A001043 -id:A001043 - cp's OEIS frontend

A096277 Sum of successive sums of successive primes: a(n) = s(n) + s(n+1) where s(n) = prime(n) + prime(n+1) (A001043).

13, 20, 30, 42, 54, 66, 78, 94, 112, 128, 146, 162, 174, 190, 212, 232, 248, 266, 282, 296, 314, 334, 358, 384, 402, 414, 426, 438, 462, 498, 526, 544, 564, 588, 608, 628, 650, 670, 692, 712, 732, 756, 774, 786, 806, 844, 884, 906, 918, 934

Offset: 1

Cino Hilliard, Jun 22 2004

The first term is the only term that has a chance of being prime.

			The sums of the first two pairs of successive primes are 5 and 8. 5+8 = 13 is the first term in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001043, A000040.

Total/@Partition[Total/@Partition[Prime[Range[60]],2,1],2,1] (* Harvey P. Dale, May 10 2011 *)
Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],2] (* Paolo Xausa, Oct 31 2023 *)

f1(n,f(n)=prime(n)+prime(n+1)) = for(x=1,n,print(f(x)+f(x+1)","))

a(n) = A001043(n) + A001043(n+1) = A000040(n) + 2*A000040(n+1) + A000040(n+2). - M. F. Hasler, Jun 02 2017

Edited by M. F. Hasler, Jun 02 2017

A134651 Numbers which are the sum of two terms from A001043.

Original entry on oeis.org

10, 13, 16, 17, 20, 23, 24, 26, 29, 30, 32, 35, 36, 38, 41, 42, 44, 47, 48, 50, 54, 57, 60, 64, 65, 66, 68, 70, 72, 73, 76, 78, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98, 102, 104, 105, 108, 110, 112, 114, 117, 118, 120, 124, 125, 126, 128, 130, 132, 133, 136, 138, 140

Offset: 1

N. J. A. Sloane, Jan 25 2008

			Recall that A001043 begins with 5,8,12, ..
So 10 (5+5), 13 (5+8), 16 (8+8), 17 (5+12), 20 (8+12) are in the sequence.

B. D. Swan, Table of n, a(n) for n = 1..60000

Cf. A001043, A134650.

issum(i, vss) = {for (j = 1, #vss, if (vss[j] > i, break); for (k = 1, #vss, sv = vss[j] + vss[k]; if (sv == i, return (1)); if (sv > i, break););); return (0);}
lista(nn) = {vec = vector(nn, i, i); vss = select(i->((precprime((i-1)/2) + nextprime(i/2) == i) && (i>2)), vec); for (i = 1, nn, if (issum(i, vss), print1(i, ", ")););} \\ Michel Marcus, Oct 14 2013

A124434 LCM (least common multiple) of A001043 (sum of consecutive primes) and A001223 (difference of consecutive primes).

Original entry on oeis.org

5, 8, 12, 36, 24, 60, 36, 84, 156, 60, 204, 156, 84, 180, 300, 336, 120, 384, 276, 144, 456, 324, 516, 744, 396, 204, 420, 216, 444, 1680, 516, 804, 276, 1440, 300, 924, 960, 660, 1020, 1056, 360, 1860, 384, 780, 396, 2460, 2604, 900, 456, 924, 1416, 480, 2460

Offset: 1

Mitch Cervinka (Mitch.Cervinka(AT)eds.com), Dec 15 2006

			a(3)=12 because prime(3)=5, prime(4)=7 and lcm(7+5, 7-5) = lcm(12,2) = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001223, A001043.

LCM[Total[#],#[[2]]-#[[1]]]&/@Partition[Prime[Range[60]],2,1] (* Harvey P. Dale, Apr 19 2013 *)
Join[{5}, Table[(Prime[n + 1]^2 - Prime[n]^2)/2, {n, 2, 59}]] (* Jon Maiga, Jan 17 2019 *)

a(n) = my(p = prime(n), q = prime(n+1)); lcm(q+p, q-p); \\ Michel Marcus, Mar 15 2018

a(n) = lcm((prime(n+1)+prime(n)), (prime(n+1)-prime(n))).

a(n) = (prime(n+1)^2 - prime(n)^2)/2 for n > 1. - Jon Maiga, Jan 17 2019

A102729 Triangle read by rows: n-th row consists of lexicographically least set of n distinct terms of A001043 whose sum is minimal prime.

