A014284 -id:A014284 - cp's OEIS frontend

A123213 a(n) = A062457(n) concatenated with A014284(n).

21, 93, 1256, 240111, 16105118, 482680929, 41033867342, 1698356304159, 180115266146378, 420707233300201101, 25408476896404831130, 6582952005840035281161, 925103102315013629321198, 73885357344138503765449239, 12063348350820368238715343282

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 05 2006

			a(1) = 21 = 2 concatenated with 1.
a(2) = 93 = 9 concatenated with 3.
a(3) = 1256 = 125 concatenated with 6.
a(4) = 240111 = 2401 concatenated with 11.

Sergio Silva, Teste Numerico.

Cf. A014284, A062457.

rng=13;A062457=Table[Prime[n]^n, {n, rng}] ;A014284=Accumulate[Join[{1}, Prime[Range[rng-1]]]];Table[FromDigits[Join[IntegerDigits[A062457[[n]]],IntegerDigits[A014284[[n]]]]],{n,rng}] (* James C. McMahon, Nov 17 2024 *)

s=1;print1(21,",");for(n=2,20,s=s+prime(n-1);print1(prime(n)^n,s,","))

A361150 a(n) = A014284(n^2) + A014284(n^2-1).

Original entry on oeis.org

1, 17, 137, 611, 1839, 4405, 9101, 16859, 28987, 46663, 71797, 105863, 151259, 209895, 284777, 378661, 493863, 634985, 804801, 1007439, 1245345, 1526369, 1851971, 2227153, 2658287, 3151447, 3711837, 4343483, 5053859, 5849959, 6739255, 7727399, 8825137, 10034745

Offset: 1

N. J. A. Sloane, Mar 02 2023

nonn

Cf. A008578, A014284.

nn = 34; s = Accumulate[{1}~Join~Prime@ Range[nn^2]]; {1}~Join~Array[Total@ s[[# - 1 ;; #]] &[#^2] &, nn - 1, 2] (* Michael De Vlieger, Aug 10 2023 *)

f(n) = if (n, 1 + vecsum(primes(n-1)), 0); \\ A014284
a(n) = f(n^2) + f(n^2-1); \\ Michel Marcus, Aug 10 2023

a(n) = 2*A014284(n^2-1) + A008578(n^2). - Michel Marcus, Aug 10 2023

A007504 Sum of the first n primes.

Original entry on oeis.org

0, 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160, 197, 238, 281, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, 3087, 3266, 3447, 3638, 3831, 4028, 4227, 4438, 4661, 4888

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

It appears that a(n)^2 - a(n-1)^2 = A034960(n). - Gary Detlefs, Dec 20 2011
This is true. Proof: By definition we have A034960(n) = Sum_{k = (a(n-1)+1)..a(n)} (2*k-1). Since Sum_{k = 1..n} (2*k-1) = n^2, it follows A034960(n) = a(n)^2 - a(n-1)^2, for n > 1. - Hieronymus Fischer, Sep 27 2012 [formulas above adjusted to changed offset of A034960 - Hieronymus Fischer, Oct 14 2012]
Row sums of the triangle in A037126. - Reinhard Zumkeller, Oct 01 2012
Ramanujan noticed the apparent identity between the prime parts partition numbers A000607 and the expansion of Sum_{k >= 0} x^a(k)/((1-x)...(1-x^k)), cf. A046676. See A192541 for the difference between the two. - M. F. Hasler, Mar 05 2014
For n > 0: row 1 in A254858. - Reinhard Zumkeller, Feb 08 2015
a(n) is the smallest number that can be partitioned into n distinct primes. - Alonso del Arte, May 30 2017
For a(n) < m < a(n+1), n > 0, at least one m is a perfect square.
Proof: For n = 1, 2, ..., 6, the proposition is clear. For n > 6, a(n) < ((prime(n) - 1)/2)^2, set (k - 1)^2 <= a(n) < k^2 < ((prime(n) + 1)/2)^2, then k^2 < (k - 1)^2 + prime(n) <= a(n) + prime(n) = a(n+1), so m = k^2 is this perfect square. - Jinyuan Wang, Oct 04 2018
For n >= 5 we have a(n) < ((prime(n)+1)/2)^2. This can be shown by noting that ((prime(n)+1)/2)^2 - ((prime(n-1)+1)/2)^2 - prime(n) = (prime(n)+prime(n-1))*(prime(n)-prime(n-1)-2)/4 >= 0. - Jianing Song, Nov 13 2022
Washington gives an oscillation formula for |a(n) - pi(n^2)|, see links. - Charles R Greathouse IV, Dec 07 2022

