A007443 -id:A007443 - cp's OEIS frontend

A302334 A weighted smoothing applied to the primes as a data set: a(n) = floor(A007443(2n-1)/2^(2n-2)), where A007443 is binomial transform of primes.

2, 3, 5, 7, 10, 13, 16, 20, 24, 28, 32, 36, 40, 44, 48, 53, 57, 61, 66, 70, 75, 79, 84, 89, 94, 98, 103, 108, 113, 119, 124, 129, 135, 140, 146, 151, 156, 162, 167, 172, 178, 183, 189, 194, 200, 205, 211, 216, 222, 228, 233, 239, 244, 250, 255, 261, 267, 273, 278, 284

Offset: 1

Peter Munn, Apr 05 2018

nonn

a(n) is the weighted average of the first 2n - 1 primes, using row 2n - 2 of Pascal's triangle as weights, with the result rounded down. a(n) is thus based on the longest ordered list of consecutive primes that has prime(n) in the central position, while giving substantially greater weight to the primes near prime(n).
A guiding aim when framing the definition was having the arithmetic mean of the first k terms close to the arithmetic mean of the first k primes. In this respect, a simplified analysis suggested the binomial weighting might perform equally well for large k as small k, and empirical results were encouraging. For all k <= 500 the difference between the means is < 0.541, with 0.5 being exceeded only for 394 <= k <= 401. (These figures become not quite as good if floor rounding is replaced by nearest-integer, though a rounding midway between the two does better than either.)
The early terms (playing the role of primes) correspond closely to A053620 (in the role of primepi function), but the correspondence gets better if nearest-integer rounding is used instead of the floor rounding used here. - Peter Munn, Feb 26 2024
Conjecture: the second differences are in [-2,2].

			For n=3, we calculate a weighted average of the first 2n - 1 = 5 primes. Row 2n - 2 = 4 of Pascal's triangle, (1,4,6,4,1), provides the weights, and its row sum is 2^4 = 16.
Specifically, using the first formula, a(3) = floor( Sum_{k=0..4}(binomial(4,k)*prime(k+1)) / 2^4 ).
The sum in the formula = 1*prime(1) + 4*prime(2) + 6*prime(3) + 4*prime(4) + 1*prime(5) = 1*2 + 4*3 + 6*5 + 4*7 + 1*11 = 2 + 12 + 30 + 28 + 11 = 83.
So a(3) = floor(83/2^4) = floor(83/16) = 5.
Comparison with the primes: (Start)
Analysis table showing the difference between the start of this sequence and the start of the list of primes. a(n) is subtracted from prime(n) to give a sense of how prime(n) is lower or higher than it might be if the primes were more smoothly distributed. The column headed "cumulative" gives the partial sums of the previous column, which are then divided by n and rounded to 3 decimal places to give the final column. The final column therefore shows the difference between the arithmetic means of the first n primes and the first n terms of this sequence.
  n        prime(n)      a(n)    difference  cumulative average
   1           2           2          0          0       0.000
   2           3           3          0          0       0.000
   3           5           5          0          0       0.000
   4           7           7          0          0       0.000
   5          11          10          1          1       0.200
   6          13          13          0          1       0.167
   7          17          16          1          2       0.286
   8          19          20         -1          1       0.125
   9          23          24         -1          0       0.000
  10          29          28          1          1       0.100
  11          31          32         -1          0       0.000
  12          37          36          1          1       0.083
  13          41          40          1          2       0.154
  14          43          44         -1          1       0.071
  15          47          48         -1          0       0.000
  16          53          53          0          0       0.000
  17          59          57          2          2       0.118
  18          61          61          0          2       0.111
  19          67          66          1          3       0.158
  20          71          70          1          4       0.200
  21          73          75         -2          2       0.095
  22          79          79          0          2       0.091
  23          83          84         -1          1       0.043
  24          89          89          0          1       0.042
  25          97          94          3          4       0.160
  26         101          98          3          7       0.269
  27         103         103          0          7       0.259
  28         107         108         -1          6       0.214
  29         109         113         -4          2       0.069
  30         113         119         -6         -4      -0.133
  31         127         124          3         -1      -0.032
  32         131         129          2          1       0.031
(End)

