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 1-5 of 5 results.

A076271 a(1) = 1, a(2) = 2, and for n > 2, a(n) = a(n-1) + gpf(a(n-1)), where gpf = greatest prime factor = A006530.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 88, 99, 110, 121, 132, 143, 156, 169, 182, 195, 208, 221, 238, 255, 272, 289, 306, 323, 342, 361, 380, 399, 418, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 696, 725, 754, 783, 812, 841
Offset: 1

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Author

Reinhard Zumkeller, Oct 04 2002

Keywords

Comments

a(n+1) is the smallest number such that the largest prime divisor of a(n) is the highest common factor of a(n) and a(n+1). - Amarnath Murthy, Oct 17 2002
Essentially the same as A036441(n) = a(n+1) and A180107(n) = a(n-1) (n > 1).
The equivalent sequence with A020639 = spf instead of A006530 = gpf begins a(1) = 1, a(2) = 2, and from then on we get all even numbers: a(n) = a(2) + 2*(n-2), n > 1. - M. F. Hasler, Apr 08 2015
From David James Sycamore, Apr 27 2017: (Start)
The sequence contains only one prime; a(2)=2, all other terms (excluding a(1)=1) being composite, since if a(n) for some n > 2 is assumed to be the first prime after 2, then a(n) = a(n-1) + gpf(a(n-1))= m*q+q = q*(m+1) for some integer m > 1 and some prime q. This number is composite; contradiction. Terms after a(3)=4 alternate between even and odd values since each is created by addition of a prime (odd term).
All terms a(n) arise as consecutive multiples of consecutive primes occurring in their natural ascending order, 2,3,5,7.... (A000040). The number of (consecutive) terms which arise as multiples of p(n)= A000040(n) is 1 + p(n+1)- p(n-1), namely n-th term of the sequence: 2,4,5,7,7,7,7,7,11, etc. Example: Number of multiples of 17, the 7th prime, is 1+p(8)-p(6) = 1+19-13 = 7.
For any pair of consecutive primes, p,q (p < q) a(p+q-1) = p*q, the (semiprime) term where multiples of p end and multiples of q start. Example a(7+11-1) = a(17) = 77 = 11*7, the last multiple of 7 and first multiple of 11. Every string of multiples of prime p contains the term p^2, located at a(2*p-1). E.g.: a(3)=4, a(5)=9, a(9)=25. (End)

Links

Crossrefs

Cf. A036441, A076272(n) = a(n+1) - a(n).
See also A180107.
Cf. A070229.

Programs

  • Haskell
    a076271 n = a076271_list !! (n-1)
a076271_list = iterate a070229 1  -- Reinhard Zumkeller, Nov 07 2015
  • Mathematica
    NestList[#+FactorInteger[#][[-1,1]]&,1,60] (* Harvey P. Dale, May 11 2015 *)
  • PARI
    print1(n=1);for(i=1,199,print1(","n+=A006530(n))) \\ M. F. Hasler, Apr 08 2015

Formula

a(A076274(n)) = A008578(n)^2 for all n.
a(n+1) = A070229(a(n)). - Reinhard Zumkeller, Nov 07 2015

Extensions

Edited by M. F. Hasler, Apr 08 2015

A076273 a(0) = 1, a(1) = 2; for n>1, a(n) = prime(n)+prime(n-1)-1.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 23, 29, 35, 41, 51, 59, 67, 77, 83, 89, 99, 111, 119, 127, 137, 143, 151, 161, 171, 185, 197, 203, 209, 215, 221, 239, 257, 267, 275, 287, 299, 307, 319, 329, 339, 351, 359, 371, 383, 389, 395, 409, 433, 449, 455, 461, 471, 479, 491, 507, 519, 531
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 04 2002

Keywords

Comments

Least m such that A076272(m) > A076272(m-1) for n>0; a(0)=1.
A076272(a(k)+j) = A008578(k) for k>0 and 0<=j < A075527(k-1).

