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.

Previous Showing 11-16 of 16 results.

A127338 Numbers that are the sum of 11 consecutive primes.

Original entry on oeis.org

160, 195, 233, 271, 311, 353, 399, 443, 491, 539, 583, 631, 677, 725, 779, 833, 883, 931, 979, 1025, 1081, 1139, 1197, 1253, 1313, 1367, 1423, 1483, 1543, 1607, 1673, 1727, 1787, 1843, 1901, 1951, 2011, 2077, 2141, 2203, 2263, 2323, 2383, 2443, 2507
Offset: 1

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

a(n) = absolute value of coefficient of x^10 of the polynomial Product_{j=0..10} (x - prime(n+j)) of degree 11; the roots of this polynomial are prime(n), ..., prime(n+10).

Links

Crossrefs

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127339, A127340.

Programs

  • Magma
    [&+[ NthPrime(n+k): k in [0..10] ]: n in [1..70] ]; // Vincenzo Librandi, Apr 03 2011
  • Mathematica
    f[n_] := Sum[Prime[n + i], {i, 0, 10}]; Array[f, 45]
Plus @@@ Partition[ Prime@ Range@ 55, 11, 1] (* Robert G. Wilson v, Jan 13 2011 *)
  • PARI
    {m=45;k=11;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007
  • PARI
    {m=45;k=11;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

Extensions

Edited by Klaus Brockhaus, Jan 13 2007

A127334 Numbers that are the sum of 7 consecutive primes.

Original entry on oeis.org

58, 75, 95, 119, 143, 169, 197, 223, 251, 281, 311, 341, 371, 401, 431, 463, 493, 523, 559, 593, 625, 659, 689, 719, 757, 791, 827, 863, 905, 947, 991, 1027, 1063, 1099, 1139, 1171, 1211, 1247, 1281, 1313, 1351, 1395, 1441, 1479, 1519, 1561, 1603, 1643
Offset: 1

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

a(n) = absolute value of coefficient of x^6 of the polynomial Product_{j=0..6} (x - prime(n+j)) of degree 7; the roots of this polynomial are prime(n), ..., prime(n+6).

Links

Crossrefs

Cf. A001043, A011974, A034961, A034963, A034964, A082246, A127333, A127335, A127336, A127337, A127338, A127339.

Programs

  • GAP
    P:=Filtered([1..1000],IsPrime);; List([0..50],n->Sum([1+n..7+n],i->P[i])); # Muniru A Asiru, Apr 16 2018
  • Magma
    [&+[ NthPrime(n+k): k in [0..6] ]: n in [1..70] ]; // Vincenzo Librandi, Apr 03 2011
  • Maple
    seq(add(ithprime(i),i=n..6+n),n=1..50); # Muniru A Asiru, Apr 16 2018
  • Mathematica
    a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 6}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],7,1] (* Harvey P. Dale, May 14 2023 *)
  • PARI
    {m=48;k=7;for(n=0,m-1,print1(a=sum(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007
  • PARI
    {m=48;k=7;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007
  • Python
    from sympy import prime
def a(x): return sum(prime(x + n) for n in range(7))
print([a(i) for i in range(1, 50)]) # Indranil Ghosh, Mar 18 2017
  • Sage
    BB = primes_first_n(62)
L = []
for i in range(55):
    L.append(sum(BB[i+j] for j in range(7)))
L
# Zerinvary Lajos, May 14 2007

Extensions

Edited by Klaus Brockhaus, Jan 13 2007

A127339 Numbers that are the sum of 12 consecutive primes.

Original entry on oeis.org

197, 236, 276, 318, 364, 412, 460, 510, 562, 612, 662, 714, 766, 822, 880, 936, 990, 1040, 1092, 1152, 1212, 1276, 1336, 1402, 1464, 1524, 1586, 1650, 1716, 1786, 1854, 1918, 1980, 2040, 2100, 2162, 2234, 2304, 2370, 2436, 2502, 2564, 2634, 2700, 2770
Offset: 1

Views

Author

Artur Jasinski, Jan 11 2007

Keywords

Comments

a(n) = absolute value of coefficient of x^11 of the polynomial Product_{j=0..11} (x - prime(n+j)) of degree 12; the roots of this polynomial are prime(n), ..., prime(n+11).