Peter Munn, Table of n, a(n) for n = 1..500
P. Marchand and L. Marmet, Binomial smoothing filter: A way to avoid some pitfalls of least square polynomial smoothing, Review of Scientific Instruments, 54, 1034-41, 1983.
Wikipedia, Smoothing

Cf. A000040, A000079, A007443, A053620, A060620.

a[n_] := Floor[ Sum[ Binomial[2n -2, k]*Prime[k +1]/2^(2n -2), {k, 0, 2n -2}]]; Array[a, 60] (* Robert G. Wilson v, Jun 10 2018 *)

a(n) = floor(sum(k=0, 2*n-2, (binomial(2*n-2,k) * prime(k+1))/2^(2*n-2))); \\ Michel Marcus, Aug 21 2018

a(n) = floor(Sum_{k=0..2n-2} (binomial(2n-2,k) * prime(k+1))/2^(2n-2)).

a(n) = floor(A007443(2n-1)/2^(2n-2)).

a(51)-a(60) from Robert G. Wilson v, Jun 10 2018

A287915 Indices of primes in A007443.

Original entry on oeis.org

1, 2, 3, 5, 9, 22, 25, 28, 32, 41, 99, 104, 138, 183, 225, 361, 641, 1636, 1719, 3191, 3590, 4144, 5340, 6372, 6893, 6915, 8429, 10024, 10546, 16401, 21636, 22612, 24813, 31416, 36065

Offset: 1

M. F. Hasler, Jun 02 2017

Sequence A007443, the binomial transform of the primes A000040, is defined as A007443(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). This is also the first column of the infinite square array T(m,n) with T(1,n) = prime(n) and T(m+1,n) = T(m,n) + T(m,n+1), as for binomial coefficients. Successive rows result from applying this operation of taking the sum of successive terms. So it would be more natural to use index 0 for the first term of this sequence (which is also the only even term, and results from applying the operation 0 times to the primes). This would yield the sequence 0, 1, 2, 4, 8, 21, 24, 27, 31, 40, 98, 103, 137, 182, ...
The next term, if it exists, is greater than 20000. - Vaclav Kotesovec, Dec 19 2020
Any subsequent terms are > 10^5. - Lucas A. Brown, Mar 18 2024

Lucas A. Brown, Python program.

Cf. A007443.

A007443 = Table[Sum[Binomial[n-1, k-1]*Prime[k], {k, 1, n}], {n, 1, 1000}]; Select[Range[Length[A007443]], PrimeQ[A007443[[#]]]&] (* Vaclav Kotesovec, Dec 19 2020 *)

for(n=1,199,isprime(A007443(n))&&print1(n","))

a(18)-a(22) from Jinyuan Wang, Dec 19 2020
a(23)-a(30) from Vaclav Kotesovec, Dec 19 2020
a(31)-a(35) from Lucas A. Brown, Mar 18 2024

A096280 Primes in A007443 (= binomial transform of primes).

Original entry on oeis.org

2, 5, 13, 83, 2707, 71475193, 674721797, 6245693407, 118543624847, 82736199371081, 72298621492552303967009812018997, 2454725173623452943975951834280921, 59966692897276736774965300014477948187539553

Offset: 1

Cino Hilliard, Jun 23 2004

nonn

Sum of reciprocals = 0.2893406979695919267175673140... Are these primes infinite?

The next term is too large to be displayed here. See A287915 for the indices k which yield these primes A007443(k). - M. F. Hasler, Jun 02 2017

Paolo Xausa, Table of n, a(n) for n = 1..20 (terms 1..17 from M. F. Hasler)

Cf. A007443, A287915, A001043, A096277, A096278, A096279.

See A287915 for the corresponding indices of A007443.

A007443[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];
With[{upto=500},Select[Array[A007443,upto],PrimeQ]] (* or *)
Module[{upto=500,b},b=Prime[Range[upto]];Join[{2},Select[Table[First[b=ListConvolve[{1,1},b]],upto-1],PrimeQ]]] (* Paolo Xausa, Oct 31 2023 *)

\\ n = terms to add, m = order.
sucsumspr(n,m) = { local(a,b,i,j,k,sr); sr=0; a = primes(1001); b = vector(1001); for(i=1,m, for(j=1,n+n, b[j] = a[j]+ a[j+1]; ); a=b; if(isprime(a[1]),print1(a[1]",");sr+=1.0/a[1]); ); print(); print(sr); }

for(n=1,999, ispseudoprime(A007443(n))&&print1(A007443(n)",")) \\ M. F. Hasler, Jun 02 2017

a(n) = A007443(A287915(n)). - M. F. Hasler, Jun 02 2017

Definition corrected, initial term 2 added, and edited by M. F. Hasler, Jun 02 2017

Name simplified by Paolo Xausa, Nov 05 2023

A007442 Inverse binomial transform of primes.

