A051349 -id:A051349 - cp's OEIS frontend

A129749 Numbers k that divide the sum of the first k nonprimes.

1, 8, 32, 44, 577, 5066, 5669, 8615, 39787, 59689, 109752, 146328, 15451380, 22173220, 28558717, 332573533, 1837410366, 3289933379, 19053646133, 370648112641

Offset: 1

J. M. Bergot, May 14 2007

A variant of A045345 and A053781.

a(21) > 10^12. - Donovan Johnson, May 20 2010

			The sum of the first 44 nonprimes is 1452 = 33*44.

Cf. A045345 (n divides sum of first n primes), A053781 (n divides sum of first n composites), A018252 (nonprimes), A051349 (sum of first n nonprimes).

np:=proc(j) if isprime(j)=false then j else fi end: NP:=[seq(np(j),j=1..50000)]: a:=proc(n) if type(add(NP[j],j=1..n)/n,integer)=true then n else fi end: seq(a(n),n=1..nops(NP)); # Emeric Deutsch, May 16 2007

Module[{nn=150000,np},np=Accumulate[Select[Range[nn],!PrimeQ[#]&]];Select[Thread[ {np,Range[Length[np]]}],Mod[#[[1]],#[[2]]]==0&]][[;;,2]] (* The program generates the first 11 terms of the sequence. To generate more, increase the nn constant. *) (* Harvey P. Dale, Jan 04 2024 *)

Edited and a(5) to a(18) added by Klaus Brockhaus, May 17 2007
a(19) from Donovan Johnson, Sep 19 2009
a(20) from Donovan Johnson, May 20 2010

A160758 Integer averages of first n nonprime numbers for some n.

Original entry on oeis.org

1, 8, 25, 33, 359, 2948, 3291, 4959, 22350, 33357, 60907, 80962, 8276347, 11856980, 15254419, 176009996, 967538242, 1729774831, 9977169279, 193005936726

Offset: 1

Daniel Tisdale, May 25 2009

A variant of A050248 for nonprimes.

Numbers n such that (1/n)*sum(j=1..n, A018252(j)) is an integer. - Robert G. Wilson v, Jun 05 2009

			The sum of the first 44 nonprimes is 1452. 1452 / 44 = 33, hence 33 is in the sequence.

Cf. A050248, integer averages of n primes for some n.

Cf. A018252, A051349, A164280, A129749.

S:=[]; a:=0; c:=0; for n in [1..40000000] do if not IsPrime(n) then a+:=n; c+:=1; if a mod c eq 0 then Append(~S, a div c); end if; end if; end for; S; // Klaus Brockhaus, Aug 11 2009

lst = {}; s = 0; c = 0; k = 1; While[k < 2700000000, If[ !PrimeQ@k, c++; s = s + k; If[Mod[s, c] == 0, AppendTo[lst, s/c]]]; k++ ]; lst (* Robert G. Wilson v, Jun 05 2009 *)
a=0;lst={};Do[If[ !PrimeQ[n],m=n;a+=m;If[a/n==IntegerPart[a/n],AppendTo[lst,a/n]]],{n,9!}];lst

a(n) = A164280(n) / A129749(n).

a(6) - a(16) from Robert G. Wilson v, Jun 05 2009
a(17) - a(19) from Donovan Johnson, Sep 16 2009
Edited by N. J. A. Sloane, May 11 2010
a(20) from Donovan Johnson, May 20 2010

A060697 The sum of the first a(n) composite numbers plus 1 is a prime.

Original entry on oeis.org

1, 2, 3, 8, 10, 16, 21, 29, 34, 45, 49, 52, 84, 104, 114, 123, 130, 161, 165, 170, 181, 185, 192, 202, 216, 227, 228, 240, 245, 246, 265, 266, 271, 281, 287, 295, 301, 325, 331, 355, 390, 395, 400, 406, 410, 416, 429, 498, 502, 517, 522, 527, 529, 538, 539

Offset: 1

Robert G. Wilson v, Apr 20 2001

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000040, A002808, A051349.

Composite[n_Integer] := Block[{k = n + 1 + PrimePi@ n}, While[k - 1 - PrimePi@ k != n, k++]; k]; lst = 1 + Accumulate@ Array[ Composite, 600]; Select[ Range@ 600, PrimeQ@ lst[[#]] &] (* Robert G. Wilson v, Nov 29 2014 *)
Position[Accumulate[Select[Range[900],CompositeQ]],?(PrimeQ[ #+1]&)]// Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jan 04 2020 *)

A071411 "Sum of n first primes" minus "sum of first n nonprimes".