E. Bach and J. Shallit, §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms, MIT Press, Cambridge, MA, 1996.
H. L. Nelson, "Prime Sums", J. Rec. Math., 14 (1981), 205-206.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 0..100000
C. Axler, On a Sequence involving Prime Numbers, J. Int. Seq. 18 (2015) # 15.7.6.
Christian Axler, New bounds for the sum of the first n prime numbers, arXiv:1606.06874 [math.NT], 2016.
P. Hecht, Post-Quantum Cryptography: S_381 Cyclic Subgroup of High Order, International Journal of Advanced Engineering Research and Science (IJAERS, 2017) Vol. 4, Issue 6, 78-86.
R. J. Mathar, Table of 100000n, a(100000n) for n = 1..10000
Romeo Meštrović, Curious conjectures on the distribution of primes among the sums of the first 2n primes, arXiv:1804.04198 [math.NT], 2018.
Vladimir Shevelev, Asymptotics of sum of the first n primes with a remainder term
Nilotpal Kanti Sinha, On the asymptotic expansion of the sum of the first n primes, arXiv:1011.1667 [math.NT], 2010-2015.
Lawrence C. Washington, Sums of Powers of Primes II, arXiv preprint (2022). arXiv:2209.12845 [math.NT]
Eric Weisstein's World of Mathematics, Prime Sums
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A000041, A034386, A111287, A013916, A013918 (primes), A045345, A050247, A050248, A068873, A073619, A034387, A014148, A014150, A178138, A254784, A254858.
See A122989 for the value of Sum_{n >= 1} 1/a(n).
Cf. A008472, A002110, A038107.

P:=Filtered([1..250],IsPrime);;
a:=Concatenation([0],List([1..Length(P)],i->Sum([1..i],k->P[k]))); # Muniru A Asiru, Oct 07 2018

a007504 n = a007504_list !! n
a007504_list = scanl (+) 0 a000040_list
-- Reinhard Zumkeller, Oct 01 2014, Oct 03 2011

[0] cat [&+[ NthPrime(k): k in [1..n]]: n in [1..50]]; // Bruno Berselli, Apr 11 2011 (adapted by Vincenzo Librandi, Nov 27 2015 after Hasler's change on Mar 05 2014)

s1:=[2]; for n from 2 to 1000 do s1:=[op(s1),s1[n-1]+ithprime(n)]; od: s1;
A007504 := proc(n)
    add(ithprime(i), i=1..n) ;
end proc: # R. J. Mathar, Sep 20 2015

Accumulate[Prime[Range[100]]] (* Zak Seidov, Apr 10 2011 *)
primeRunSum = 0; Table[primeRunSum = primeRunSum + Prime[k], {k, 100}] (* Zak Seidov, Apr 16 2011 *)

A007504(n) = sum(k=1,n,prime(k)) \\ Michael B. Porter, Feb 26 2010

a(n) = vecsum(primes(n)); \\ Michel Marcus, Feb 06 2021

from itertools import accumulate, count, islice
from sympy import prime
def A007504_gen(): return accumulate(prime(n) if n > 0 else 0 for n in count(0))
A007504_list = list(islice(A007504_gen(),20)) # Chai Wah Wu, Feb 23 2022