Original entry on oeis.org

2, 1, 1, -1, 3, -9, 23, -53, 115, -237, 457, -801, 1213, -1389, 445, 3667, -15081, 41335, -95059, 195769, -370803, 652463, -1063359, 1570205, -1961755, 1560269, 1401991, -11023119, 36000427, -93408425, 214275735, -450374071, 879254493, -1599245737, 2695465017

Offset: 1

N. J. A. Sloane

a(n) is the (n-1)-st difference of the first n primes. Although the magnitude of the terms appears to grow exponentially, a plot shows that the sequence a(n)/2^n has quite a bit of structure. See A082594 for an interesting application. - T. D. Noe, May 09 2003
Graph this divided by A122803 using plot2! - Franklin T. Adams-Watters
From Robert G. Wilson v, Jan 28 2020: (Start)
a(n) is odd for all n>1.
As opposed to A331573, there are terms where abs(a(n)) >= abs(a(n+1)). (End)

			a(4) = 7 - 3*5 + 3*3 - 2 = -1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..3321 (first 1000 from Franklin T. Adams-Watters)
Vaclav Kotesovec, Plot of |a(n)|^(1/n) for n = 1..10000
T. D. Noe, Plot of A007442
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Binomial Transform

Cf. A007443, A082594, A095195, A122803, A331573.

Diff[lst_List] := Table[lst[[i + 1]] - lst[[i]], {i, Length[lst] - 1}]; n=1000; dt = Prime[Range[n]]; a = Range[n]; a[[1]] = 2; Do[dt = Diff[dt]; a[[i]] = dt[[1]], {i, 2, n}]; a
u = Table[Prime[Range[k]], {k, 1, 100}];Flatten[Table[Differences[u[[k]], k - 1], {k, 1, 100}]] (* Clark Kimberling, May 15 2015 *)
t = Array[Prime, 30]; f[x_] := Rest[x] - Most[x];
Flatten[Last /@ (NestList[f, t[[1 ;; #]], (# - 1)] & /@ Range[1, 29])] (* Horst H. Manninger, Mar 22 2021 *)

vector(50, n, sum(k=0, n-1,(-1)^(n-k-1)*binomial(n-1, k)*prime(k+1))) \\ Altug Alkan, Oct 17 2015

a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) binomial(n-1, k) prime(k+1).
a(n) = A095195(n,n-1). - Alois P. Heinz, Sep 25 2013
G.f.: Sum_{k>=1} prime(k)*x^k/(1 + x)^k. - Ilya Gutkovskiy, Apr 23 2019

Incorrect conjecture concerning the sign of even terms removed by Glen Whitney, Nov 10 2024

A096279 Sums of successive sums of successive sums of successive sums of successive primes.

Original entry on oeis.org

83, 122, 168, 216, 264, 316, 378, 446, 514, 582, 644, 700, 766, 846, 924, 994, 1062, 1126, 1188, 1258, 1340, 1434, 1528, 1602, 1656, 1704, 1764, 1860, 1984, 2094, 2178, 2260, 2348, 2432, 2514, 2598, 2682, 2766, 2848, 2932, 3018, 3090, 3152, 3242, 3378

Offset: 1

Cino Hilliard, Jun 22 2004

The first term is always odd and may be prime. The first terms of more and more successions produce A007443.

			2 3 5 7 11 successive primes
5 8 12 18 sums of successive primes
13 20 30 sums of successive sums of successive primes
33 50 sums of successive sums of successive sums of successive primes
83 sums of successive sums of successive sums of successive sums of successive primes

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A096277, A096278.

Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],4] (* Paolo Xausa, Oct 31 2023 *)

\\ Iterated successive sums of successive sums... of successive primes.
\\ Input number of terms n and the order m. m=0 yields the primes.
sucsums(n, m) = { my(a, b, i, j, k); a = primes(n+m); b = vector(#a); for(i=1, m, for(j=1, n+m-i, b[j] = a[j]+ a[j+1]; ); a=b; ); for(k=1, n, print1(a[k], ", "); ) }

A111107 Lexicographically smallest increasing sequence of primes whose binomial transform consists only of primes.

