-1))); while(1, while(!ispseudoprime(-1+denominator(s+=1/(prime(i++)-1))),);print1(i","))}

Extensions

Edited by T. D. Noe, Oct 30 2008
a(43)-a(60) from Jason Yuen, Sep 26 2024

A137690 Primes of the form A128646(k)-1 for some k (listed in A137689), where A128646 = denominators of partial sums of 1/(prime(i)-1).

Original entry on oeis.org

3, 11, 59, 79, 719, 7919, 55439, 425039, 5525519, 19709529839, 197095298399, 999294451257532807016639, 2823006824802530179822007999, 2649530397357338361250749788962714016407928543999
Offset: 1

Views

Author

M. F. Hasler, Feb 07 2008

Keywords

Comments

The next term is A128646(243)-1, which has 148 decimal digits. New terms should be added to A137689, not here.

Crossrefs

Cf. A128646, A137689-A137692, A092063.

Programs

  • PARI
    A137689_v=[3,4,5,7,8,9,10,11,15,16,23,24,26,47,48,54,78,79,80,243]/*see there*/;
vecsort(vector(#A137689_v,k,denominator(sum(i=1,A137689_v[k],1/(prime(i)-1)))/*i.e.
A128646(A137689_v[k])*/-1),0,8) /* ...,8 removes duplicate entries in PARI > 2.4.1 */

A092065 Numbers m such that numerator of Sum_{k=1..m} 1/(prime(k)-k) is prime.

Original entry on oeis.org

2, 3, 4, 5, 7, 14, 21, 22, 26, 27, 32, 43, 51, 58, 62, 65, 82, 131, 148, 207, 229, 249, 257, 320, 334, 386, 423, 440, 481, 747, 823, 1181, 1314, 1915, 2025, 2269, 2700, 2717, 2801, 2865, 4548, 6015, 6364, 8532, 10612, 10863, 11960, 15156, 15898, 19186, 19622, 22203, 25345
Offset: 1

Views

Author

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 20 2004; corrected Apr 24 2006

Keywords

Comments

Note that the definition here is subtly different from that of A092063.

Crossrefs

Cf. A092066.

Programs

  • Maple
    count:= 0:
S:= 0: p:= 0;
for n from 1 to 2500 do
  p:= nextprime(p);
  S:= S + 1/(p - n);
  if isprime(numer(S)) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Sep 07 2014
  • Mathematica
    f=0; Do[ p=Prime[n]; f=f+1/(p-n); g=Numerator[f]; If[ PrimeQ[g], Print[n]], {n,1,500} ]
  • PARI
    S=1;for(n=2,100,S=S+1/(prime(n)-n);if(isprime(numerator(S)),print1(n,","))) \\ Edward Jiang, Sep 08 2014

Extensions

Sequence and Mathematica program corrected by Alexander Adamchuk, Jul 29 2007
a(30)-a(34) from Vincenzo Librandi, Nov 26 2012
a(35)-a(36) from Robert Israel, Sep 07 2014
a(37)-a(53) from Michael S. Branicky, Aug 26 2024

A137691 Indices m such that A128646(m)+1 is prime, where A128646 = denominators of partial sums of 1/(prime(i)-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 18, 38, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 475, 476, 477, 478, 479, 488, 489, 490, 491, 492, 493, 858, 859, 860, 861, 862, 863, 864, 2670, 3261, 3262, 3263, 3264, 3265, 4819, 6034, 6035, 6036, 6037, 6038
Offset: 1

Views

Author

M. F. Hasler, Feb 07 2008

Keywords

Comments

Terms corresponding to indices m = a(k) > 1000 are not certified primes but at least probable primes. Is there a simple explanation for the large gaps between a(k)=38 and a(k+1)=376; a(k)=864 and a(k+1)=2670, etc.?

Examples

			n=3 is in this sequence because A128646(n)+1 = 5 is a prime (where A128646(3) is the denominator of 1/(2-1) + 1/(3-1) + 1/(5-1) = 7/4).

Crossrefs

Cf. A128646, A137689, A137690, A137692, A092063.

Programs

  • PARI
    print_A137691(i=0/*start checking at i+1*/)={my(s=sum(j=1,i,1/(prime(j)-1))); while(1, while(!ispseudoprime(1+denominator(s+=1/(prime(i++)-1))),);print1(i","))}

Extensions

Edited by T. D. Noe, Oct 30 2008
Showing 1-6 of 6 results.

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