Links

Crossrefs

Cf. A001043.

Programs

  • Mathematica
    nxt[{a_,b_}]:={a+1,Prime[a+1]+Prime[a]-1}; Join[{1},Transpose[ NestList[ nxt,{1,2},60]][[2]]] (* or *) Join[{1,2},Total/@Partition[Prime[ Range[ 60]],2,1]-1] (* Harvey P. Dale, Jun 12 2012 *)
  • PARI
    a(n)=if(n<1,1,if(n==1,2,prime(n)+prime(n-1)-1)) \\ Lambert Klasen, Jan 14 2005; corrected by Michel Marcus, Nov 05 2023

Formula

a(n) = A001043(n-1)-1, n>1. - R. J. Mathar, Jun 04 2020

Extensions

Simpler description from Vladeta Jovovic, Mar 29 2003
More terms from Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 14 2005

A075527 a(n) = A008578(n+3) - A008578(n+1).

Original entry on oeis.org

2, 3, 4, 6, 6, 6, 6, 6, 10, 8, 8, 10, 6, 6, 10, 12, 8, 8, 10, 6, 8, 10, 10, 14, 12, 6, 6, 6, 6, 18, 18, 10, 8, 12, 12, 8, 12, 10, 10, 12, 8, 12, 12, 6, 6, 14, 24, 16, 6, 6, 10, 8, 12, 16, 12, 12, 8, 8, 10, 6, 12, 24, 18, 6, 6, 18, 20, 16, 12, 6, 10, 14, 14, 12, 10, 10, 14, 12, 12, 18, 12
Offset: 0

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Author

Reinhard Zumkeller, Sep 22 2002

Keywords

Comments

For n>0: a(n) = A031131(n) and a(n) - a(n-1) = A075526(n).

Crossrefs

Cf. A000040, A076272, A076273.
Cf. A008578, A031131, A075526.

Extensions

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A076973 Starting with 2, largest prime divisor of the sum of all previous terms.

Original entry on oeis.org

2, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37
Offset: 1

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Author

Amarnath Murthy, Oct 22 2002

Keywords

Comments

Conjecture: start from any initial value a(1) = m >= 2 and define a(n) to be the largest prime factor of a(1)+a(2)+...+a(n-1); then a(n) = n/2 + O(log(n)) and there are infinitely many primes p such that a(2p)=p. - Benoit Cloitre, Jun 04 2003

Links

Crossrefs

From the third term onwards the sequence coincides with A076272.

Programs

  • Mathematica
    nxt[{t_,a_}]:=Module[{c=FactorInteger[t][[-1,1]]},{t+c,c}]; NestList[nxt,{2,2},80][[All,2]] (* Harvey P. Dale, May 21 2017 *)

Formula

a(n) = p(m) (the m-th prime), where m is the smallest index such that n <= p(m+1) + p(m) - 2. - Max Alekseyev, Oct 21 2008

Extensions

More terms from Sascha Kurz, Jan 22 2003

A180101 a(0)=0, a(1)=1; thereafter a(n) = largest prime factor of sum of all previous terms.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41
Offset: 0

Views

Author

N. J. A. Sloane, Jan 16 2011

Keywords

Comments

More precisely, a(n) = A006530 applied to sum of previous terms.
Inspired by A175723.
Except for initial terms, same as A076272, but the simple definition warrants an independent entry.

Crossrefs

Cf. A006530, A076272, A175723, A180107 (partial sums).

Formula

For the purposes of this paragraph, regard 0 as the (-1)st prime and 1 as the 0th prime. Conjectures: All primes appear; the primes appear in increasing order; the k-th prime p(k) appears p(k+1)-p(k-1) times (cf. A031131); and p(k) appears for the first time at position A164653(k) (sums of two consecutive primes). These assertions are stated as conjectures only because I have not written out a formal proof, but they are surely true.
Showing 1-5 of 5 results.

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