Original entry on oeis.org

2, 3, 5, 11, 13, 29, 43, 53, 59, 71, 79, 83, 103, 113, 139, 173, 181, 227, 269, 277, 317, 383, 463, 509, 673, 701, 751, 863, 967, 977, 1187, 1201, 1493, 1531, 1609, 1637, 1801, 2153, 2221, 2239, 2371, 2377, 2543, 2557, 2683, 2687, 2791, 2837, 3067, 3229, 3257

Offset: 0

Daniel Joyce, Oct 14 2005

In the standard binomial transform of the primes most of the terms are composite.

			The binomial transform of this sequence gives: 2, 5, 13, 37, 101, 271, 727, 1931, 5003, 12547, 30449, 71761, ... = A384676.
The prime 7 and various larger primes are missing from the new sequence because the transform would not consist of primes. For example,
  2,5,13,33
  3,8,20
  5,12
  7
and 33 is not prime, so we must eliminate 7.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000040, A007443, A384676.

a(n) = Sum_{i=0..n} A384676(n-i)*binomial(n,i)*(-1)^i. - Alois P. Heinz, Jun 06 2025

Offset set to 0 by Alois P. Heinz, Jun 06 2025

A164738 Triangle read by rows. Row 0 = {2}; left half of row n+1 = row n, right half = row n reversed with each term replaced by the next prime.

Original entry on oeis.org

2, 2, 3, 2, 3, 5, 3, 2, 3, 5, 3, 5, 7, 5, 3, 2, 3, 5, 3, 5, 7, 5, 3, 5, 7, 11, 7, 5, 7, 5, 3, 2, 3, 5, 3, 5, 7, 5, 3, 5, 7, 11, 7, 5, 7, 5, 3, 5, 7, 11, 7, 11, 13, 11, 7, 5, 7, 11, 7, 5, 7, 5, 3, 2, 3, 5, 3, 5, 7, 5, 3, 5, 7, 11, 7, 5, 7, 5, 3, 5, 7, 11, 7, 11, 13, 11, 7, 5, 7, 11, 7, 5, 7, 5, 3, 5, 7, 11, 7

Offset: 0

Gary W. Adamson, Aug 23 2009

Row n has 2^n terms.

Row sums = (2, 5, 13, 33, 83, 205, 495,...) = A007443, the binomial transform of the primes.

			The triangle begins:
  2;
  2, 3;
  2, 3, 5, 3;
  2, 3, 5, 3, 5, 7, 5, 3;
  2, 3, 5, 3, 5, 7, 5, 3, 5, 7, 11, 7, 5, 7, 5, 3;
  ...

Alois P. Heinz, Rows n = 0..13, flattened

Cf. A000040, A007443, A088696.

Flatten[NestList[Join[ #, NextPrime[Reverse[ # ]]] &, {2}, 7]] (* Zak Seidov, Aug 24 2009 *)

More terms from Zak Seidov, Aug 24 2009
Edited by Zak Seidov and N. J. A. Sloane, Aug 25 2009
Offset 0 from Alois P. Heinz, Oct 31 2023

A178167 Binomial transform of odd primes.

Original entry on oeis.org

3, 8, 20, 50, 122, 290, 674, 1538, 3462, 7720, 17104, 37708, 82768, 180880, 393488, 851914, 1835470, 3935808, 8402158, 17865738, 37858268, 79991512, 168605710, 354649382, 744605426, 1560666296, 3265699888, 6822355756, 14229565634

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 21 2010

nonn

03 05 07 11 13 08 12 18 24 20 30 42 50 72 122

First differences of A007443.

q=33;lst=Prime[Range[2,q]]; lst1={First[lst]};Do[lst=Table[Abs[lst[[n]]+lst[[n+1]]],{n,1,Length[lst]-1}];AppendTo[lst1,First[lst]],{k,q-2}];lst1
Differences[Table[Sum[Binomial[n-1, k-1]*Prime[k], {k, 1, n}], {n, 1, 30}]] (* Vaclav Kotesovec, Dec 19 2020 *)

A333176 a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * prime(k).