Original entry on oeis.org

5, 5, 8, 5, 8, 18, 5, 8, 12, 18, 5, 8, 12, 18, 24, 5, 8, 12, 18, 24, 30, 5, 8, 12, 18, 24, 30, 42, 5, 8, 12, 18, 24, 30, 42, 52, 5, 8, 12, 18, 24, 30, 36, 42, 52, 5, 8, 12, 18, 24, 30, 36, 42, 52, 84, 5, 8, 12, 18, 24, 30, 36, 42, 52, 68, 78, 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78

Offset: 1

Giovanni Teofilatto, Feb 07 2005

A001043 gives sums of consecutive primes.

			5
5,8
5,8,18
5,8,12,18
5,8,12,18,24

Cf. A001043, A102724; A102725 gives row sum.

Edited and extended by Ray Chandler, Feb 12 2005

A134650 Numbers that are the sum of two consecutive primes (i.e., in A001043) but are not the sum of two sums of two consecutive primes.

Original entry on oeis.org

5, 8, 12, 18, 52, 100, 946

Offset: 1

N. J. A. Sloane, Jan 25 2008

Numbers in A001043 but not in A134651.
Conjectured to be finite, may be complete.
a(8), if it exists, is greater than 20100000. - R. J. Mathar, Jan 26 2008

R. K. Guy, ed., Unsolved Problems, Western Number Theory Meeting, Las Vegas, 1988.

Cf. A001043, A134651, A135045.

with(numtheory): Sset := {}: for i to 15000 do Sset := `union`(Sset, {ithprime(i) + ithprime(i + 1)}) od: Sset := convert(Sset, list): for n from 1 to nops(Sset) do count := 0: s := Sset[n]: for i from 1 to n do if member(s - Sset[i], Sset) and s-Sset[i] >= s/2 then count := count + 1 fi; od: if count = 0 then printf(`%d,`, Sset[n]) fi; od: # James Sellers, Jan 28 2008

946 found by James Sellers, Jan 25 2008

A135045 Numbers in A001043 which are the sum of two terms from A001043 in a unique way.

Original entry on oeis.org

24, 30, 68, 268, 434, 520

Offset: 1

Manuel Valdivia, Feb 10 2008; definition corrected Feb 15 2008

A subsequence of A001043 INTERSECT A134651.

			24 = 12 + 12, A001043(5) = A001043(3) + A001043(3)
30 = 12 + 18, A001043(6) = A001043(3) + A001043(4)
68 = 8 + 60, A001043(11) = A001043(2) + A001043(10)
268 = 52 + 216, A001043(32) = A001043(9) + A001043(28)
434 = 24 + 410, A001043(47) = A001043(5) + A001043(46)
520 = 12 + 508, A001043(55) = A001043(3) + A001043(54)

Cf. A134651, A001043, A134650, A135548.

A171743 a(n) = A000010(A001043(n)).

Original entry on oeis.org

4, 4, 4, 6, 8, 8, 12, 12, 24, 16, 32, 24, 24, 24, 40, 48, 32, 64, 44, 48, 72, 54, 84, 60, 60, 64, 48, 72, 72, 64, 84, 132, 88, 96, 80, 120, 128, 80, 128, 160, 96, 120, 128, 96, 120, 160, 180, 120, 144, 120, 232, 128, 160, 252, 192, 216, 144, 272, 180, 184, 192, 160

Offset: 1

Giovanni Teofilatto, Dec 17 2009

Cf. A000010, A001043.

A000010 := proc(n) numtheory[phi](n); end proc: A001043 := proc(n) ithprime(n)+ithprime(n+1) ; end proc: A171743 := proc(n) A000010(A001043(n)) ; end proc: for n from 1 to 100 do printf("%d,", A171743(n)) ; od: # R. J. Mathar, Jan 21 2010

EulerPhi[Total[#]]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Mar 05 2013 *)

p=2;forprime(q=3,1e3,print1((p-1)*(q-1)", ");p=q) \\ Charles R Greathouse IV, Feb 21 2013

a(n) = phi(prime(n) + prime(n+1)).

Corrected (a(36)=204 removed) by R. J. Mathar, Jan 21 2010

A135548 Numbers in A134651 which are the sum of two terms from A001043 in a unique way.

Original entry on oeis.org

10, 13, 16, 17, 20, 23, 24, 26, 29, 30, 32, 35, 38, 41, 44, 47, 50, 57, 64, 65, 68, 70, 73, 80, 82, 83, 88, 89, 94, 95, 105, 110, 117, 118, 125, 133, 140, 143, 148, 149, 154, 157, 167, 176, 177, 178, 182, 191, 192, 200, 203, 208, 209, 215, 221, 227, 236, 242

Offset: 1

Zak Seidov, Feb 15 2008

nonn

Numbers that are the sum of two terms of A001043, but not sum of a different pair of two terms.