Original entry on oeis.org

1, 0, -1, -2, 0, 3, 8, 13, 21, 34, 47, 64, 84, 105, 128, 156, 189, 223, 262, 303, 344, 390, 439, 493, 554, 617, 681, 748, 815, 884, 966, 1051, 1140, 1230, 1329, 1429, 1534, 1643, 1755, 1872, 1994, 2117, 2248, 2379, 2513, 2648, 2794, 2951, 3110, 3270, 3433, 3600, 3767, 3943, 4124, 4310, 4501, 4692, 4888

Offset: 1

Benoit Cloitre, Jun 23 2002

for(n=1, 100, s=2; while(sum(i=1, s, 1-isprime(i))

a(n) = A007504(n) - A051349(n). Cf. A024850.

Partial sums of A014237.

Corrected and edited by Jaroslav Krizek, Jun 17 2009

A133784 Positive integers n such that the sum of all primes <= n divides n(n+1)/2.

Original entry on oeis.org

4, 21, 24, 71

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 21 2008

nonn

Also, positive integers n for which the sum of all primes <= n divides the sum of all nonprimes <= n.
Is this sequence finite?
No other terms below 10^13. - Max Alekseyev, Nov 28 2017
Sequence is probably complete; integral_{x=10^10..infinity} (2 log x)/x^2 dx = 4.80517... * 10^-9. - Charles R Greathouse IV, Mar 21 2013

			n=21 is in this sequence since the sum of the primes <=21 is 2+3+5+7+11+13+17+19=77 and 77 divides 21*22/2=231.

Cf. A007504, A000217, A051349.

P:=proc(n) local i,snp,sp; snp:=1; sp:=2; for i from 3 by 1 to n do if isprime(i) then sp:=sp+i; else snp:=snp+i; fi; if trunc(snp/sp)=snp/sp then print(i); fi; od; end: P(10000000);

Select[Range[2,100],Divisible[(#(#+1))/2,Total[Prime[Range[ PrimePi[ #]]]]]&] (* Harvey P. Dale, May 31 2012 *)

A=1;P=0;for(n=2,1e9,A+=n;P+=isprime(n)*n;if(A%P==0,print1(n", "))) \\ Charles R Greathouse IV, Mar 21 2013

Edited by Max Alekseyev, Feb 13 2009

A140235 Partial sum of non-semiprimes A100959.

Original entry on oeis.org

1, 3, 6, 11, 18, 26, 37, 49, 62, 78, 95, 113, 132, 152, 175, 199, 226, 254, 283, 313, 344, 376, 412, 449, 489, 530, 572, 615, 659, 704, 751, 799, 849, 901, 954, 1008, 1064, 1123, 1183, 1244, 1307, 1371, 1437, 1504, 1572, 1642, 1713, 1785, 1858, 1933, 2009

Offset: 1

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A051352 is to primes A000040. Equivalently, this is to non-semiprimes A100959 as A051349 is to nonprimes A018252.

			a(5) = 18 = 1 + 2 + 3 + 5 + 7.

Cf. A000217, A001358, A018252, A034387, A051349, A051352, A062198, A072000, A100959, A140234.

Accumulate[Select[Range[100],PrimeOmega[#]!=2&]] (* Harvey P. Dale, Aug 22 2021 *)

a(n) = Sum{k=1..n} A100959(k).

Corrected and edited by Giovanni Resta, Jun 20 2016

A163061 Sum of the first n primes plus the first n nonprimes.

Original entry on oeis.org

2, 6, 15, 28, 47, 69, 96, 127, 164, 208, 255, 310, 371, 435, 504, 581, 665, 752, 846, 945, 1048, 1159, 1275, 1398, 1530, 1667, 1808, 1954, 2103, 2258, 2429, 2605, 2788, 2975, 3173, 3374, 3582, 3797, 4018, 4246, 4481, 4719, 4968, 5221, 5480, 5742

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2009

nonn

			a(1)=2+0=2. a(2)=5+1=6. a(3)=10+5=15. a(4)=17+11=28. a(5)=28+19=47. a(6)=41+28=69. a(7)=58+38=96.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A141468, A007504, A051349.

