A071216 -id:A071216 - cp's OEIS frontend

A202063 Position of first appearance of n-th prime in A071216.

2, 3, 1, 8, 25, 9, 11, 21, 19, 69, 24, 29, 46, 23, 60, 115, 51, 111, 32, 82, 129, 185, 132, 71, 106, 155, 63, 116, 84, 203, 54, 77, 58, 145, 108, 87, 289, 93, 67, 443, 254, 460, 292, 76, 350, 300, 447, 86, 397, 124, 284, 808, 128, 335, 136, 547, 742, 361

Offset: 1

Zak Seidov, Dec 10 2011

nonn

Zak Seidov, Table of n, a(n) for n = 1..2000

Cf. A071216 (largest prime factor of sum of 2 successive primes).

a(n) = primepi(A188815(n)) (conjectured). - Michel Marcus, Jul 05 2017

A071215 Number of distinct prime factors of sum of 2 successive primes.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 2, 3, 1, 3, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 3, 3, 3, 4, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 2, 3, 4, 3, 3, 4, 3, 2, 2, 3, 2, 3, 3, 4, 4, 3, 4, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 2, 4, 3

Offset: 1

Labos Elemer, May 17 2002

			Prime(6) = 13 and prime(7) = 17. 13 + 17 = 30 = 2 * 3 * 5, which has three distinct prime factors, hence a(6) = 3.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001043, A001221, A071216, A251609 (greedy inverse).

Table[PrimeNu[Prime[n] + Prime[n + 1]], {n, 105}] (* Jean-François Alcover, Oct 21 2013 *)

A071215(n)=omega(prime(n)+prime(n+1)) \\ M. F. Hasler, Jul 23 2007

a(n) = omega(prime(n) + prime(n + 1)) = A001221(A001043(n)), where omega is the number of distinct prime factors function.

A230518 Smallest prime p = a(n) such that the sum of p and the next prime has n distinct prime factors.

Original entry on oeis.org

2, 5, 13, 103, 1783, 15013, 285283, 9699667, 140645501, 4127218087, 100280245063, 5625398263453, 202666375276361, 11602324073775431, 438272504610946003, 21828587281891445047, 1156915125940246587913, 66595945348137856405747, 4632891063696575353839163

Offset: 1

Jean-François Alcover, Oct 22 2013

nonn

			30 = 13+17 is the earliest case with 3 prime divisors, so a(3) = 13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..100

Cf. A001043, A071215, A071216, A098037, A105418.

Clear[a]; a[_] = 0; Do[p = Prime[k]; q = Prime[k+1]; n = PrimeNu[p+q]; If[a[n] == 0, a[n] = p; Print["a(", n, ") = p = ", p, ", q = ", q]], {k, 1, 10^9}]; Table[a[n], {n, 1, 10}]

a(n) = {p = 2; while (omega(p+nextprime(p+1)) != n, p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 22 2013

step(Fvec)=my([n,f]=Fvec,v=List(),t);for(i=1,#f~,t=f;t[i,2]++;listput(v,[n*f[i,1],t]);t=f;t[i,1]=nextprime(t[i,1]+1);if(i==#f~||t[i,1]1,1,prime(i))),v=[[factorback(f),f]],t); if(!bad(v[1][1]),return(precprime(v[1][1]/2))); v=vecsort(step(v[1]),1); while(bad(v[1][1]), v=vecsort(concat(step(v[1]),v[2..#v]),1,8)); precprime(v[1][1]/2); \\ Charles R Greathouse IV, Oct 22 2013

a(n) > (1/2 + o(1)) n^n. - Charles R Greathouse IV, Oct 22 2013

a(11)-a(19) from Charles R Greathouse IV, Oct 22 2013

A071217 Numbers k such that the largest prime factor of the sum of successive primes p(k) + p(k+1) is greater than k.

Original entry on oeis.org

1, 9, 11, 12, 19, 23, 24, 29, 31, 32, 51, 54, 58, 63, 67, 71, 75, 76, 77, 84, 86, 87, 93, 95, 97, 103, 108, 110, 124, 128, 136, 151, 158, 159, 164, 169, 188, 191, 192, 200, 202, 205, 208, 210, 211, 216, 227, 232, 241, 243, 245, 246, 247, 252, 265, 273, 278, 282

Offset: 1

Labos Elemer, May 17 2002

nonn

			p(9) + p(10) = 23 + 29 = 52 = 2*2*13 and 13 > 10, so index 9 is here; it is the 2nd term.

Cf. A001043, A006530, A071216.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=pf[Prime[n+1]+Prime[n]]; If[Greater[s, n], Print[n]], {n, 1, 1000}]

A071216(k) > k.

A071218 Numbers k such that the largest prime factor of the sum of the two consecutive primes prime(k) + prime(k+1) is at most k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 13, 14, 15, 16, 17, 18, 20, 21, 22, 25, 26, 27, 28, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 78, 79, 80, 81, 82, 83, 85, 88, 89, 90, 91, 92, 94

Offset: 1

Labos Elemer, May 17 2002

nonn

			k=25: prime(25) + prime(26) = 97 + 101 = 198 = 2*3*3*11 and 11 < 25, so 25 is in this sequence;

Cf. A006530, A001043, A071216, A071217, A071219.

Select[Range[94], FactorInteger[Prime[#] + Prime[# + 1]][[-1, 1]] <= # &] (* Giovanni Resta, Jul 13 2018 *)

isok(n) = vecmax(factor(prime(n)+prime(n+1))[,1]) <= n; \\ Michel Marcus, Jul 09 2018

If A071216(k) <= k, then k is in this sequence.