a(n) ~ n^2 * log(n) / 2. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 24 2001 (see Bach & Shallit 1996)
a(n) = A014284(n+1) - 1. - Jaroslav Krizek, Aug 19 2009
a(n+1) - a(n) = A000040(n+1). - Jaroslav Krizek, Aug 19 2009
a(A051838(n)) = A002110(A051838(n)) / A116536(n). - Reinhard Zumkeller, Oct 03 2011
a(n) = min(A068873(n), A073619(n)) for n > 1. - Jonathan Sondow, Jul 10 2012
a(n) = A033286(n) - A152535(n). - Omar E. Pol, Aug 09 2012
For n >= 3, a(n) >= (n-1)^2 * (log(n-1) - 1/2)/2 and a(n) <= n*(n+1)*(log(n) + log(log(n))+ 1)/2. Thus a(n) = n^2 * log(n) / 2 + O(n^2*log(log(n))). It is more precise than in Fares's comment. - Vladimir Shevelev, Aug 01 2013
a(n) = (n^2/2)*(log n + log log n - 3/2 + (log log n - 3)/log n + (2 (log log n)^2 - 14 log log n + 27)/(4 log^2 n) + O((log log n/log n)^3)) [Sinha]. - Charles R Greathouse IV, Jun 11 2015
G.f: (x*b(x))/(1-x), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 10 2016
a(n) = A008472(A002110(n)), for n > 0. - Michel Marcus, Jul 16 2020

More terms from Stefan Steinerberger, Apr 11 2006

a(0) = 0 prepended by M. F. Hasler, Mar 05 2014

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88

Offset: 1

N. J. A. Sloane

d(a(n)) != 2 (cf. A000005). - Juri-Stepan Gerasimov, Oct 17 2009
Number of prime divisors of a(n) (counted with multiplicity) != 1. - Juri-Stepan Gerasimov, Oct 30 2009
Largest nonprime < n-th composite. - Juri-Stepan Gerasimov, Oct 29 2009
The nonnegative nonprimes A141468 without zero; the natural nonprimes; the whole nonprimes; the counting nonprimes. If the nonprime numbers A141468 which are also the nonnegative integers A001477, then the nonprimes A141468 also called the nonnegative nonprimes. If the nonprime numbers A018252 which are also the natural (or whole or counting) numbers A000027, then the nonprimes A018252 also called the natural nonprimes, the whole nonprimes and the counting nonprimes. - Juri-Stepan Gerasimov, Nov 22 2009
Smallest nonprime > n-th nonnegative nonprime. - Juri-Stepan Gerasimov, Dec 04 2009
a(n) = A175944(A014284(n)) = A175944(A175965(n)). - Reinhard Zumkeller, Mar 18 2011

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

N. J. A. Sloane, List of nonprimes up to 20000: Table of n, a(n) for n = 1..17738
Eric Weisstein's World of Mathematics, Monica Set
Eric Weisstein's World of Mathematics, Suzanne Set
Index entries for "core" sequences

Cf. A000040 (complement), A002808.
Cf. A000005, A001222, A141468.
Boustrophedon transforms: A230955, A230954.
Cf. A010051, A239968.

A018252 := Difference([1..10^5], Filtered([1..10^5], IsPrime)); # Muniru A Asiru, Oct 21 2017

a018252 n = a018252_list !! (n-1)
a018252_list = filter ((== 0) . a010051) [1..]
-- Reinhard Zumkeller, Mar 31 2014

Magma
```
[n : n in [1..100] | not IsPrime(n) ];
```

with(numtheory); sort(convert(convert([ seq(i,i=1..541) ],set) minus convert([ seq(ithprime(i),i=1..100) ],set),list));
seq(`if`(not isprime(n),n,NULL),n=1..88); # Peter Luschny, Jul 29 2009
A018252 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do; end if; end proc: # R. J. Mathar, Oct 22 2010

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@ n]; Array[ nonPrime, 75] (* Robert G. Wilson v, Jan 29 2015, based on the algorithm by Labos Elemer in A006508 *)
max = 90; Complement[Range[max], Prime[Range[PrimePi[max]]]] (* Harvey P. Dale, Aug 12 2011 *)
Join[{1}, Select[Range[100], CompositeQ]] (* Jean-François Alcover, Nov 07 2021 *)

isA018252(n) = !isprime(n)
A018252(n) = {local(a,b);b=n;a=1;while(a!=b,a=b;b=n+primepi(a));b} \\ Michael B. Porter, Nov 06 2009

a(n) = my(k=0); while(-n+n-=k-k=primepi(n), ); n; \\ Ruud H.G. van Tol, Jul 15 2024 (after code in A002808)

from sympy import isprime
def ok(n): return not isprime(n)
print([k for k in range(1, 89) if ok(k)]) # Michael S. Branicky, Nov 10 2022

from sympy import composite
def A018252(n): return 1 if n == 1 else composite(n-1) # Chai Wah Wu, Nov 15 2022

def A018252_list(n) :
    return [k for k in (1..n) if not k.is_prime()]
A018252_list(88)  # Peter Luschny, Feb 03 2012

