be the sum of the first

*
primes (i.e., the amount analog the the primorial function). The first couple of terms space 2, 5, 10, 17, 28, 41, 58, 77, ... (OEIS A007504). Bach and Shallit (1996) display that


*

and administer a general an approach for estimating such sums.

You are watching: Sum of prime numbers from 1 to 100

The first couple of values the

*
such that
*
is prime are 1, 2, 4, 6, 12, 14, 60, 64, 96, 100, ... (OEIS A013916). The equivalent values of
*
space 2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, ... (OEIS A013918).

The first few values that

*
such that
*
space 1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, ... (OEIS A045345). The equivalent values the
*
*
room 2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, ... (OEIS A050248; Rivera).

In 1737, Euler proved that the harmonic seriesof primes, (i.e., sum of the reciprocals of the primes) diverges


*

(3)

(Nagell 1951, p.59; Hardy and Wright 1979, pp.17 and 22), although it does so really slowly.

A quickly converging collection for the Mertens constant


*
" />

(4)

is offered by


*
, " />

(5)

where

*
is the Euler-Mascheroni constant,
*
is the Riemann zeta function, and
*
is the Möbius function (Flajolet and also Vardi 1996, Schroeder 1997, Knuth 1998).

Dirichlet verified the also stronger an outcome that


*

(6)

(Davenport 1980, p.34). Regardless of the divergence of the amount of mutual primes,the alternating series


(7)

(OEIS A078437) converges (Robinson and Potter1971), but it is not recognized if the sum


(8)

does (Guy 1994, p.203; Erdős 1998; Finch 2003).

There are also classes the sums of reciprocal primes v sign established by congruences top top

*
, because that example


(9)

(OEIS A086239), where


(10)

(Glaisher 1891b; Finch 2003; Jameson 2003, p.177),


(11)

(OEIS A086240; Glaisher 1893, Finch 2003),and


(12)

(OEIS A086241), where


(13)

(Glaisher 1891c; Finch 2003; Jameson 2003, p.177).

Although

*
diverges, Brun (1919) confirmed that


*


*


(18)
(19)
(20)
(21)
*
*
*

*
*
*

*
*
*

(24)

(OEIS A093597 and A093598).

Consider the analogous sum where, in addition, the terms consisted of must have actually an odd variety of distinct prime factors, i.e.,

*
is odd and also
*
. The first couple of such numbers room 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, ... (OEIS A030059), which encompass the composite numbers 30, 42, 66, 70, 78, 102, ... (OEIS A093599). Then

*
*
*

*
*
*

*
*
*
^2-zeta(2p))/(2zeta(p)zeta(2p))," />

(Gourdon and Sebah). The first few terms room then

*
*
*

(28)
*
*
*

*
*
*

*
*
*

(31)

(OEIS A093595 and A093596).

The sum

*
*
*

*
*
*
" />

*
*
*

(OEIS A086242) is additionally finite (Glaisher 1891a;Cohen; Finch 2003), where


*

(35)

*
is the totient function, and also
*
is the Riemann zeta function.

Some curious sums satisfied by primes

*
include


(36)

*


*


(39)
(40)
*
2)lnpsum_(k=1)^infty1/(e^(p^kx)+1) " />


(Berndt 1994, p.114).

Let

*
be the variety of ways an integer
*
have the right to be composed as a amount of 2 or much more consecutive primes. For example,
*
, for this reason
*
and
*
, for this reason
*
. The succession of worths of
*
because that
*
, 2, ... Is given by 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, ... (OEIS A084143). The complying with table provides the first few
*
such the
*
=k" /> for small
*
.

*
OEISvalues of
*
such the
*
=k" />
1A0509365, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 36, ...
2A06737236, 41, 60, 72, 83, 90, 100, 112, 119, ...
3A067373240, 287, 311, 340, 371, 510, 660, 803, ...

Similarly, the complying with table offers the first couple of

*
such the
*
for little
*
.

*
OEISvalues that
*
such the
*
1A0841465, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 39, ...
2A08414736, 41, 60, 72, 83, 90, 100, 112, 119, 120, 138, ...
*

Now take into consideration instead the number

*
of ways in which a number
*
have the right to be represented as a sum of one or more consecutive primes (i.e., the exact same sequence as prior to but one larger for every prime number). Amazingly, it then turns out that


(42)

(Moser 1963; Le Lionnais 1983, p.30).


SEE ALSO: Bruns Constant, Harmonic series of Primes, Mertens Constant, Mertens 2nd Theorem, element Formulas, element Number, prime Products, element Zeta Function, Primorial, amount of prime Factors

Portions of this entry contributed by Jean-ClaudeBabois


REFERENCES:

Bach, E. And Shallit, J. §2.7 in Algorithmic Number Theory, Vol.1: effective Algorithms. Cambridge, MA: MIT Press, 1996.