Edited by Jon E. Schoenfield, Jul 08 2018

A071219 Numbers m such that the largest prime factor of prime(m) + prime(m+1) equals m.

Original entry on oeis.org

2, 3, 439

Offset: 1

Labos Elemer, May 17 2002

a(4), if it exists, is larger than 10^7.

a(4) > 5*10^14, if it exists. - Giovanni Resta, Jul 14 2018

			Numbers x such that A006530(A001043(x)) = x.
x = 2 is a term: p(2) + p(3) = 3 + 5 = 8 with largest factor = 2 = x.
x = 3 is a term: p(3) + p(4) = 5 + 7 = 12 with largest factor = 3 = x.
x = 439 is a term: p(439) + p(440) = 3067 + 3079 = 6146 = 2*7*439 = 14x.

Cf. A006530, A001043, A071216-A071218.

DeleteCases[#, 0] &@ MapIndexed[Boole[#1 == First@ #2] First@ #2 &, Map[FactorInteger[Total@ #][[-1, 1]] &, Partition[Prime@ Range[10^6], 2, 1]]] (* Michael De Vlieger, Aug 09 2017 *)

Edited by N. J. A. Sloane, Aug 09 2017

A076559 Greatest prime divisor of n-th interprime: (prime(n) + prime(n+1))/2.

Original entry on oeis.org

2, 3, 3, 3, 5, 3, 7, 13, 5, 17, 13, 7, 5, 5, 7, 5, 2, 23, 3, 19, 3, 43, 31, 11, 17, 7, 3, 37, 5, 43, 67, 23, 3, 5, 11, 5, 11, 17, 11, 5, 31, 3, 13, 11, 41, 31, 5, 19, 11, 59, 5, 41, 127, 13, 19, 5, 137, 31, 47, 3, 5, 103, 13, 7, 3, 167, 19, 29, 13, 89, 11, 37, 47, 127, 193, 131, 19

Offset: 2

Zak Seidov, Oct 19 2002

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A071216.

A076559 := proc(n)
    A006530((ithprime(n)+ithprime(n+1))/2) ;
end proc:
seq(A076559(n),n=2..120) ;  # R. J. Mathar, May 10 2023

gpf[n_] := FactorInteger[n][[-1, 1]]; p = Select[Range[405], PrimeQ]; gpf /@ ((p[[2 ;; -2]] + p[[3 ;; -1]])/2) (* Amiram Eldar, Aug 29 2019 *)

a(n) = A006530(A024675(n-1)). - R. J. Mathar, May 10 2023

A361170 The leading column of the table of primes in the top row and subsequent rows defined by the GPF of Pascal-alike sums of previous rows.

Original entry on oeis.org

2, 5, 7, 3, 5, 5, 3, 2, 3, 3, 2, 7, 3, 2, 7, 3, 5, 2, 7, 3, 5, 3, 5, 5, 3, 2, 7, 7, 7, 5, 7, 5, 2, 7, 7, 3, 5, 7, 3, 2, 2, 2, 2, 7, 3, 5, 3, 3, 2, 3, 2, 2, 3, 2, 7, 3, 2, 7, 7, 5, 5, 7, 3, 2, 3, 3, 2, 5, 2, 7, 7, 3, 5, 3, 2, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2

Offset: 1

R. J. Mathar, May 10 2023

nonn

			The table starts
  2  3  5  7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
  5  2  3  3  3  5  3  7 13  5 17 13  7  5  5  7  5  2 23  3
  7  5  3  3  2  2  5  5  3 11  5  5  3  5  3  3  7  5 13 11
  3  2  3  5  2  7  5  2  7  2  5  2  2  2  3  5  3  3  3 11
  5  5  2  7  3  3  7  3  3  7  7  2  2  5  2  2  3  3  7  7
  5  7  3  5  3  5  5  3  5  7  3  2  7  7  2  5  3  5  7  3
  3  5  2  2  2  5  2  2  3  5  5  3  7  3  7  2  2  3  5  2
  2  7  2  2  7  7  2  5  2  5  2  5  5  5  3  2  5  2  7  5

Mihai Caragiu, Alexandru Zaharescu and Mohammad Zaki, An analogue of the Proth-Gilbreath conjecture. Far East Journal of Mathematical Sciences, 81(1) (2013), 1-12.

Cf. A006530, A071216 (row n=2).

T361170 := proc(n,k)
    option remember ;
    if n = 1 then
        ithprime(k) ;
    else
        A006530(procname(n-1,k+1)+procname(n-1,k)) ;
    end if;
end proc:
A361170 := proc(n)
    T361170(n,1) ;
end proc:
seq(A361170(n),n=1..120) ;

a(n) = T(n,1) where T(1,k) = prime(k), T(n,k) = A006530( T(n-1,k+1) + T(n-1,k)).

cp's OEIS Frontend

A202063 Position of first appearance of n-th prime in A071216.

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A071215 Number of distinct prime factors of sum of 2 successive primes.

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A230518 Smallest prime p = a(n) such that the sum of p and the next prime has n distinct prime factors.

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A071217 Numbers k such that the largest prime factor of the sum of successive primes p(k) + p(k+1) is greater than k.

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A071218 Numbers k such that the largest prime factor of the sum of the two consecutive primes prime(k) + prime(k+1) is at most k.

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A071219 Numbers m such that the largest prime factor of prime(m) + prime(m+1) equals m.

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A076559 Greatest prime divisor of n-th interprime: (prime(n) + prime(n+1))/2.

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A361170 The leading column of the table of primes in the top row and subsequent rows defined by the GPF of Pascal-alike sums of previous rows.

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