A001043 -id:A001043 - cp's OEIS frontend

A160830 Integer part of the product of two consecutive primes divided by their sum.

1, 1, 2, 4, 5, 7, 8, 10, 12, 14, 16, 19, 20, 22, 24, 27, 29, 31, 34, 35, 37, 40, 42, 46, 49, 50, 52, 53, 55, 59, 64, 66, 68, 71, 74, 76, 79, 82, 84, 87, 89, 92, 95, 97, 98, 102, 108, 112, 113, 115, 117, 119, 122, 126, 129, 132, 134, 136, 139, 140, 143, 149, 154, 155, 157

Offset: 1

Cino Hilliard, May 27 2009

nonn

The differences a(n+1) - a(n) appear to grow without bound while the difference 2 appears to occur infinitely often.

			a(5) = floor(prime(5)*prime(6)/(prime(5)+prime(6))) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A006094.

[Floor(NthPrime(n)*NthPrime(n+1)/(NthPrime(n)+NthPrime(n+1))): n in [1..100]]; // G. C. Greubel, Apr 30 2018

a:= n-> (l-> floor(mul(i,i=l)/add(i,i=l)))([ithprime(n+i)$i=0..1]):
seq(a(n), n=1..65);  # Alois P. Heinz, Sep 20 2024

Table[Floor[Prime[n]*Prime[n+1]/(Prime[n] +Prime[n+1])], {n, 1, 100}] (* G. C. Greubel, Apr 30 2018 *)
Floor[Times@@#/Total[#]&/@Partition[Prime[Range[100]],2,1]] (* Harvey P. Dale, Sep 20 2024 *)

g(x) = p1=prime(x);p2=prime(x+1);y=p1*p2/(p1+p2);floor(y);
g1(n) = for(j=1,n,print1(g(j)","))

a(n) = floor(prime(n)*prime(n+1)/(prime(n)+prime(n+1))) where prime(.) = A000040(.).

a(n) = floor( A006094(n)/A001043(n) ). - R. J. Mathar, May 29 2009.

Inserted "two" in definition - R. J. Mathar, May 29 2009

A162571 Palindromes which are sums of two consecutive primes.

Original entry on oeis.org

5, 8, 222, 434, 696, 828, 2112, 2992, 4224, 4554, 6336, 8448, 8888, 20202, 21712, 21812, 22722, 23832, 25652, 25952, 26862, 27672, 29092, 29292, 41114, 42024, 42724, 43334, 43734, 44544, 47174, 47974, 60106, 61116, 62526, 62626, 63936, 64146, 64446, 64946

Offset: 1

Claudio Meller, Jul 06 2009

			a(3) = 222= A002113(32) = A001043(29). a(4) = 434= A002113(53) = A001043(47).

Harvey P. Dale and Chai Wah Wu, Table of n, a(n) for n = 1..10000, first 1500 term from Harvey P. Dale.

isA002113 := proc(n) dgs := convert(n,base,10) ; for i from 1 to nops(dgs)/2 do if op(i,dgs) <> op(-i,dgs) then RETURN(false); end if; end do: true; end proc:
A001043 := proc(n) ithprime(n)+ithprime(n+1) ; end proc:
for n from 1 do ps := A001043(n) ; if isA002113(ps) then printf("%d,\n",ps) ; fi; end do: # R. J. Mathar, Aug 14 2009

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Select[Total/@ Partition[Prime[Range[5000]],2,1],palQ] (* Harvey P. Dale, Dec 30 2013 *)

A001043 INTERSECT A002113.

Keyword:base added by R. J. Mathar, Aug 14 2009

A167597 The isolated nonprimes that are the sum of two successive primes.

Original entry on oeis.org

12, 18, 30, 42, 60, 138, 198, 240, 462, 600, 618, 810, 828, 882, 1230, 1290, 1320, 1428, 1482, 1620, 1668, 1722, 1878, 2088, 2112, 2688, 2970, 3330, 3390, 3768, 4002, 4092, 4242, 4260, 4482, 4518, 5100, 5280, 5418, 5502, 5520, 5652, 6090, 6198, 6300, 6450

Offset: 1

Juri-Stepan Gerasimov, Nov 07 2009

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001043, A001097.

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(i,i=3..N+1,2)]):
A001043:= convert(Primes[1..-2] + Primes[2..-1],set):
Primes:= convert(Primes,set):
A:= map(`+`,Primes intersect map(`-`,Primes,2),1) intersect A001043:
sort(convert(A,list)); # Robert Israel, Jan 18 2016

A001043 INTERSECT A014574. [R. J. Mathar, May 30 2010]

Corrected (60 inserted, 178 replaced by 198, 462 inserted....) by R. J. Mathar, May 30 2010

A180617 Sum of divisors of the product of two consecutive primes.

Original entry on oeis.org

12, 24, 48, 96, 168, 252, 360, 480, 720, 960, 1216, 1596, 1848, 2112, 2592, 3240, 3720, 4216, 4896, 5328, 5920, 6720, 7560, 8820, 9996, 10608, 11232, 11880, 12540, 14592, 16896, 18216, 19320, 21000, 22800, 24016, 25912, 27552, 29232, 31320, 32760, 34944, 37248, 38412

Offset: 1

Thomas Kellar, Sep 12 2010

nonn

			a(1) = sigma(2*3) = 12, a(2) = sigma(3*5) = 24.

