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 1-8 of 8 results.

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 */

A128649 Numbers m such that A128646(m) = A128648(m).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 15, 16, 17, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 539, 540, 541, 542, 543, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

Terms of this sequence are 1..5, 7..11, 14..17, 21..35, 65..66, 71..77, 81..93, 539..543, 600..639, 644..650, 707..818, 1152..1185, 4502..4577, 4601..4823, 4893..5003, 7483..7633, ...

Links

  • Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A128646 (denominator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128647, A120271, A119686, A006093, A000040.

Programs

  • Mathematica
    f=0;g=0;Do[p=Prime[n];f=f+1/(p-1);g=g+(-1)^(n+1)*1/(p-1);kf=Denominator[f];kg=Denominator[g];If[Equal[kf,kg],Print[n]],{n,1,10000}]

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 */

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

A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

Original entry on oeis.org

1, 3, 7, 23, 121, 21, 173, 1597, 17927, 127469, 129317, 43619, 44081, 44521, 1033223, 13538159, 395369371, 132680013, 400467919, 402757063, 1214947859, 1221110939, 50305908619, 50529880549, 101470376303, 509322834499, 8691337402883
Offset: 1

Views

Author

Alexander Adamchuk, Jul 01 2006

Keywords

Comments

a(n) is squarefree except for n = 5, 14, 49, ... where squared prime factors are 11, 211, 479, ...
a(n)/A128646(n) is the asymptotic mean over the positive integers of the number of prime divisors that are not greater than prime(n), counted with multiplicity (cf. A007814, A169611, A356006). - Amiram Eldar, Jul 23 2022

Links

Crossrefs

Cf. A128646 (denominators), A119686, A006093, A000040.
Cf. A135212, A078456.
Cf. A007814, A169611, A356006.

Programs

  • Maple
    R:= [seq(1/(ithprime(k)-1),k=1..40)]:
S:= ListTools:-PartialSums(R):
A:= map(numer,S); # Robert Israel, Jan 12 2025
  • Mathematica
    Numerator[Table[Sum[1/(Prime[i]-1),{i,1,n}],{n,1,50}]]
Accumulate[1/(Prime[Range[30]]-1)]//Numerator (* Harvey P. Dale, May 03 2025 *)
  • PARI
    a(n) = numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

Formula

a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).
a(n) = A078456(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007

A128647 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

Original entry on oeis.org

1, 1, 3, 7, 41, 3, 53, 437, 5167, 34189, 36037, 3833, 3987, 11521, 274223, 3458639, 103063291, 100392623, 34273501, 33510453, 308270747, 302107667, 12626774467, 12402802537, 25216220279, 124110148411, 2142721739387, 111888942151111
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

Numbers m such that A128648(m) = A128646(n) are listed in A128649.

Links

  • Eric Weisstein's World of Mathematics, Prime sums

Crossrefs

Cf. A128648 (denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A120271 (numerator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128646, A128649, A119686, A006093, A000040.

Programs

  • Mathematica
    Table[Numerator[Sum[(-1)^(k+1)*1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

A128648 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).

Original entry on oeis.org

1, 2, 4, 12, 60, 5, 80, 720, 7920, 55440, 55440, 6160, 6160, 18480, 425040, 5525520, 160240080, 160240080, 53413360, 53413360, 480720240, 480720240, 19709529840, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400
Offset: 1

Views

Author

Alexander Adamchuk, Mar 18 2007

Keywords

Comments

Numbers m such that a(m) equals A128646(m) are listed in A128649.

Links

  • Eric Weisstein's World of Mathematics, Prime Sums

Crossrefs

Cf. A128647 (numerator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1))).
Cf. A128646 (denominator(Sum_{k=1..n} 1/(prime(k)-1))).
Cf. A128649, A120271, A119686, A006093, A000040.

Programs

  • Mathematica
    Table[Denominator[Sum[(-1)^(k+1)*1/(Prime[k]-1),{k,1,n}]],{n,1,36}]

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/(prime(k)-1)).
Showing 1-8 of 8 results.

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