A076273 -id:A076273 - cp's OEIS frontend

A076272 Largest prime factor of A076271(n): A006530(A076271(n)).

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

Offset: 1

Reinhard Zumkeller, Oct 04 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

See A180101 for a variant.

Differences[NestList[#+FactorInteger[#][[-1,1]]&,1,100]] (* Paolo Xausa, Dec 09 2023 *)

a(n) = A076271(n+1) - A076271(n) for all n;

a(A076273(k)+j) = A008578(k) for k>0 and 0 <= j < A075527(k-1).

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

Reinhard Zumkeller, Sep 22 2002

nonn

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

Cf. A000040, A076272, A076273.

Cf. A008578, A031131, A075526.

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

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

Jaroslav Krizek, Aug 19 2009

nonn

Basically these are the sums of two successive primes. - N. J. A. Sloane, Nov 16 2018

Essentially the same as A001043, A011974 and A069102.

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

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

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

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

A226534 a(n) = (p(n+1) + p(n) - 1) mod (p(n+1) - p(n) + 1) where p(n) is the n-th prime.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 1, 2, 2, 4, 2, 2, 4, 1, 6, 2, 1, 2, 2, 4, 1, 3, 5, 2, 2, 4, 2, 1, 14, 2, 1, 2, 1, 2, 6, 4, 4, 3, 1, 2, 8, 2, 4, 2, 6, 4, 4, 2, 1, 2, 2, 7, 3, 1, 6, 2, 1, 2, 2, 3, 14, 2, 2, 4, 2, 2, 1, 2, 1, 4, 5, 4, 2, 1, 1, 2, 2, 8, 2, 2, 4, 2, 3, 1, 2, 5, 2, 2, 4, 9, 2, 2, 8, 1, 3, 2

Offset: 1

Juri-Stepan Gerasimov, Aug 31 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Table[Mod[Prime[p + 1] + Prime[p] - 1, Prime[p + 1] - Prime[p] + 1], {p, 100}] (* Alonso del Arte, Jan 18 2014 *)
Mod[Total[#]-1,#[[2]]-#[[1]]+1]&/@Partition[Prime[Range[100]],2,1] (* Harvey P. Dale, Mar 16 2023 *)

a(n)=lift(Mod(prime(n+1)+prime(n)-1,prime(n+1)-prime(n)+1)) /* Ralf Stephan, Sep 03 2013 */

a(n) = A076273(n+1) mod A076368(n+1).

A366274 a(n) is the least k such that prime(n+1+k) >= prime(n)+prime(n+1).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 9, 10, 10, 10, 13, 13, 13, 14, 14, 15, 15, 16, 18, 20, 20, 19, 19, 18, 22, 24, 24, 25, 27, 27, 27, 29, 28, 29, 30, 31, 31, 33, 33, 32, 34, 37, 39, 38, 39, 40, 40, 41, 42, 42, 43, 42, 43, 43, 43

Offset: 1

Patrick Butler, Oct 05 2023

nonn

a(n) is the number of primes between prime(n) and prime(n) + prime(n+1).

Conjecture: for n >= 3, a(n) < n.

			For n = 5 prime(n) = 11. prime(5) + prime(6) = 11+13=24.  The 4th prime after 13 is 29 which is the next prime after 13 greater than or equal to 24. So a(5) = 4.

Patrick Butler, Table of n, a(n) for n = 1..10000

Cf. A000720, A001043, A076273, A098084, A098085.

R:= 1: pn:= 2: pn1:= 3: p:=5: m:= 4: pp:= 7:
for n from 2 to 100 do
  pn:= pn1; pn1:= nextprime(pn1);
  while pp <= pn + pn1 do m:= m+1; pp:= nextprime(pp); od;
  R:= R, m-n-1;
od:
R; # Robert Israel, Oct 31 2023

A366274[n_]:=PrimePi[Prime[n]+Prime[n+1]-1]-n;Array[A366274,100] (* Paolo Xausa, Dec 09 2023 *)

a(n) = my(k=1, q=prime(n)+prime(n+1)); while(prime(n+k) < q, k++); k; \\ Michel Marcus, Oct 06 2023

m=0
#list here is a list of prime numbers A000040.
def a(n):
    global list
    sum= list[n]+list[n+1]
    i=n+2
    while True:
        if(list[i]>=sum):
            m=i
            break
        i=i+1
    k = m-(n+1)
    return k
#
#calculate the terms of the sequence a(n).
seq = []
for n in range(0,firstN):
   seq.append(a(n))

a(n) = A000720(A001043(n)-1)-n = A000720(A076273(n+1))-n. - Paolo Xausa, Dec 09 2023

cp's OEIS Frontend

A076272 Largest prime factor of A076271(n): A006530(A076271(n)).

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

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

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A226534 a(n) = (p(n+1) + p(n) - 1) mod (p(n+1) - p(n) + 1) where p(n) is the n-th prime.

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A366274 a(n) is the least k such that prime(n+1+k) >= prime(n)+prime(n+1).

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