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 1-5 of 5 results.

A082915 Numbers n for which the first difference sequence of A054040 decreases.

Original entry on oeis.org

27, 53, 179, 305, 431, 557, 683, 809, 835, 935, 961, 1087, 1213, 1339, 1465, 1591, 1717, 1743, 1843, 1869, 1995, 2121, 2247, 2373, 2499, 2525, 2625, 2651, 2777, 2903, 3029, 3155, 3281, 3407, 3433, 3533, 3559, 3685, 3811, 3937, 4063, 4189, 4215, 4315
Offset: 1

Views

Author

Robert G. Wilson v, Apr 14 2003

Keywords

Comments

Same as asking when the second difference of A054040 is -1.

Examples

			A054040(27)-A054040(26) = 13 while A054040(26)-A054040(25) = 14, therefore the first entry is 27.

Crossrefs

Cf. A054040.

Programs

  • Mathematica
    k = s = a = b = c = 0; Do[m = n; While[s < n, k++; s = N[s + 1/Sqrt[k], 50]]; a = b; b = c; c = k; If[2b - a > c, Print[n]], {n, 1, 4400}]

A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 202, 217, 232, 247, 263, 280, 297, 314, 332, 351, 370, 389, 409, 430, 451, 472, 494, 517, 540, 563, 587, 612, 637, 662, 688, 715, 741, 769, 797, 825
Offset: 0

Views

Author

Robert G. Wilson v

Keywords

Examples

			Let b(k) = 1 + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(k):
.k.......1....2.....3.....4.....5.....6.....7
-------------------------------------------------
b(k)...1.00..1.71..2.28..2.78..3.23..3.64..4.01
For A019529 we have:
  n=0: smallest k is a(0) = 1 since 1.00 > 0
  n=1: smallest k is a(1) = 2 since 1.71 > 1
  n=2: smallest k is a(2) = 3 since 2.28 > 2
  n=3: smallest k is a(3) = 5 since 3.23 > 3
  n=4: smallest k is a(4) = 7 since 4.01 > 4
For A054040 we have:
  n=1: smallest k is a(1) = 1 since 1.00 >= 1
  n=2: smallest k is a(2) = 3 since 2.28 >= 2
  n=3: smallest k is a(3) = 5 since 3.23 >= 3
  n=4: smallest k is a(4) = 7 since 4.01 >= 4

Crossrefs

A054040 is another version. See also A002387, A004080.

Programs

Extensions

Edited by N. J. A. Sloane, Sep 01 2009

A292774 a(n) = smallest m such that Sum_{i=1..m} 1/sqrt(prime(i)) >= n.

Original entry on oeis.org

2, 4, 8, 13, 21, 30, 43, 58, 76, 97, 121, 149, 180, 214, 252, 294, 340, 390, 444, 502, 564, 630, 700, 775, 854, 937, 1025, 1118, 1215, 1317, 1423, 1535, 1650, 1771, 1897, 2027, 2162, 2303, 2448, 2598, 2753, 2914, 3079, 3250, 3426, 3607, 3793, 3984, 4181, 4383, 4591, 4803, 5022, 5245, 5474, 5709
Offset: 1

Views

Author

N. J. A. Sloane, Sep 30 2017

Keywords

Links

Crossrefs

Cf. A292775; A019529, A054040.

Programs

  • Maple
    Digits:=50;
s0:=0; k:=1; lisi:=[]; lisP:=[];
for i from 1 to 10000 do p:=ithprime(i);
s0:=s0+evalf(1/sqrt(p));
if s0 >= k then k:=k+1; lisi:=[op(lisi),i]; lisP:=[op(lisP),p]; fi;
od:
lisi; # A292774
lisP; # A292775
  • Mathematica
    f[n_]:=Block[{k=0, s=0}, While[sVincenzo Librandi, Oct 01 2017 *)

Formula

a(n) ~ (n^2*log(n))/2. - Benoit Cloitre, Oct 01 2017 [This follows from the asymptotics for A292775]

A292775 a(n) = smallest prime q such that Sum_{primes p <= q} 1/sqrt(p) >= n.

Original entry on oeis.org

3, 7, 19, 41, 73, 113, 191, 271, 383, 509, 661, 859, 1069, 1307, 1601, 1931, 2287, 2687, 3119, 3583, 4093, 4657, 5279, 5881, 6607, 7351, 8167, 9001, 9851, 10837, 11867, 12899, 13967, 15161, 16361, 17627, 19031, 20389, 21821, 23297, 24917, 26557, 28279, 30059, 31891, 33647, 35617, 37607, 39779
Offset: 1

Views

Author

N. J. A. Sloane, Sep 30 2017

Keywords

Links

Crossrefs

Cf. A292774; A019529, A054040.

Programs

  • Maple
    Digits:=50;
s0:=0; k:=1; lisi:=[]; lisP:=[];
for i from 1 to 10000 do p:=ithprime(i);
s0:=s0+evalf(1/sqrt(p));
if s0 >= k then k:=k+1; lisi:=[op(lisi),i]; lisP:=[op(lisP),p]; fi;
od:
lisi; # A292774
lisP; # A292775
  • Mathematica
    f[n_]:=Block[{k=0, s=0}, While[sVincenzo Librandi, Oct 01 2017 *)

Formula

a(n) ~ prime(n)^2. - Benoit Cloitre, Oct 01 2017 [See link]

A231405 Least integer j such that Sum_{i=1..j} 1/i^(1/3) >= n.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 10, 12, 15, 17, 20, 23, 25, 28, 32, 35, 38, 41, 45, 49, 52, 56, 60, 64, 68, 72, 76, 81, 85, 89, 94, 98, 103, 108, 113, 117, 122, 127, 132, 138, 143, 148, 153, 159, 164, 170, 175, 181, 187, 192, 198, 204, 210, 216, 222, 228, 234, 240, 247, 253
Offset: 0

Views

Author

Carmine Suriano, Nov 08 2013

Keywords

Examples

			a(7)=12 since Sum_{i=1..12} 1/i^(1/3) = 7.106248... and Sum_{i=1..11} 1/i^(1/3) = 6.669458... .

Links

Crossrefs

Cf. A004080, A054040, A054041 (condition >n).
Cf. A067086.

Programs

  • JavaScript
    s=0;n=1;
for (i=1;i<30;i++) {
s+=1/Math.pow(i,1/3);
if (s>=n) {n++;document.write(Math.floor(i)+", ");}
}
  • Mathematica
    s = 0; i = 0; Table[i++; While[s = s + 1/(i^(1/3)); s < n, i++]; i, {n, 100}] (* T. D. Noe, Nov 09 2013 *)
Module[{nn=300,c},c=Accumulate[1/Surd[Range[nn],3]];Table[Position[ c,?(#>=n&),1,1],{n,0,60}]]//Flatten (* _Harvey P. Dale, Aug 14 2021 *)

Extensions

a(0) added by Jon Perry, Nov 10 2013
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.