Original entry on oeis.org

2, 3, 10, 7, 20, 23, 58, 19, 44, 51, 112, 63, 140, 151, 328, 53, 114, 117, 250, 131, 276, 287, 604, 161, 342, 355, 742, 383, 798, 825, 1720, 131, 270, 273, 566, 289, 596, 607, 1252, 323, 664, 675, 1392, 711, 1458, 1481, 3046, 407, 832, 839, 1718, 875, 1782

Offset: 1

Ilya Gutkovskiy, Mar 10 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A007443, A007504, A010060, A030015, A050513.

N:= 200: # for a(1) .. a(N)
P:= [seq(ithprime(i),i=1..N)]:
B:= [1,1]: R:= 2:
for n from 2 to N do
  B:= [1,op(B[2..-1]+B[1..-2] mod 2),1];
  R:= R, convert(P[select(t -> B[t+1] = 1,[$1..n])],`+`);
od:
R; # Robert Israel, Jan 29 2025

Table[Sum[Mod[Binomial[n, k], 2] Prime[k], {k, 1, n}], {n, 1, 53}]

a(n) = sum(k=1, n, if (binomial(n, k) % 2, prime(k))); \\ Michel Marcus, Mar 10 2020

from sympy import prime
def A333176(n): return sum(prime(k) for k in range(1,n+1) if not ~n&k) # Chai Wah Wu, Jul 22 2025

Sum_{k=1..n} (-1)^A010060(n-k) * (binomial(n,k) mod 2) * a(k) = prime(n).

A287919 Square array T(0,n) = prime(n) and T(m+1,n) = T(m,n) + T(m,n+1), m >= 0, n >= 1, read by falling antidiagonals.

Original entry on oeis.org

2, 3, 5, 5, 8, 13, 7, 12, 20, 33, 11, 18, 30, 50, 83, 13, 24, 42, 72, 122, 205, 17, 30, 54, 96, 168, 290, 495, 19, 36, 66, 120, 216, 384, 674, 1169, 23, 42, 78, 144, 264, 480, 864, 1538, 2707, 29, 52, 94, 172, 316, 580, 1060, 1924, 3462, 6169

Offset: 0

M. F. Hasler, Jun 02 2017

			The array starts:
[0]    2    3    5     7    11    13    17    19    23    29 ... (A000040)
[1]    5    8   12    18    24    30    36    42    52    60 ... (A001043)
[2]   13   20   30    42    54    66    78    94   112   128 ... (A096277)
[3]   33   50   72    96   120   144   172   206   240   274 ... (A096278)
[4]   83  122  168   216   264   316   378   446   514   582 ... (A096279)
[5]  205  290  384   480   580   694   824   960  1096  1226 ...
[6]  495  674  864  1060  1274  1518  1784  2056  2322  2570 ...
[7] 1169 1538 1924  2334  2792  3302  3840  4378  4892  5380 ...
...
First column is A007443: binomial transform of primes.
Second column is A178167: binomial transform of odd primes.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Rows and columns include A001043, A096277-A096279, A007443, A178167.

A287919list[dmax_]:=With[{a=Reverse[NestList[ListConvolve[{1,1},#]&,Prime[Range[dmax]],dmax-1]]},Array[Reverse[Diagonal[a,#]]&,dmax,1-dmax]];
A287919list[10] (* Generates 10 antidiagonals *) (* Paolo Xausa, Oct 31 2023 *)

A287919(m,n)=sum(k=0,m,prime(n+k)*binomial(m,k))
/* read by antidiagonals */ for(m=0,13,for(n=0,m,print1(A287919(n,m-n+1)",")))

cp's OEIS Frontend

A302334 A weighted smoothing applied to the primes as a data set: a(n) = floor(A007443(2n-1)/2^(2n-2)), where A007443 is binomial transform of primes.

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A287915 Indices of primes in A007443.

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A096280 Primes in A007443 (= binomial transform of primes).

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A007442 Inverse binomial transform of primes.

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A096279 Sums of successive sums of successive sums of successive sums of successive primes.

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A111107 Lexicographically smallest increasing sequence of primes whose binomial transform consists only of primes.

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A164738 Triangle read by rows. Row 0 = {2}; left half of row n+1 = row n, right half = row n reversed with each term replaced by the next prime.

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