A096277 -id:A096277 - cp's OEIS frontend

A007443 Binomial transform of primes.

2, 5, 13, 33, 83, 205, 495, 1169, 2707, 6169, 13889, 30993, 68701, 151469, 332349, 725837, 1577751, 3413221, 7349029, 15751187, 33616925, 71475193, 151466705, 320072415, 674721797, 1419327223, 2979993519, 6245693407, 13068049163

Offset: 1

N. J. A. Sloane

Equals row sums of triangle A164738. Example: a(4) = 33 = sum of terms in row 4 of triangle A164738: (2, 3, 5, 3, 5, 7, 5, 3). - Gary W. Adamson, Aug 23 2009

It might have been more natural to define this sequence with offset 0, which would also make the formula simpler. Then a(n) would be the first term of the sequence obtained from the primes by applying n times the operation "take sums of successive terms", Ts(k) = s(k)+s(k+1). - M. F. Hasler, Jun 02 2017

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

Seiichi Manyama, Table of n, a(n) for n = 1..3000 (terms 1..1000 from Vincenzo Librandi)
Vaclav Kotesovec, Plot of a(n) / (2^n * n * log(n)) for n = 2..20000
N. J. A. Sloane, Transforms

Cf. A164738.
Cf. A001043, A096277, A096278, A096279. See A287915 for indices of primes.
First differences give A178167.

a:=n->add(binomial(n-1,k-1)*ithprime(k),k=1..n): seq(a(n),n=1..30); # Muniru A Asiru, Oct 23 2018

A007443[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];Array[A007443,50] (* or *)
Module[{nmax=50,b},b=Prime[Range[nmax]];Join[{2},Table[First[b=ListConvolve[{1,1},b]],nmax-1]]] (* Paolo Xausa, Oct 31 2023 *)

A007443(n)=sum(k=1,n,binomial(n-1,k-1)*prime(k)) \\ M. F. Hasler, Jun 02 2017

a(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). - M. F. Hasler, Jun 02 2017

G.f.: Sum_{k>=1} prime(k)*x^k/(1 - x)^k. - Ilya Gutkovskiy, Apr 21 2019

More terms from Vladimir Joseph Stephan Orlovsky, May 21 2010

A096278 Sums of successive sums of successive sums of successive primes.

Original entry on oeis.org

33, 50, 72, 96, 120, 144, 172, 206, 240, 274, 308, 336, 364, 402, 444, 480, 514, 548, 578, 610, 648, 692, 742, 786, 816, 840, 864, 900, 960, 1024, 1070, 1108, 1152, 1196, 1236, 1278, 1320, 1362, 1404, 1444, 1488, 1530, 1560, 1592, 1650, 1728, 1790, 1824

Offset: 1

Cino Hilliard, Jun 22 2004

If we consider the m-fold iterated "take sums of successive terms" operation acting on the primes, then for all m >= 1, the first term is always odd (and the only odd term); it is prime for m=1, 2, 4, 8, 21, 24, 27, 31, 40, 98,..., but not for m=3 (the present sequence). [Edited by M. F. Hasler, Jun 02 2017]

			The first two terms of SS order 1 is 13 and 20. 13+20 = 33 the first term of the sequence.

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

Cf. A000040, A096277, A001043.

Ss:= L -> L[1..-2]+L[2..-1]:
(Ss@@3)([seq(ithprime(i),i=1..100)]); # Robert Israel, Dec 28 2022

Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],3] (* Paolo Xausa, Oct 31 2023 *)

f(n) = return(prime(n)+prime(n+1))
f1(n) = return(f(n)+f(n+1))
f2(n) = return(f1(n)+f1(n+1))
g(n) = for(x=1,n,print1(f2(x)","))

A096278(n,m=3)=for(k=0,m,prime(n+k)*binomial(m,k)) \\ or, to get a list:
A096278_vec(Nmax,m=3,v=primes(Nmax+m))=sum(k=0,m,binomial(m,k)*v[1+k,k-1-m]) \\ Alternatively, do m times v=v[^1]+v[^-1]. - M. F. Hasler, Jun 02 2017

Let f(n) = prime(n) + prime(n+1) f1(n) = f(n)+f(n+1) : SS of order 1 Then f2(n) = f1(n)+f1(n) : SS of order 2 is the general term of this sequence.
a(n) = A096277(n) + A096277(n+1). - M. F. Hasler, Jun 02 2017
a(n) = prime(n)+3*prime(n+1)+3*prime(n+2)+prime(n+3). - Robert Israel, Dec 28 2022

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], ", "); ) }

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

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)",")))

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A007443 Binomial transform of primes.

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A096278 Sums of successive sums of successive sums of successive primes.

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

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

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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.

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