A051349 -id:A051349 - cp's OEIS frontend

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

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.

A122998 Decimal expansion of Sum_{n>=1} 1/A051349(n).

Original entry on oeis.org

1, 5, 8, 1, 9, 3, 4, 8, 2, 2, 5, 4, 2, 0

Offset: 1

Pierre CAMI, Oct 28 2006

a(15) is either 3 or 4. - Robert Price, Jul 27 2011

			1/1 + 1/5 + 1/11 + 1/19 + 1/28 + 1/38 + 1/50 + 1/64 + 1/79 + 1/95 + 1/113 + ... = 1.5819348...

Cf. A051349.

Corrected a(9) from 1 to 2; added a(10)-a(14)=2,5,4,2,0, Robert Price, Jul 27 2011

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

A053767 Sum of first n composite numbers.

Original entry on oeis.org

0, 4, 10, 18, 27, 37, 49, 63, 78, 94, 112, 132, 153, 175, 199, 224, 250, 277, 305, 335, 367, 400, 434, 469, 505, 543, 582, 622, 664, 708, 753, 799, 847, 896, 946, 997, 1049, 1103, 1158, 1214, 1271, 1329, 1389, 1451, 1514, 1578, 1643, 1709, 1777, 1846, 1916, 1988

Offset: 0

G. L. Honaker, Jr., Mar 29 2000

nonn

a(A196415(n)) = A036691(A196415(n)) / A141092(n). - Reinhard Zumkeller, Oct 03 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
C. Rivera, See also

Cf. A002808, A051349.

First differences of A023539.

a053767 n = a053767_list !! (n-1)
a053767_list = scanl1 (+) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A053767 := proc(n)
        add(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011
ListTools[PartialSums](remove(isprime,[$2..1000])); # Robert Israel, Jan 09 2015

lst={};s=0;Do[If[ !PrimeQ[n], s=s+n;AppendTo[lst, s]], {n, 2, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)
Accumulate[Complement[Range[2,100],Prime[Range[PrimePi[100]]]]] (* Harvey P. Dale, Dec 28 2010 *)
Accumulate[Select[Range[2, 100], ! PrimeQ[#] &]]

lista(nn) = {my(s=0); forcomposite(n=0, nn, print1(s, ", "); s += n;);} \\ Michel Marcus, Jan 09 2015

a(n) = A000217(A002808(n)) - A034387(A002808(n)) - 1 . - Robert Israel, Jan 09 2015

a(n) = A051349(n+1) - 1. - Michel Marcus, Feb 16 2018

a(0)=0 prepended by Max Alekseyev, Feb 10 2018

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.

A175967 Lexicographically earliest sequence with first differences as increasing sequence of nonprimes A018252.

Original entry on oeis.org

1, 2, 6, 12, 20, 29, 39, 51, 65, 80, 96, 114, 134, 155, 177, 201, 226, 252, 279, 307, 337, 369, 402, 436, 471, 507, 545, 584, 624, 666, 710, 755, 801, 849, 898, 948, 999, 1051, 1105, 1160, 1216, 1273, 1331, 1391, 1453, 1516, 1580, 1645, 1711, 1779

Offset: 1

Jaroslav Krizek, Oct 31 2010

nonn

Complement of A175968.

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

Cf. A175965, A175966, A175968, A014284, A175969, A051349, A175970.

Cf. A018252.

a175967 n = a175967_list !! n
a175967_list = scanl (+) 1 a018252_list
-- Reinhard Zumkeller, Mar 26 2015

a(n) - a(n-1) = A018252(n-1) for n >= 2.

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.

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

Original entry on oeis.org

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.

A154587 Numbers that can be expressed both as the sum of first prime numbers and as the sum of first nonprime numbers.

Original entry on oeis.org

0, 5, 28, 71208, 74139, 9260197734, 12374540078, 7574780746329, 11101148723618, 102581905748236, 3325997869054417, 2886018916559052244845, 46437379006448216748610, 120197329614203475099994

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 15 2009

Is this sequence finite?
Intersection of A007504 and A053767 generates A294174. - R. J. Mathar, Jan 17 2009
Heuristically, the sequence is infinite with about 2 sqrt(log x) members up to x. - Charles R Greathouse IV, Aug 14 2013

			5 = 2+3 = 1+4. 28 = 2+3+5+7+11 = 1+4+6+8+9.

Giovanni Resta, C program

Intersection of A007504 and A051349. - R. J. Mathar, Jan 17 2009

Cf. A133784, A294174.

P:=proc(q) local a,b,c,d,n; a:=0; b:=0; c:=0; d:=0; print(a);
for n from 1 to q do b:=nextprime(b); a:=a+b;
while cPaolo P. Lava, Feb 23 2018

With[{p = Prime@ Range[10^7]}, {0}~Join~Intersection[Accumulate@ p, Accumulate@ Complement[Range@ Max@ p, p]]] (* Michael De Vlieger, Feb 25 2018 *)

Corrected definition and a(6)-a(7) from R. J. Mathar, Jan 17 2009
a(8)-a(11) from Donovan Johnson, Feb 19 2009
a(12)-a(14) from Giovanni Resta, Aug 14 2013
Edited and a(1)=0 prepended by Max Alekseyev, Feb 10 2018

cp's OEIS Frontend

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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A122998 Decimal expansion of Sum_{n>=1} 1/A051349(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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A053767 Sum of first n composite numbers.

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

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Haskell

Mathematica

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A175967 Lexicographically earliest sequence with first differences as increasing sequence of nonprimes A018252.

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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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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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