A014284 -id:A014284 - cp's OEIS frontend

A237589 Sum of first n odd noncomposite numbers.

1, 4, 9, 16, 27, 40, 57, 76, 99, 128, 159, 196, 237, 280, 327, 380, 439, 500, 567, 638, 711, 790, 873, 962, 1059, 1160, 1263, 1370, 1479, 1592, 1719, 1850, 1987, 2126, 2275, 2426, 2583, 2746, 2913, 3086, 3265, 3446, 3637, 3830, 4027, 4226, 4437, 4660, 4887

Offset: 1

Omar E. Pol, Feb 21 2014

Partial sums of A006005.

			For n = 5 the first five odd noncomposite numbers are 1, 3, 5, 7, 11, so a(5) = 1 + 3 + 5 + 7 + 11 = 27.

Cf. A000040, A006005, A007504, A008578, A014284.

a := proc(n) option remember; `if`(n=1, 1, a(n-1) + ithprime(n)) end:
seq(a(n), n=1..49); # Peter Luschny, Sep 20 2018

a[1]=1; a[n_]:=a[n]=a[n-1]+Prime[n]; Table[a[n], {n,1,49}] (* Robert P. P. McKone, Jan 18 2022 *)

terms(n) = my(s=1, i=0); forprime(p=3, , if(i >= n, break, print1(s, ", "); i++; s=s+p))
/* Print initial 50 terms as follows */
terms(50) \\ Felix Fröhlich, Sep 20 2018

a(n) = A007504(n) - 1 = A014284(n+1) - 2.

A343809 Divide the positive integers into subsets of lengths given by successive primes, then reverse the order of terms in each subset.

Original entry on oeis.org

2, 1, 5, 4, 3, 10, 9, 8, 7, 6, 17, 16, 15, 14, 13, 12, 11, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59

Offset: 1

Paolo Xausa, Apr 30 2021

From Omar E. Pol, Apr 30 2021: (Start)
Irregular triangle read by rows T(n,k) in which row n lists the next p positive integers in decreasing order, where p is the n-th prime, with n >= 1.
The triangle has the following properties:
Column 1 gives the nonzero terms of A007504.
Column 2 gives A237589.
Column 3 gives A071148.
Column 4 gives the terms > 2 of A343859.
Column 5 gives the absolute values of the terms < -1 of A282329.
Column 6 gives the terms > 7 of A082548.
Column 7 gives the terms > 6 of A115030.
Records are in the column 1.
Indices of records are in the right border.
Right border gives A014284.
Row lengths give A000040.
Row products give A078423.
Row sums give A034956. (End)

			From _Omar E. Pol_, Apr 30 2021: (Start)
Written as an irregular triangle in which row lengths give A000040 the sequence begins:
   2,  1;
   5,  4,  3;
  10,  9,  8,  7,  6;
  17, 16, 15, 14, 13, 12, 11;
  28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18;
  41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29;
  58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42;
  77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59;
  ...
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10887 (rows n = 1..70 of triangle, flattened)
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A000040, A007504, A014284, A034956, A038722, A071148, A073612 (fixed points), A078423, A082548, A115030, A237589, A282329, A343859, A344891.

R:= NULL: t:= 1:
for i from 1 to 20 do
  p:= ithprime(i);
  R:= R, seq(i,i=t+p-1..t,-1);
  t:= t+p;
od:
R; # Robert Israel, Apr 30 2021

With[{nn=10},Reverse/@TakeList[Range[Total[Prime[Range[nn]]]],Prime[Range[nn]]]]//Flatten (* Harvey P. Dale, Apr 27 2022 *)

T(n,k) = A007504(n) - k + 1, with n >= 1 and 1 <= k <= A000040(n). - Omar E. Pol, May 01 2021

A118482 Partial sums of Chen primes (starting with 1).

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 42, 59, 78, 101, 130, 161, 198, 239, 286, 339, 398, 465, 536, 619, 708, 809, 916, 1025, 1138, 1265, 1396, 1533, 1672, 1821, 1978, 2145, 2324, 2505, 2696, 2893, 3092, 3303, 3530, 3763, 4002, 4253, 4510, 4773, 5042, 5323, 5616, 5923

Offset: 0

Jani Melik, May 05 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A014284, A109611.

ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end: ts_partsum_chenprime:=proc(n) local i,ans,tren; ans:=1: tren:=1: for i from 1 to n do if (ischenprime(i)='true') then tren := tren+i: ans:=[op(ans), tren]: fi od; RETURN(ans) end: ts_partsum_chenprime(500);

Accumulate[Join[{1},Select[Prime[Range[70]],PrimeOmega[#+2]<3&]]] (* Harvey P. Dale, May 26 2014 *)

A158976 a(n) = sum of numbers k <= n such that not all proper divisors of k are divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 10, 6, 18, 23, 37, 10, 49, 45, 54, 66, 94, 75, 112, 90, 123, 149, 175, 120, 199, 220, 241, 251, 305, 236, 335, 307, 358, 396, 409, 385, 505, 501, 534, 499, 622, 568, 664, 630, 632, 749, 799, 688, 847, 857, 937, 959, 1049, 985, 1078, 1039, 1205

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

For primes p, a(p) = A000217(p) - A158662(p) = A000217(p) - A014284(A036234(p)).

			For n = 7 we have the following proper divisors for k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}. Only 4 and 6 have proper divisors that are not divisors of 7, viz. 2 and 2, 3. Hence a(7) = 4 + 6 = 10.

Cf. A000040, A000217, A158662, A014284, A036234, A158974.

[ IsEmpty(S) select 0 else &+S where S is [ k: k in [1..n] | exists(t){ d: d in Divisors(k) | d ne k and d notin Divisors(n) } ]: n in [1..57] ];

Edited and extended by Klaus Brockhaus, Apr 06 2009

A159073 Sum of the k in the range 1

Original entry on oeis.org

0, 2, 5, 9, 10, 20, 17, 29, 26, 31, 28, 67, 41, 59, 65, 69, 58, 95, 77, 119, 107, 103, 100, 179, 125, 130, 136, 154, 129, 228, 160, 220, 202, 198, 220, 280, 197, 239, 245, 320, 238, 334, 281, 359, 402, 331, 328, 487, 377, 417, 388, 418, 381, 499, 461, 556, 447, 443, 440

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

Here proper divisors include 1 but not the argument (k or n, respectively) in the divisor set, as defined in A032741.

Terms of the sum are counted in A159070.

			a(8) = 29 is the sum of the following six k: 2 {1}, 3 {1}, 4 {1, 2}, 5 {1}, 7 {1}, 8 {1, 2, 4} with subsets of the proper divisors {1, 2, 4} of n = 8. 2 + 3 + 4 + 5 + 7 + 8 = 29.

Cf.: A158975, A000040, A014284, A036234.

a(n) = A158975(n) - 1.

If p = prime, element of A000040, a(p) = A158662(p) - 1 = A014284(A036234(p)) - 1.

Edited and extended by R. J. Mathar, Apr 06 2009

A179859 Numbers k that divide the sum of the first k noncomposites.

Original entry on oeis.org

1, 3, 7, 225, 487, 735, 50047, 142835, 170209, 249655, 316585343, 374788043, 2460457827, 2803329305, 6860334657, 65397031525, 78658228039

Offset: 1

Ray Chandler, Jul 29 2010

A variant of A045345 (primes), A053781 (composites) and A129749 (nonprimes).

			The sum of the first 7 noncomposites is 42 = 6*7, so 7 is in the sequence.

Cf. A008578 (noncomposites), A014284 (sum of first n noncomposites).

Cf. A045345, A053781, A129749, A179860, A179861.

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

Numbers k such that k | A014284(k).
a(n) = A179861(n) / A179860(n).
a(n+1) = A158682(n) + 1.

a(16)-a(17) from Amiram Eldar, May 24 2024

A179860 Integer averages of first n noncomposites for some n.

Original entry on oeis.org

1, 2, 6, 636, 1592, 2574, 292656, 917042, 1108972, 1678508, 3334890730, 3981285760, 28567166356, 32739591796, 83332116034, 871263881618, 1055495274756

Offset: 1

Ray Chandler, Jul 29 2010

A variant of A050248 (primes), A073263 (composites) and A160758 (nonprimes).