Module[{nn=80,np,len},np=Select[Range[0,nn],!PrimeQ[#]&];len=Length[np];Total/@ Thread[{Accumulate[np],Accumulate[Prime[Range[len]]]}]] (* Harvey P. Dale, Jun 14 2019 *)

a(n)=A007504(n)+A051349(n-1).

Formula index corrected, 1119 replaced by 1159 - R. J. Mathar, Jul 27 2009

A163116 Partial sums of A161671.

Original entry on oeis.org

2, 4, 5, 6, 9, 13, 20, 27, 36, 50, 65, 84, 105, 127, 152, 181, 215, 250, 290, 333, 376, 423, 473, 528, 590, 655, 720, 788, 857, 928, 1011, 1097, 1188, 1279, 1379, 1480, 1586, 1697, 1810, 1928, 2051, 2175, 2308, 2441, 2576, 2712

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2009

nonn

The n-th partial sum of the primes minus the n-th partial sum of the nonprimes.

			a(1) = A161671(1) = 2.
a(2) = a(1) + A161671(2) = 2 + 2 = 4.
a(3) = a(2) + A161671(3) = 4 + 1 = 5.

Cf. A000040, A141468, A161671.

a(n) = A007504(n) - A051349(n-1). - R. J. Mathar, Jul 31 2009

Corrected from a(9) on by R. J. Mathar, Jul 31 2009

A165239 Integers of the form "partial sum of the first k nonprimes divided by the k-th nonprime.".

Original entry on oeis.org

1, 8, 9, 46, 172, 1188, 4708, 28611, 480644, 785709, 1061769, 24947740, 31585866, 110275647, 149612794, 159273153, 396643326, 586740520, 622899752, 769642427, 979286966, 2603393493, 4102885015, 4313575355, 5075807081, 5890950206, 16152068952, 62853548947

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 09 2009

nonn

Integers of the form A051349(k)/A018252(k), generated by k = 1, 14, 16, 88, 334, 2345, 9328, ... - R. J. Mathar, Oct 29 2011

			(1 + 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18 + 20 + 21 + 22)/22 = 8.
(1 + 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18 + 20 + 21 + 22 + 24 + 25)/25 = 9.
(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 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 98 + 99 + 100 + 102 + 104 + 105 + 106 + 108 + 110 + 111 + 112 + 114 + 115 + 116 + 117 + 118)/118 = 46.

Cf. A051349, A160758, A018252.

lst={};a=0;Do[If[ !PrimeQ[n],m=n;a+=m;If[IntegerQ[a/n]&&a!=n,AppendTo[lst,a/n]]],{n,5*9!}];lst

lista(kmax) = {my(s = 1); print1(1, ", "); forcomposite(k = 1, kmax, s += k; if(!(s%k), print1(s/k, ", ")));} \\ Amiram Eldar, May 24 2024

a(11)-a(28) from Amiram Eldar, May 24 2024

A257392 Number of ways of representing n as the sum of one or more consecutive nonprime numbers (A018252).

Original entry on oeis.org

1, 2, 0, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 4, 2, 1, 2, 1, 1, 1, 4, 2, 0, 1, 4, 3, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 0, 2, 4, 3, 2, 1, 3, 2, 3, 1, 2, 1, 0, 3, 4, 2, 2, 3, 3, 1, 3, 2, 1, 2, 3, 2, 2, 1, 1, 4, 2, 2, 1, 4, 5

Offset: 1

Juri-Stepan Gerasimov, Apr 21 2015

nonn

			a(2) = 2 because n = 2 itself is already a nonprime number (sum of 1 term), and 1 can in addition be written as A018252(1) + A018252(2), a sum of 2 consecutive nonprime numbers.

Cf. A051349, A053767, A018252, A166037, A166039, A242667, A257393.

cp's OEIS Frontend

A129749 Numbers k that divide the sum of the first k nonprimes.

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A160758 Integer averages of first n nonprime numbers for some n.

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A060697 The sum of the first a(n) composite numbers plus 1 is a prime.

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A071411 "Sum of n first primes" minus "sum of first n nonprimes".

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A133784 Positive integers n such that the sum of all primes <= n divides n(n+1)/2.

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A140235 Partial sum of non-semiprimes A100959.

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A163061 Sum of the first n primes plus the first n nonprimes.

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A163116 Partial sums of A161671.

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