A001043 -id:A001043 - cp's OEIS frontend

A126199 a(n) = prime(n)*prime(n+1) + prime(n) + prime(n+1).

11, 23, 47, 95, 167, 251, 359, 479, 719, 959, 1215, 1595, 1847, 2111, 2591, 3239, 3719, 4215, 4895, 5327, 5919, 6719, 7559, 8819, 9995, 10607, 11231, 11879, 12539, 14591, 16895, 18215, 19319, 20999, 22799, 24015, 25911, 27551, 29231, 31319, 32759

Offset: 1

Jonathan Vos Post, Mar 08 2007

Cf. A000040, A001043, A006094, A126148, A096342, A180617.

f[n_] := Block[{p = Prime[n], q = Prime[n + 1]}, p*q + p + q]; Array[f, 42] (* Robert G. Wilson v, Mar 09 2007 *)
Times@@#+Total[#]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Nov 01 2017 *)

a(n) = A180617(n) - 1. - Omar E. Pol, Dec 08 2019

More terms from Robert G. Wilson v, Mar 09 2007

A206329 Squarefree sums of 2 successive primes.

Original entry on oeis.org

5, 30, 42, 78, 138, 186, 210, 222, 258, 330, 390, 410, 434, 462, 618, 762, 786, 798, 906, 930, 946, 966, 978, 1002, 1030, 1230, 1290, 1334, 1374, 1410, 1446, 1482, 1518, 1542, 1606, 1722, 1758, 1770, 1794, 1830, 1866, 1878, 1938, 1974, 2006, 2022, 2190, 2226

Offset: 1

Zak Seidov, Feb 06 2012

nonn

Intersection of A001043 and A005117, both infinite, but is their intersection infinite?

Also note that the only prime is a(1)=5 and there are no semiprimes (products of 2 primes A001358).

			a(1)=5=A001043(1)=A005117(4), a(2)=30=A001043(6)=A005117(19), a(3)=42=A001043(8)=A005117(28).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001043, A001358, A005117, A206462.

N:= 1000: # to get the first N terms
count:= 0:
p:= 2:
while count < N do
  pp:= nextprime(p);
  if numtheory:-issqrfree(p+pp) then
    count:= count+1;
    A[count]:= p+pp;
  fi;
  p:= pp;
od:
seq(A[i],i=1..N);
# Robert Israel, Jul 20 2014

Select[Table[Prime[n] + Prime[n + 1], {n, 300}], SquareFreeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

p=2;forprime(q=3,1e4,if(issquarefree(p+q),print1(p+q", "));p=q) \\ Charles R Greathouse IV, Feb 08 2012

A247245 Least multiple of n that is the sum of 2 successive primes.

Original entry on oeis.org

5, 8, 12, 8, 5, 12, 42, 8, 18, 30, 198, 12, 52, 42, 30, 112, 68, 18, 152, 60, 42, 198, 138, 24, 100, 52, 162, 84, 696, 30, 186, 128, 198, 68, 210, 36, 222, 152, 78, 120, 410, 42, 172, 308, 90, 138, 564, 144, 882, 100, 204, 52, 1272, 162, 330, 112, 456, 696, 472, 60, 1220, 186, 630, 128, 390

Offset: 1

Zak Seidov, Nov 28 2014

nonn

a(n) = n if n is a term of A001043 (on graph this corresponds to the lower bound).

			5 is a term because prime(1) + prime(2) = 2 + 3 = 5 = 5*1 (k = 5),
8 is a term because prime(2) + prime(3) = 3 + 5 = 8 = 4*2 (k = 4),
198 is a term because prime(25) + prime(26) = 97 + 101 = 198 = 18*11 (k = 18).

Michel Marcus, Table of n, a(n) for n = 1..10000
Zak Seidov, Table of {n, i, a(n), k=a(n)/n} for n = 1..1000

Cf. A001043.

is(n) = (precprime((n-1)/2) + nextprime(n/2) == n) && (n>2); \\ A001043
a(n) = my(k=1); while (!is(k*n), k++); k*n; \\ Michel Marcus, Oct 06 2021

a(n) = smallest number, of the form k*n (k >= 1), that is the sum of 2 successive primes.

A336371 Numbers k such that gcd(k, prime(k) + prime(k-1)) > 1.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Clark Kimberling, Oct 04 2020

nonn

Cf. A000040, A001043, A336366, A336370, A336372, A336373.

p[n_] := Prime[n];
u = Select[Range[2, 200], GCD[#, p[#] + p[# - 1]] == 1 &]  (* A336370 *)
v = Select[Range[2, 200], GCD[#, p[#] + p[# - 1]] > 1 &]   (* A336371 *)
Prime[u]  (* A336372 *)
Prime[v]  (* A336373 *)

In the following table, p(k) = A000040(k) = prime(k).
  k    p(k)   p(k)+p(k-1)   gcd
   3         5          1
   5         8          1
   7        12          4
  11        18          1
  13        24          6
and 3 are in A336370; 4 and 6 are in this sequence; 3 and 5 are in A336372; 7 and 13 are in A336373.

