A019529 -id:A019529 - cp's OEIS frontend

A054040 a(n) terms of series {1/sqrt(j)} are >= n.

1, 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: 1

Asher Auel, Apr 13 2000

nonn

In many cases the first differences have the form {2k, 2k, 2k, 2k+1} (A004524). In such cases the second differences are {0, 0, 1, 1}. See A082915 for the exceptions. In as many as these, the first differences have the form {2k-1, 2k-1, 2k-1, 2k}. - Robert G. Wilson v, Apr 18 2003 [Corrected by Carmine Suriano, Nov 08 2013]
a(100)=2574, a(1000)=250731 & a(10000)=25007302 which differs from Sum{i=4..104}A004524(i)=2625, Sum{i=4..1004}A004524(i)=251250 & Sum{i=4..10004}A004524(i)=25012500. - Robert G. Wilson v, Apr 18 2003
A054040(n) <= A011848(n+2), A054040(10000)=25007302 and A011848(n+2)=25007500. - Robert G. Wilson v, Apr 18 2003

			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 this sequence 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

Sela Fried, On the partial sums of the Zeta function Sum_{n>=1} 1/n^s for 0 < s < 1, 2024.
Hugo Pfoertner, Plot of a(n)-(1/4)*(n^2-2*zeta(1/2)*n), n=1..1000.

Cf. A002387, A004080, A059750, A082915, A054044, A011848, A082915.

See A019529 for a different version.

f[n_] := Block[{k = 0, s = 0}, While[s < n, k++; s = N[s + 1/Sqrt[k], 50]]; k]; Table[f[n], {n, 1, 60}]

a(n)=if(n<0,0,t=1;z=1;while(zBenoit Cloitre, Sep 23 2012

Let f(n) = (1/4)*(n^2-2*zeta(1/2)*n) then we have a(n) = f(n) + O(1). More precisely we claim that for n >= 2 we have a(n) = floor(f(n)+c) where c > Max{a(n)-f(n) : n>=1} = a(153) - f(153) = 1.032880076066608813953... and we believe we can take c = 1.033. - Benoit Cloitre, Sep 23 2012

Definition and offset modified by N. J. A. Sloane, Sep 01 2009

A054041 Sum of a(n) terms of 1/k^(1/3) first exceeds n.

Original entry on oeis.org

1, 2, 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, 259

Offset: 0

Asher Auel, Apr 13 2000

nonn

Cf. A002387, A054043, A019529.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(1/3), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]

Corrected and extended by Robert G. Wilson v, Aug 01 2000

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

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

N. J. A. Sloane, Sep 30 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Cf. A292775; A019529, A054040.

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

f[n_]:=Block[{k=0, s=0}, While[sVincenzo Librandi, Oct 01 2017 *)

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

N. J. A. Sloane, Sep 30 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Benoit Cloitre, Asymptotics for A292775

Cf. A292774; A019529, A054040.

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

f[n_]:=Block[{k=0, s=0}, While[sVincenzo Librandi, Oct 01 2017 *)

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

A056176 Sum of a(n) terms of 1/k^(1/4) first exceeds n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 12, 14, 16, 18, 20, 23, 25, 27, 29, 32, 34, 37, 39, 42, 44, 47, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 77, 80, 83, 86, 90, 93, 96, 99, 102, 105, 108, 112, 115, 118, 122, 125, 128, 132, 135, 138, 142, 145, 149, 152, 156, 159, 163, 167, 170

Offset: 0

Robert G. Wilson v, Aug 01 2000

nonn

Cf. A019529.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(1/4), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]

A056177 Sum of a(n) terms of 1/k^(1/5) first exceeds n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 36, 38, 40, 42, 44, 46, 48, 50, 53, 55, 57, 59, 62, 64, 66, 68, 71, 73, 75, 78, 80, 83, 85, 88, 90, 92, 95, 97, 100, 102, 105, 108, 110, 113, 115, 118, 120, 123, 126, 128, 131, 134, 136, 139

Offset: 0

Robert G. Wilson v, Aug 01 2000

nonn

Cf. A019529.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(1/5), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]

A056178 Sum of a(n) terms of 1/k^(2/3) first exceeds n.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 22, 31, 42, 56, 71, 90, 112, 137, 165, 197, 233, 272, 317, 365, 419, 477, 541, 610, 685, 766, 853, 946, 1045, 1152, 1265, 1386, 1514, 1650, 1793, 1945, 2105, 2274, 2451, 2637, 2833, 3038, 3252, 3477, 3711, 3956, 4212, 4478, 4755, 5043

Offset: 0

Robert G. Wilson v, Aug 01 2000

nonn

Cf. A019529 and A002387.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(2/3), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]
Flatten[Table[Position[Accumulate[Table[1/k^(2/3),{k,5100}]],?(#>n&),{1}, 1],{n, 0,50}]] (* _Harvey P. Dale, Mar 16 2015 *)

A056179 Sum of a(n) terms of 1/k^(3/4) first exceeds n.

Original entry on oeis.org

1, 2, 3, 7, 12, 20, 31, 46, 67, 94, 128, 170, 222, 285, 361, 452, 558, 682, 826, 991, 1179, 1394, 1637, 1909, 2215, 2556, 2935, 3354, 3817, 4327, 4885, 5496, 6163, 6889, 7676, 8530, 9453, 10449, 11521, 12674, 13911, 15237, 16656, 18171, 19787, 21509

Offset: 0

Robert G. Wilson v, Aug 01 2000

nonn

Cf. A019529 and A002387.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(3/4), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 60} ]

A056180 Sum of a(n) terms of 1/k^(4/5) first exceeds n.

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 40, 63, 95, 140, 201, 281, 384, 516, 682, 888, 1141, 1449, 1820, 2263, 2789, 3408, 4133, 4976, 5951, 7074, 8360, 9826, 11492, 13376, 15499, 17884, 20554, 23533, 26849, 30528, 34600, 39095, 44045, 49485, 55450, 61976, 69103

Offset: 0

Robert G. Wilson v, Aug 01 2000

nonn

Cf. A019529 and A002387.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(4/5), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 50} ]
Join[{1},Table[Position[Accumulate[Table[N[1/k^(4/5)],{k,100000}]],?(#>n&),{1},1],{n,50}]//Flatten] (* _Harvey P. Dale, Sep 23 2016 *)

A056181 Sum of a(n) terms of 1/k^(5/6) first exceeds n.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 48, 79, 126, 194, 290, 422, 602, 841, 1155, 1561, 2079, 2733, 3550, 4562, 5803, 7314, 9140, 11330, 13941, 17035, 20681, 24956, 29944, 35735, 42432, 50143, 58988, 69097, 80610, 93679

Offset: 0

Robert G. Wilson v, Aug 01 2000

nonn

Cf. A019529 and A002387.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/k^(5/6), 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 35} ]

cp's OEIS Frontend

A054040 a(n) terms of series {1/sqrt(j)} are >= n.

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A054041 Sum of a(n) terms of 1/k^(1/3) first exceeds n.

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

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

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A056176 Sum of a(n) terms of 1/k^(1/4) first exceeds n.

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A056177 Sum of a(n) terms of 1/k^(1/5) first exceeds n.

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A056178 Sum of a(n) terms of 1/k^(2/3) first exceeds n.

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A056179 Sum of a(n) terms of 1/k^(3/4) first exceeds n.

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A056180 Sum of a(n) terms of 1/k^(4/5) first exceeds n.

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A056181 Sum of a(n) terms of 1/k^(5/6) first exceeds n.

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