			Recall that A001043 begins with 5,8,12,18,24,30,36,42,52,60,68,78,84,90, ...
So 48 is not in the sequence because 48 = 18+30 = 24+24.
But 88 is in the sequence as it is only = 52+36.
And 89 is there too because = 84+5.

Cf. A001043, A134651, A135045.

issum(i, vss) = {nb = 0; for (j = 1, #vss, if (vss[j] > i, break); for (k = 1, j, sv = vss[j] + vss[k]; if (sv == i, nb++); if (sv > i, break););); return (nb == 1);}
lista(nn) = {vec = vector(nn, i, i); vss = select(i->((precprime((i-1)/2) + nextprime(i/2) == i) && (i>2)), vec); for (i = 1, nn, if (issum(i, vss), print1(i, ", ")););} \\ Michel Marcus, Oct 14 2013

Example, corrected and extended by Michel Marcus, Oct 14 2013

A162569 Primes of the form A001043(j)-3.

Original entry on oeis.org

2, 5, 97, 109, 149, 317, 337, 349, 431, 709, 769, 1009, 1061, 1117, 1129, 1217, 1297, 2003, 2029, 2069, 2129, 2153, 2237, 2377, 2411, 2437, 2777, 2909, 2927, 3089, 3109, 3229, 3359, 3533, 3557, 3631, 4129, 4337, 4603, 4789, 4903, 4937, 5021, 5167, 5563, 5737

Offset: 1

Claudio Meller, Jul 06 2009

nonn

Primes 3 less than the sum of two consecutive primes.

			2 = prime(1)+prime(2)-3 = 2+3-3 has the requested format and is added to the sequence.
5= prime(2)+prime(3)-3= 3+5-3 has the requested format and is added to the sequence.
97 = prime(15)+prime(16)-3 = 47+53-3.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Select[Table[Prime[n] + Prime[n + 1] - 3, {n, 600}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2012 *)
Select[Total/@Partition[Prime[Range[500]],2,1]-3,PrimeQ] (* Harvey P. Dale, Feb 15 2025 *)

Definition rephrased by R. J. Mathar, Aug 07 2009

A283621 Smallest term A001043(k) with k>n such that A001043(k) is multiple of A001043(n).

Original entry on oeis.org

30, 24, 24, 36, 120, 60, 144, 84, 520, 120, 204, 390, 840, 360, 300, 1120, 240, 384, 276, 288, 456, 648, 2064, 372, 396, 1428, 630, 648, 2886, 480, 1290, 3216, 828, 576, 600, 924, 1920, 990, 1360, 2112, 3240, 2604, 1920, 1560, 1584, 1230, 2604, 1350, 2280, 924

Offset: 1

Zak Seidov, Mar 12 2017

nonn

			a(1)=30 because A001043(1)=5 and A001043(6)=30=6*5,
a(2)=24 because A001043(2)=8 and A001043(5)=24=3*8,
a(5)=120 because A001043(1)=24 and A001043(17)=120=5*24.

Zak Seidov, Table of {n, m=A001043(n), k, a(n)=A001043(k), a(n)/m} for n = 1..200.

Cf. A001043 (sums of 2 successive primes).

With[{s = #},Table[k = n + 1; While[! Divisible[s[[k]], s[[n]]], k++]; s[[k]], {n, 50}]] &@ Map[Total, Partition[Prime@ Range[10^4 + 1], 2, 1]] (* Michael De Vlieger, Mar 13 2017, after Harvey P. Dale at A001043 *)

cp's OEIS Frontend

A096277 Sum of successive sums of successive primes: a(n) = s(n) + s(n+1) where s(n) = prime(n) + prime(n+1) (A001043).

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A134651 Numbers which are the sum of two terms from A001043.

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A124434 LCM (least common multiple) of A001043 (sum of consecutive primes) and A001223 (difference of consecutive primes).

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A102729 Triangle read by rows: n-th row consists of lexicographically least set of n distinct terms of A001043 whose sum is minimal prime.

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A134650 Numbers that are the sum of two consecutive primes (i.e., in A001043) but are not the sum of two sums of two consecutive primes.

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A135045 Numbers in A001043 which are the sum of two terms from A001043 in a unique way.

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A171743 a(n) = A000010(A001043(n)).

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A135548 Numbers in A134651 which are the sum of two terms from A001043 in a unique way.

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