Offset corrected by Mohammed Yaseen, Jun 02 2023

A347137 a(n) = Sum_{d|n} phi(d) * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes, and phi is Euler totient function.

Original entry on oeis.org

1, 4, 7, 14, 11, 28, 17, 46, 41, 44, 23, 98, 29, 68, 77, 146, 35, 164, 41, 154, 119, 92, 51, 322, 97, 116, 223, 238, 59, 308, 67, 454, 161, 140, 187, 574, 77, 164, 203, 506, 83, 476, 89, 322, 451, 204, 99, 1022, 229, 388, 245, 406, 111, 892, 253, 782, 287, 236, 119, 1078, 127, 268, 697, 1394, 319, 644, 137, 490

Offset: 1

Antti Karttunen, Aug 24 2021

Dirichlet convolution of Euler phi (A000010) with the prime shift function (A003961). Multiplicative because both A000010 and A003961 are.
Dirichlet convolution of the identity function (A000027) with the prime shifted phi (A003972).
Möbius transform of A347136.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000027, A000040, A001043, A003961, A003972, A008683, A151800, A347122, A347136 (inverse Möbius transform).

Cf. also A018804, A347237.

f[p_, e_] := (q = NextPrime[p])^e + (p - 1)*(q^e - p^e)/(q - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347137(n) = sumdiv(n,d,eulerphi(n/d)*A003961(d));

a(n) = Sum_{d|n} A000010(n/d) * A003961(d).
a(n) = Sum_{d|n} d * A003972(n/d).
a(n) = Sum_{d|n} A008683(n/d) * A347136(d).
a(n) = A347122(n) + 2*A000010(n).
a(A000040(n)) = A001043(n) - 1.
Multiplicative with a(p^e) = q(p)^e + (p-1)*(q(p)^e - p^e)/(q(p) - p), where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Sep 16 2023

A349660 Numbers which are the sum of a prime and the square of the next prime.

Original entry on oeis.org

11, 28, 54, 128, 180, 302, 378, 548, 864, 990, 1400, 1718, 1890, 2252, 2856, 3534, 3780, 4550, 5108, 5400, 6314, 6968, 8004, 9498, 10298, 10710, 11552, 11988, 12878, 16242, 17288, 18900, 19458, 22340, 22950, 24800, 26726, 28052, 30096, 32214, 32940, 36662

Offset: 1

Karl-Heinz Hofmann, Nov 24 2021

			a(2) = 3 + 5^2 = 28; a(3) = 5 + 7^2 = 54.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A140511, A000040, A001248, A036690, A001223, A001043, A036689, A349661, A124129.

nterms=100;Table[Prime[n]+Prime[n+1]^2,{n,nterms}] (* Paolo Xausa, Nov 24 2021 *)

a(n) = prime(n) + prime(n+1)^2; \\ Michel Marcus, Nov 24 2021

from sympy import sieve;
for n in range(1,10001): print(sieve[n] + sieve[n+1]**2)

a(n) = prime(n) + prime(n+1)^2.
a(n) = A000040(n) + A001248(n+1).
a(n) = A036690(n+1) - A001223(n).
a(n) = A001043(n) + A036689(n+1). - Michel Marcus, Nov 24 2021

A096215 Greatest primes not greater than the sum of two succeeding primes.

Original entry on oeis.org

5, 7, 11, 17, 23, 29, 31, 41, 47, 59, 67, 73, 83, 89, 97, 109, 113, 127, 137, 139, 151, 157, 167, 181, 197, 199, 199, 211, 211, 239, 257, 263, 271, 283, 293, 307, 317, 317, 337, 349, 359, 367, 383, 389, 389, 409, 433, 449, 449, 461, 467, 479, 491, 503, 509, 523

Offset: 1

Reinhard Zumkeller, Jul 28 2004

nonn

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

Cf. A001043, A007917.

NextPrime[ListConvolve[{1,1},Prime[Range[100]]]+1,-1] (* Paolo Xausa, Nov 02 2023 *)

a(n) = A007917(A001043(n)).

A100479 a(n) = prime(2n-1) + prime(2n).

Original entry on oeis.org

5, 12, 24, 36, 52, 68, 84, 100, 120, 138, 152, 172, 198, 210, 222, 258, 276, 300, 320, 340, 360, 384, 396, 434, 456, 472, 492, 520, 540, 558, 576, 618, 630, 668, 696, 712, 740, 762, 786, 810, 840, 864, 882, 906, 924, 946, 978, 1002, 1030, 1064, 1104, 1132

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Nov 22 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000
C. Caldwell Lists of small primes.

Cf. A092163, A001043, A031215, A031368.