Crossrefs

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338.

Programs

  • Magma
    [&+[ NthPrime(n+k): k in [0..11] ]: n in [1..100] ]; // Vincenzo Librandi, Apr 03 2011
  • Mathematica
    a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 11}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],12,1] (* Harvey P. Dale, May 05 2018 *)
  • PARI
    {m=45;k=12;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007
  • PARI
    {m=45;k=12;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

Extensions

Edited by Klaus Brockhaus, Jan 13 2007

A061802 Sum of n-th row of triangle of primes: 2; 2 3 2; 2 3 5 3 2; 2 3 5 7 5 3 2; ...; where n-th row contains 2n+1 terms.

Original entry on oeis.org

2, 7, 15, 27, 45, 69, 99, 135, 177, 229, 289, 357, 435, 519, 609, 709, 821, 941, 1069, 1207, 1351, 1503, 1665, 1837, 2023, 2221, 2425, 2635, 2851, 3073, 3313, 3571, 3839, 4115, 4403, 4703, 5011, 5331, 5661, 6001, 6353, 6713, 7085, 7469, 7859, 8255, 8665
Offset: 0

Views

Author

Amarnath Murthy, May 28 2001

Keywords

Comments

Row sums of A138143. - Omar E. Pol, Feb 13 2014
For n = 3..9, a(n) = 3*(n^2 - 3*n + 5). - Nicholas Drozd, Apr 10 2021

Links

Crossrefs

Cf. A001043 (first differences), A007504, A138143.
Partial sums of A011974.

Programs

  • Mathematica
    Accumulate[Join[{2},ListConvolve[{1,1},Prime[Range[100]]]]] (* Paolo Xausa, Oct 31 2023 *)
  • PARI
    { n=-1; a=q=0; forprime (p=2, prime(1001), write("b061802.txt", n++, " ", a+=p + q); q=p ) } \\ Harry J. Smith, Jul 28 2009

Formula

a(n) = a(n-1) + prime(n) + prime(n-1).
a(n) = A007504(n) + A007504(n+1) so we have the asymptotic expansion a(n) ~ n^2*log(n). - Henry Bottomley, May 30 2001

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 29 2001

A069102 a(1) = 1; a(2) = 1; a(n) = Prime[n-1] + Prime[n-2] if n > 2.

Original entry on oeis.org

1, 1, 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508
Offset: 1

Views

Author

Joseph L. Pe, Apr 06 2002

Keywords

Examples

			a(3) = Prime[2] + Prime[1] = 3 + 2 = 5.

Links

Crossrefs

Cf. A001043, A011974.

Programs

  • Mathematica
    Join[{1,1},Total/@Partition[Prime[Range[60]],2,1]] (* Harvey P. Dale, Feb 01 2024 *)
  • PARI
    print1("1,1,"); for(n=1,100,print1(prime(n)+prime(n+1)","))

Extensions

More terms from Ralf Stephan, Mar 20 2003

A164653 a(1) = 1, for n>=2: a(n) = sum of two consecutive noncomposite numbers A008578.

Original entry on oeis.org

1, 3, 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508
Offset: 1

Views

Author

Jaroslav Krizek, Aug 19 2009

Keywords

Comments

Basically these are the sums of two successive primes. - N. J. A. Sloane, Nov 16 2018
Essentially the same as A001043, A011974 and A069102.

Links

Crossrefs

Cf. A000040, A001043, A008578, A011974, A069102, A076273, A158611.

Programs

  • Mathematica
    ListConvolve[{1,1},Join[{0,1},Prime[Range[100]]]] (* Paolo Xausa, Nov 02 2023 *)

Formula

a(n) = A158611(n) + A158611(n+1).
a(n) = A008578(n-1) + A008578(n) for n >= 2.
a(n) = A076273(n-1) + 1 for n >= 2.
a(n) = A000040(n-1) + A008578(n-1) for n >= 2. - Jaroslav Krizek, Dec 13 2009

Extensions

Edited by R. J. Mathar, Aug 21 2009
Correction for change of offset in A158611 and A008578 in Aug 2009 by Jaroslav Krizek, Jan 27 2010
Formulas edited by Paolo Xausa, Nov 04 2023
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.