A175967 -id:A175967 - cp's OEIS frontend

A175968 Complement of A175967(n), where A175967(n) = the lexicographically earliest sequence with first differences as increasing sequence of nonprimes A018252(n).

3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175966, A175967, A014284, A175969, A051349, A175970.

A014284 Partial sums of primes, if 1 is regarded as a prime (as it was until quite recently, see A008578).

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 42, 59, 78, 101, 130, 161, 198, 239, 282, 329, 382, 441, 502, 569, 640, 713, 792, 875, 964, 1061, 1162, 1265, 1372, 1481, 1594, 1721, 1852, 1989, 2128, 2277, 2428, 2585, 2748, 2915, 3088, 3267, 3448, 3639, 3832, 4029

Offset: 1

Deepan Majmudar (dmajmuda(AT)esq.com)

Lexicographically earliest sequence whose first differences are an increasing sequence of primes. Complement of A175969. - Jaroslav Krizek, Oct 31 2010
A175944(a(n)) = A018252(n). - Reinhard Zumkeller, Mar 18 2011
Partial sums of noncomposite numbers (A008578). - Omar E. Pol, Aug 09 2012

			a(7) = 42 because the first six primes (2, 3, 5, 7, 11, 13) add up to 41, and 1 + 41 = 42.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007504.
Equals A036439(n) - 1.
Cf. A175965, A175966, A175967, A175968, A175969, A051349, A175970. - Jaroslav Krizek, Oct 31 2010
Cf. A008578.

a014284 n = a014284_list !! n
a014284_list = scanl1 (+) a008578_list
-- Reinhard Zumkeller, Mar 26 2015

[1] cat [1 + (&+[NthPrime(j): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Jun 18 2019

A014284 := proc(n)
    add(A008578(i),i=1..n) ;
end proc:
seq(A014284(n),n=1..60) ; # R. J. Mathar, Mar 05 2017

Join[{1}, Table[1+Sum[Prime[j], {j,1,n}], {n,1,50}]] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2009, modified by G. C. Greubel, Jun 18 2019 *)
Accumulate[Join[{1}, Prime[Range[45]]]] (* Alonso del Arte, Oct 09 2012 *)

concat([1], vector(50, n, 1 + sum(j=1,n, prime(j)) )) \\ G. C. Greubel, Jun 18 2019

[1]+[1 + sum(nth_prime(j) for j in (1..n)) for n in (1..50)] # G. C. Greubel, Jun 18 2019

a(n) = Sum_{k <= n} [(A158611(k + 1)) * (A000012(n - k + 1))] = Sum_{k <= n} [(A158611(k + 1)) * (A000012(k))] = Sum_{k <= n} [(A008578(k)) * (A000012(n - k + 1))] = Sum_{k <= n} [(A008578(k)) * (A000012(k))] for n, k >= 1. - Jaroslav Krizek, Aug 05 2009
a(n + 1) = A007504(n) + 1. a(n + 1) - a(n) = A000040(n) = n-th primes. - Jaroslav Krizek, Aug 19 2009
a(n) = a(n-1) + prime(n-1), with a(1)=1. - Vincenzo Librandi, Jul 27 2013
G.f: (x*(1+b(x)))/(1-x) = c(x)/(1-x), where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - Mario C. Enriquez, Dec 10 2016

Correction for Aug 2009 change of offset in A158611 and A008578 by Jaroslav Krizek, Jan 27 2010

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.

A175966 Complement of A175965(n), where A175965(n) = the lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

Original entry on oeis.org

3, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175967, A175968, A014284, A175969, A051349, A175970.

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

Original entry on oeis.org

2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175966, A175967, A175968, A014284, A051349, A175970.

A175970 Complement of A051349(n), where A051349(n) = the lexicographically earliest sequence with first differences as increasing sequence of composites A002808(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Cf. A175965, A175966, A175967, A175968, A175969, A014284, A051349.

A277375 Each nonprime integer "n" is followed by n prime integers.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 11, 6, 13, 17, 19, 23, 29, 31, 8, 37, 41, 43, 47, 53, 59, 61, 67, 9, 71, 73, 79, 83, 89, 97, 101, 103, 107, 10, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 12, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 14, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 15, 313

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Oct 11 2016

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that does not lead to a contradiction.
From Michael De Vlieger, Oct 12 2016 (Start):
Each nonprime c = A018252(n) followed by c primes starting with prime(A175967(n)).
The position of nonprimes in a(n) is {1, 3, 8, 15, 24, 34, 45, 58, 73, 89, 106, 125, ...}. (End)

			As a(1) = 1, we take for a(2) the prime "2"; we then extend the sequence with a(3) which must be the smallest nonprime not yet present: this is "4"; we take for a(4), a(5), a(6) and a(7) the 4 smallest primes not yet present: they are 3, 5, 7 and 11; we then extend the sequence with the smallest nonprime available, which is a(8) = 6. Etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..7162

Cf. A018252 (nonprimes), A175967.

c = Select[Range@ 120, ! PrimeQ@ # &]; Table[Prepend[Prime@ Most[Range @@ Take[FoldList[Plus, 0, Take[c, n]] + 1, -2]], c[[n]]], {n, 9}] // Flatten (* Michael De Vlieger, Oct 12 2016 *)

cp's OEIS Frontend

A175968 Complement of A175967(n), where A175967(n) = the lexicographically earliest sequence with first differences as increasing sequence of nonprimes A018252(n).

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A014284 Partial sums of primes, if 1 is regarded as a prime (as it was until quite recently, see A008578).

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A051349 Sum of first n nonprimes.

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

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A175966 Complement of A175965(n), where A175965(n) = the lexicographically earliest sequence with first differences as increasing sequence of noncomposites A008578.

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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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A175970 Complement of A051349(n), where A051349(n) = the lexicographically earliest sequence with first differences as increasing sequence of composites A002808(n).

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A277375 Each nonprime integer "n" is followed by n prime integers.

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