Berndt, B.C. "Ramanujan"s concept of element Numbers." Ch.24 in Ramanujan"s Notebooks, component IV. New York: Springer-Verlag, 1994.

Brun, V. "La serie

*
, les dénominateurs sont nombres premiers jumeaux est convergente où finie." Bull. Sci. Math. 43, 124-128, 1919.

Cohen, H. "High Precision Computation that Hardy-Littlewood Constants." Preprint.http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi.

Davenport, H. MultiplicativeNumber Theory, 2nd ed. New York: Springer-Verlag, 1980.

Doster, D. "Problem 10346." Amer. Math. Monthly 100, 951,1993.

Erdős, P. "Some that My brand-new and Almost brand-new Problems and Results in Combinatorial Number Theory." In Number Theory: Diophantine, Computational and Algebraic Aspects. Proceedings of the worldwide Conference hosted in Eger, July 29-August 2, 1996 (Ed. K.Győry, A.Pethő and V.T.Sós). Berlin: de Gruyter, pp.169-180, 1998.

Finch, S.R. "Meissel-Mertens Constants." §2.2 in mathematical Constants. Cambridge, England: Cambridge university Press, pp.94-98, 2003.

Finch, S. "Two Asymptotic Series." December 10, 2003. Http://algo.inria.fr/bsolve/.

Flajolet, P. And also Vardi, I. "Zeta role Expansions of timeless Constants."Unpublished manuscript. 1996. Http://algo.inria.fr/flajolet/Publications/landau.ps.

Glaisher, J.W.L. "On the Sums of the Inverse powers of the PrimeNumbers." Quart. J. Pure Appl. Math. 25, 347-362, 1891a.

Glaisher, J.W.L. "On the series

*
." Quart. J. Pure Appl. Math. 25, 375-383, 1891b.

Glaisher, J.W.L. "On the series

*
." Quart. J. Pure Appl. Math. 25, 48-65, 1891c.

Glaisher, J.W.L. "On the collection

*
." Quart. J. Pure Appl. Math. 26, 33-47, 1893.

Gourdon, X. And Sebah, P. "Collection of collection for

*
." http://numbers.computation.free.fr/Constants/Pi/piSeries.html.

Guy, R.K. "A series and a Sequence involving Primes." §E7 in Unsolved difficulties in Number Theory, second ed. Brand-new York: Springer-Verlag, p.203, 1994.

Hardy, G.H. And Wright, E.M. "Prime Numbers" and also "The succession of Primes." §1.2 and 1.4 in An arrival to the concept of Numbers, 5th ed. Oxford, England: Clarendon Press, pp.1-4, 17, 22, and 251, 1979.

Jameson, G.J.O. The prime Number Theorem. Cambridge, England: Cambridge college Press, p.177, 2003.

Knuth, D.E. The art of computer Programming, Vol.2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.

Le Lionnais, F. Lesnombres remarquables. Paris: Hermann, pp.26, 30, and 46, 1983.

Moree, P. "Approximation that Singular collection and Automata." ManuscriptaMath. 101, 385-399, 2000.

Moser, L. "Notes top top Number concept III. ~ above the sum of consecutive Primes."Can. Math. Bull. 6, 159-161, 1963.

Nagell, T. Introductionto Number Theory. New York: Wiley, 1951.

Ramanujan, S. "Irregular Numbers." J. Indian Math. Soc. 5, 105-106, 1913. Ramanujan, S. Gathered Papers the Srinivasa Ramanujan (Ed. G.H.Hardy, P.V.S.Aiyar, and B.M.Wilson). Providence, RI: Amer. Math. Soc., pp.20-21, 2000.

Rivera, C. "Problems & Puzzles: Puzzle 031-The median Prime Number,

*
." http://www.primepuzzles.net/puzzles/puzz_031.htm.

Robinson, H.P. And also Potter, E. MathematicalConstants. Report UCRL-20418. Berkeley, CA: college of California, 1971.

Schroeder, M.R. Number concept in Science and also Communication, v Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, third ed. New York: Springer-Verlag, 1997.

Sloane, N.J.A. Sequences A007504/M1370, A013916, A013918, A030059, A045345, A046024, A050247, A050248, A050936, A065421, A067372, A067373, A078437, A078837, A078838, A084143, A084146, A084147, A086239, A086240, A086241, A086242, A093595, A093596, A093597, A093598, A093599, and A331764 in "The On-Line Encyclopedia of creature Sequences."



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Babois, Jean-Claude and Weisstein, Eric W. "Prime Sums." indigenous stclairdrake.net--A stclairdrake.net internet Resource. Https://stclairdrake.net/PrimeSums.html



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