[(&+[NthPrime(2*n+j-1): j in [0..1]]): n in [1..70]]; // G. C. Greubel, Apr 05 2023

A100479 := proc(n)
    ithprime(2*n-1)+ithprime(2*n) ;
end proc:
seq(A100479(n),n=1..50) ; # R. J. Mathar, Jan 20 2025

Total/@Partition[Prime[Range[110]],2] (* Harvey P. Dale, Apr 20 2016 *)

[sum(nth_prime(2*n+j-1) for j in range(2)) for n in range(1, 71)] # G. C. Greubel, Apr 05 2023

a(n) = A001043(2n-1). - R. J. Mathar, Apr 20 2009

a(n) = A031368(n) + A031215(n). - G. C. Greubel, Apr 05 2023

A103271 a(n) = (prime(n)+prime(n+1)) mod 4.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Yasutoshi Kohmoto, Jan 27 2005

nonn

The number of 2's among the first N terms are: count(10^3) = 381, count (10^4) = 4137, count(10^5) = 42638, count(10^6) = 437423, count(10^7) = 4448503. - M. F. Hasler, Apr 27 2016

In terms of vectors a = (p(n),p(n+1)) mod 4, as considered in the preprint arxiv:1603.03720, the 2's group together the cases a = (1,1) and (3,3) and 0's cumulate cases (1,3) and (3,1). Assuming that the two subcases of each case have roughly the same probabilities, the above counts (i.e., percentage of 44.5% : 55.5% at 10^7) are compatible with the data in the 2nd table on bottom of p.14 where respective percentages vary from 44.8% : 55.1% (at 10^10) to 46% : 54% (at 10^12). I found that at p(n) ~ 10^80, the percentages become closer than 49% : 51%. - M. F. Hasler, May 12 2016

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.

Cf. A001043, A039702.

seq(ithprime(n)+ithprime(n+1) mod 4, n=1..150); # Emeric Deutsch, May 31 2005

Table[Mod[Prime@ n + Prime[n + 1], 4], {n, 120}] (* Michael De Vlieger, Apr 27 2016 *)
Mod[Total[#],4]&/@Partition[Prime[Range[120]],2,1] (* Harvey P. Dale, Mar 16 2025 *)

a(n) = (prime(n) + prime(n+1)) % 4; \\ Michel Marcus, Apr 14 2016

a(n) = A001043(n) mod 4. - Michel Marcus, Apr 14 2016

More terms from Emeric Deutsch, May 31 2005

Prepended a(1) = 1, Joerg Arndt, Apr 14 2016

A154634 Numbers that are the first of two consecutive primes having a sum that is the product of two consecutive numbers.

Original entry on oeis.org

5, 13, 19, 43, 103, 113, 229, 293, 349, 463, 739, 773, 859, 1171, 1429, 1483, 3079, 3229, 3319, 3823, 4003, 4273, 5449, 6781, 6899, 7129, 7369, 7499, 7873, 7993, 10729, 11173, 11321, 11779, 12241, 12553, 13523, 13693, 14533, 14699, 17203, 17389

Offset: 1

J. M. Bergot, Jan 13 2009

Is the sequence mostly uniformly distributed or do clusters occur for the products? One could also find sums of 2n consecutive primes equaling the product of 2n numbers.

			For the pair of consecutive primes 1429 and 1433, their sum is 2862=53*54.
773 and 787 are consecutive primes. 773+787 = 1560 = 39*40, hence 773 is in the sequence. - _Klaus Brockhaus_, Jan 15 2009

Klaus Brockhaus, Table of n, a(n) for n=1..1000 [From _Klaus Brockhaus_, Jan 15 2009]

[ p: p in PrimesUpTo(18000) | r*(r+1) eq s where r is Iroot(s, 2) where s is p+NextPrime(p) ]; // Klaus Brockhaus, Jan 15 2009

isA002378 := proc(n) local a; a := floor(sqrt(n)) ; RETURN( a*(a+1) = n ) ; end: for i from 1 to 5000 do p := ithprime(i) ; a001043 := p+nextprime(p) ; if isA002378(a001043) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jan 15 2009
a := proc (n) local p, s: p := ithprime(n): s := p+nextprime(p): if type((1/2)*sqrt(1+4*s)-1/2, integer) = true then p else end if end proc: seq(a(n), n = 1 .. 3000); # Emeric Deutsch, Jan 15 2009

sp2Q[{a_,b_}]:=Module[{s=Floor[Sqrt[a+b]]},a+b==s(s+1)]; Select[Partition[ Prime[ Range[2100]],2,1],sp2Q][[All,1]] (* Harvey P. Dale, Jun 28 2020 *)

{A000040(i): A001043(i) in A002378}. - R. J. Mathar, Jan 15 2009

Corrected and extended by several correspondents, Jan 15 2009

cp's OEIS Frontend

A126199 a(n) = prime(n)*prime(n+1) + prime(n) + prime(n+1).

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A206329 Squarefree sums of 2 successive primes.

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A247245 Least multiple of n that is the sum of 2 successive primes.

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A336371 Numbers k such that gcd(k, prime(k) + prime(k-1)) > 1.

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A347137 a(n) = Sum_{d|n} phi(d) * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes, and phi is Euler totient function.

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Mathematica

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A349660 Numbers which are the sum of a prime and the square of the next prime.

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A096215 Greatest primes not greater than the sum of two succeeding primes.

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Mathematica

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A100479 a(n) = prime(2n-1) + prime(2n).

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