			Sum of first 7 noncomposites is 42; 42 / 7 = 6 is in the sequence.

Cf. A008578 (noncomposites), A014284 (sum of first n noncomposites).

Cf. A179859, A179861, A158682.

a(n) = A179861(n) / A179859(n) = A014284(A179859(n)) / A179859(n).

a(16)-a(17) from Robert Price, Apr 21 2013

A179861 a(n) is the sum of the first A179859(n) noncomposites.

Original entry on oeis.org

1, 6, 42, 143100, 775304, 1891890, 14646554832, 130985694070, 188757015148, 419047914740, 1055777525624570390, 1492138298614167680, 70288308055831268412, 91779857115464381780, 571686203669195590338, 56978071532766214007450, 83023388015844408083484

Offset: 1

Ray Chandler, Jul 29 2010

nonn

A variant of A050247 (primes), A073262 (composites) and A164280 (nonprimes).

			A179859(3) = 7; sum of first 7 noncomposites is 42, so a(3) = 42.

Cf. A008578 (noncomposites), A014284 (sum of first n noncomposites).

Cf. A050247, A073262, A164280, A179859, A179860.

a(n) = A014284(A179859(n)) = A179859(n) * A179860(n).

a(16)-a(17) from Amiram Eldar, May 24 2024

A180302 Sequence of primes separated by [sequence of prime] elements. 2, [find 2nd prime after 2 = ] 5, [find 3rd prime after 5 =] 13, [find 5th prime after 13 =] 61, ..., etc.

Original entry on oeis.org

2, 5, 13, 31, 61, 109, 181, 277, 397, 547, 733, 947, 1213, 1499, 1831, 2207, 2633, 3083, 3583, 4133, 4751, 5407, 6073, 6793, 7589, 8513, 9397, 10313, 11353, 12409, 13451, 14713, 15889, 17299, 18593, 20129, 21613, 23167, 24851, 26561, 28387, 30203

Offset: 1

Daniel Tisdale, Aug 24 2010

nonn

Similar to A006450.

Cf. A006450, A014284.

NestList[n = 0; (n++; NextPrime[ #, Prime@ n]) &, 2, 41] (* Robert G. Wilson v, Aug 25 2010 *)
Prime[Accumulate[Join[{1}, Prime[Range[45]]]]] (* Alonso del Arte, Oct 09 2012 *)

a(8) onwards from Robert G. Wilson v, Aug 25 2010

A023538 Convolution of natural numbers with (1, p(1), p(2), ... ), where p(k) is the k-th prime.

Original entry on oeis.org

1, 4, 10, 21, 39, 68, 110, 169, 247, 348, 478, 639, 837, 1076, 1358, 1687, 2069, 2510, 3012, 3581, 4221, 4934, 5726, 6601, 7565, 8626, 9788, 11053, 12425, 13906, 15500, 17221, 19073, 21062, 23190, 25467, 27895, 30480, 33228, 36143, 39231, 42498, 45946, 49585

Offset: 1

Clark Kimberling

nonn

F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.

Cf. A007504, A014148, A014150, A014284, A023538, A178138.

Nest[Accumulate,Join[{1},Select[Range@200,PrimeQ]],2] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)

a(n) = Sum_{k<=n} [(A158611(k+1)) * (A000027(n-k+1))] = Sum_{k<=n} [(A008578(k)) * (A000027(n-k+1))]. [Jaroslav Krizek, Aug 05 2009; Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010]

cp's OEIS Frontend

A237589 Sum of first n odd noncomposite numbers.

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A343809 Divide the positive integers into subsets of lengths given by successive primes, then reverse the order of terms in each subset.

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Maple

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A118482 Partial sums of Chen primes (starting with 1).

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Maple

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A158976 a(n) = sum of numbers k <= n such that not all proper divisors of k are divisors of n.

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A159073 Sum of the k in the range 1

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A179859 Numbers k that divide the sum of the first k noncomposites.

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A179860 Integer averages of first n noncomposites for some n.

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A179861 a(n) is the sum of the first A179859(n) noncomposites.

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