cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A007443 Binomial transform of primes.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

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

References

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

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, May 21 2010

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

Views

Author

Cino Hilliard, Jun 23 2004

Keywords

Comments

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

Links

Crossrefs

Cf. A007443, A287915, A001043, A096277, A096278, A096279.
See A287915 for the corresponding indices of A007443.

Programs

  • Mathematica
    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 *)
  • PARI
    \\ 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); }
  • PARI
    for(n=1,999, ispseudoprime(A007443(n))&&print1(A007443(n)",")) \\ M. F. Hasler, Jun 02 2017

Formula

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

Extensions

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

Views

Author

M. F. Hasler, Jun 02 2017

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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)",")))
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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