A distant relative of A054640.

Cf. A000203, A001043, A006094, A126199.

[(1+NthPrime(n))*(1+NthPrime(n+1)): n in [1..50]]; // Vincenzo Librandi, Feb 16 2015

DivisorSigma[1,#]&/@(Times@@@Partition[Prime[Range[50]],2,1]) (* Harvey P. Dale, Apr 04 2015 *)
Table[Prime[n]*Prime[n+1]+Prime[n]+Prime[n+1]+1,{n,1,30}] (* Metin Sariyar, Dec 08 2019 *)

for (n=1,10, i=prod(x=n,n+1,prime(x)); p=sigma(i); print1(p, ", "); )

a(n)=my(p=prime(n)); (p+1)*(nextprime(p+1)+1) \\ Charles R Greathouse IV, Feb 16 2015

a(n) = A000203(A006094(n)). - Omar E. Pol, Dec 08 2019
a(n) = A006094(n) + A001043(n) + 1. - Metin Sariyar, Dec 08 2019
a(n) = A126199(n) + 1 (after above formula). - Omar E. Pol, Dec 08 2019

More terms from Vincenzo Librandi, Feb 16 2015

Name simplified by Omar E. Pol, Dec 08 2019

A191583 Sum of the distinct prime divisors of prime(n) + prime(n+1).

Original entry on oeis.org

5, 2, 5, 5, 5, 10, 5, 12, 15, 10, 19, 18, 12, 10, 7, 9, 10, 2, 28, 5, 21, 5, 45, 36, 16, 22, 17, 5, 42, 10, 48, 69, 28, 5, 10, 20, 7, 21, 24, 13, 10, 36, 5, 23, 16, 48, 40, 10, 24, 23, 61, 10, 46, 129, 20, 28, 10, 139, 36, 52, 5, 10, 108, 18, 17, 5, 169, 24

Offset: 1

Michel Lagneau, Jun 07 2011

nonn

a(n) = A008472(A001043(n)). [Reinhard Zumkeller, Jun 28 2011]

			a(6) = 10 because prime(6) + prime(7) = 13+17 = 30 = 2*3*5 and 2+3+5 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

with(numtheory):for n from 1 to 100 do:x:=ithprime(n)+ithprime(n+1):y:=factorset(x):n1:=nops(y):s:=0:for  k from 1 to n1 do:s:=s+y[k]:od:printf(`%d, `,s):od:

sdpd[n_]:=Total[Transpose[FactorInteger[n]][[1]]]; sdpd/@(Total/@ Partition[ Prime[Range[70]],2,1]) (* Harvey P. Dale, Mar 18 2012 *)

vecsum(v)=sum(i=1,#v,v[i])
p=2;forprime(q=3,1e3,print1(vecsum(factor(p+q)[,1])", ");p=q)
\\ Charles R Greathouse IV, Jun 12 2011

A206462 Primes p such that p + nextprime(p) is a squarefree number (A005117).

Original entry on oeis.org

2, 13, 19, 37, 67, 89, 103, 109, 127, 163, 193, 199, 211, 229, 307, 379, 389, 397, 449, 463, 467, 479, 487, 499, 509, 613, 643, 661, 683, 701, 719, 739, 757, 769, 797, 859, 877, 883, 887, 911, 929, 937, 967, 983, 997, 1009, 1093, 1109, 1163, 1201, 1237, 1279

Offset: 1

Zak Seidov, Feb 07 2012

nonn

			13 + 17 = 30 = A206329(2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001043, A001358, A005117, A206329.

a206462 n = a206462_list !! (n-1)
a206462_list = map (a000040 . (+ 1)) $
                   elemIndices 1 $ map a008966 a001043_list
-- Reinhard Zumkeller, Feb 08 2012

Prime[Select[Range[200], Abs[MoebiusMu[Prime[#] + Prime[# + 1]]] == 1 &]] (* Alonso del Arte, Feb 08 2012 *)

a(n) + nextprime(a(n)) = A206329(n).

A008966(A001043(A049084(a(n)))) = 1. [Reinhard Zumkeller, Feb 08 2012]

A240052 2nd arithmetic derivative of products of 2 successive prime numbers (A006094).

Original entry on oeis.org

1, 12, 16, 21, 44, 31, 60, 41, 56, 92, 72, 71, 124, 123, 140, 240, 244, 448, 121, 384, 236, 297, 176, 161, 249, 284, 247, 540, 191, 608, 221, 272, 380, 912, 520, 380, 1024, 371, 428, 912, 852, 508, 1472, 433, 696, 297, 293, 705, 860, 493, 716, 1456, 668, 512, 924, 636, 1188, 552, 669, 764, 2112, 1340, 521, 1504, 951, 1836, 672, 1176, 1300, 1107, 1076, 737, 908, 1520, 641, 776, 661, 821, 1647, 1416, 1828

Offset: 1

Freimut Marschner, Mar 31 2014

nonn

The first arithmetic derivative of products of 2 successive prime numbers (A006094) is the sum of 2 successive prime numbers (A001043). A001043 = (A006094)’. The second arithmetic derivative is a(n)=( A001043)’ = (A006094)’’.