Let b(0) = n + pi(n) and b(n+1) = n + pi(b(n)), with pi(n) = A000720(n); then a(n) is the limit value of b(n). - Floor van Lamoen, Oct 08 2001
a(n) = A137621(A137624(n)). - Reinhard Zumkeller, Jan 30 2008
A010051(a(n)) = 0. - Reinhard Zumkeller, Mar 31 2014
A239968(a(n)) = n. - Reinhard Zumkeller, Dec 02 2014

A051349 Sum of first n nonprimes.

Original entry on oeis.org

0, 1, 5, 11, 19, 28, 38, 50, 64, 79, 95, 113, 133, 154, 176, 200, 225, 251, 278, 306, 336, 368, 401, 435, 470, 506, 544, 583, 623, 665, 709, 754, 800, 848, 897, 947, 998, 1050, 1104, 1159, 1215, 1272, 1330, 1390, 1452, 1515, 1579, 1644, 1710, 1778, 1847, 1917

Offset: 0

Armand Turpel (armandt(AT)unforgettable.com)

Partial sums of A141468 or A018252. - R. J. Mathar, Mar 01 2009

The lexicographically earliest sequence with first differences as increasing sequence of composites A002808. Complement of A175970. See A175965, A175966, A175967, A014284, A175969, A175970. - Jaroslav Krizek, Oct 31 2010

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A014284, A018252, A122998, A141468, A175965, A175966, A175967, A175969, A175970.

ithnonprime := proc(n)local k: option remember: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(not isprime(k))then return k fi: od: end: A051349 := proc(n) option remember: local k: if(n<=1)then return n: fi: return ithnonprime(n)+procname(n-1): end: seq(A051349(n),n=0..51); # Nathaniel Johnston, May 25 2011

lst={};s=0;Do[If[ !PrimeQ[n], s=s+n;AppendTo[lst, s]], {n, 0, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)

Sum_{n>=1} 1/a(n) = A122998. - Amiram Eldar, Nov 17 2020

A175965 Lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 43, 60, 79, 102, 131, 162, 199, 240, 283, 330, 383, 442, 503, 570, 641, 714, 793, 876, 965, 1062, 1163, 1266, 1373, 1482, 1595, 1722, 1853, 1990, 2129, 2278, 2429, 2586, 2749, 2916, 3089, 3268, 3449, 3640, 3833, 4030, 4229, 4440, 4663

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Complement of A175966.

A175944(a(n)) = A018252(n). - Reinhard Zumkeller, Mar 18 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008578, A014284, A018252, A036439, A051349.

Cf. A175944, A175966, A175967, A175968, A175969, A175970.

a175965 n = a175965_list !! n
a175965_list = scanl (+) 1 a008578_list
-- Reinhard Zumkeller, Mar 26 2015

FoldList[Plus,1,Join[{1},Prime[Range[50]]]] (* or *) Accumulate[ Join[ {1,1},Prime[Range[50]]]] (* Harvey P. Dale, Sep 28 2016 *)

a(n) = A036439(n-1) for n > 1.
a(n) - a(n-1) = A008578(n-1) for n > 1.
a(n) = A014284(n-1) + 1 for n > 1.

cp's OEIS Frontend

A175969 Complement of A014284(n), where A014284(n) = the lexicographically earliest sequence with first differences as increasing sequence of primes A000040.

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A024521 Erroneous version of A014284.

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A086741 a(n) = A026905(n) - A014284(n).

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A123213 a(n) = A062457(n) concatenated with A014284(n).

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A361150 a(n) = A014284(n^2) + A014284(n^2-1).

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A007504 Sum of the first n primes.

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A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

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A051349 Sum of first n nonprimes.

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