			(2*3)’ = 1*3+2*1 = 5; (5)’ = 1; (2^2)’ = 2*2^1 = 2*2 = 4.

Freimut Marschner, Table of n, a(n) for n = 1..429
Wikipedia, Arithmetic derivative
Wikipedia, p-derivation

Cf. A006094, A001043.

Cf. A003415 (1st derivative), A068346(2nd derivative).

a240052 = a068346 . a006094  -- Reinhard Zumkeller, Apr 15 2014

with(numtheory); P:=proc(q) local a,b,c,p,n;
for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]);
c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
print(c); od; end: P(10^3); # Paolo P. Lava, Apr 01 2014

a(n) = (A006094(n))''.

a(n) = A068346(A006094(n))= A003415(A003415(A006094(n))).

A288632 Primes p such that (p+q)/6 is prime, q is the next prime after p.

Original entry on oeis.org

5, 7, 13, 19, 37, 67, 89, 109, 127, 307, 379, 389, 449, 487, 499, 683, 719, 769, 877, 929, 937, 1009, 1163, 1297, 1523, 1559, 1567, 1831, 1933, 1979, 2153, 2213, 2221, 2269, 2389, 2459, 2659, 2803, 2857, 2909, 3037, 3089

Offset: 1

Zak Seidov, Jun 12 2017

nonn

Among first 2*10^6 primes there are 61953 terms of the sequence; e.g., a(60000)=31320053 and (31320053+31320061)/6=10440019=A000040(691876).

Except for the case 7-5=2 the minimal value of first differences is 4.

			5+7=6*2, 7+11=6*3, 13+17=6*5, 19+23=6*7, 37+41=6*11, 67+71=6*23.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043.

Select[Prime[Range[1000]], PrimeQ[(# + NextPrime[#])/6] &]

A291339 Primes p such that p^3*q^3 + p^3 + q^3 is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 7, 19, 37, 47, 83, 89, 107, 137, 181, 251, 257, 349, 379, 569, 631, 653, 677, 691, 797, 823, 839, 863, 883, 919, 1009, 1021, 1223, 1229, 1361, 1423, 1571, 1609, 1831, 1873, 1907, 1993, 2053, 2113, 2207, 2239, 2293, 2309, 2579, 2833, 3137, 3319, 3593, 3673

Offset: 1

K. D. Bajpai, Aug 22 2017

nonn

			a(2) = 3 is prime; 5 is the next prime: 3^3*5^3 + 3^3 + 5^3 = 27*125 + 27 + 125 = 3527 that is a prime.
a(3) = 7 is prime; 11 is the next prime: 7^3*11^3 + 7^3 + 11^3 = 343*1331 + 343 + 1331 = 458207 that is a prime.

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

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241.

[p: p in PrimesUpTo (5000) | IsPrime(p^3*q^3+p^3+q^3)];

select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^3+nextprime(p)^3)]), [seq(p, p=1..10^4)]);

Prime@Select[Range[1000], PrimeQ[Prime[#]^3*Prime[# + 1]^3 + Prime[#]^3 + Prime[# + 1]^3] &]
Select[Partition[Prime[Range[600]],2,1],PrimeQ[Times@@(#^3)+Total[#^3]]&][[;;,1]] (* Harvey P. Dale, Apr 28 2025 *)

is(n) = my(q=nextprime(n+1)); ispseudoprime(n^3*q^3+n^3+q^3)
forprime(p=1, 3700, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 22 2017

list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p3+q3), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017

A336370 Numbers k such that gcd(k, prime(k) + prime(k-1)) = 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 119, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 169, 171

Offset: 1

Clark Kimberling, Oct 04 2020

nonn

			In the following table, p(k) = A000040(k) = prime(k).
  k    p(k)   p(k)+p(k-1)   gcd
  2     3         5          1
  3     5         8          1
  4     7        12          4
  5    11        18          1
  6    13        24          6
2 and 3 are in this sequence; 4 and 6 are in A336371; 3 and 5 are in A336372; 7 and 13 are in A336373.

Cf. A000040, A001043, A336366, A336371, A336372, A336373.

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

Offset corrected by Mohammed Yaseen, Jun 02 2023

cp's OEIS Frontend

A160830 Integer part of the product of two consecutive primes divided by their sum.

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A162571 Palindromes which are sums of two consecutive primes.

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A167597 The isolated nonprimes that are the sum of two successive primes.

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A180617 Sum of divisors of the product of two consecutive primes.

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A191583 Sum of the distinct prime divisors of prime(n) + prime(n+1).

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A206462 Primes p such that p + nextprime(p) is a squarefree number (A005117).

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A240052 2nd arithmetic derivative of products of 2 successive prime numbers (A006094).

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