A001043 -id:A001043 - cp's OEIS frontend

A037165 a(n) = prime(n)*prime(n+1) - prime(n) - prime(n+1).

1, 7, 23, 59, 119, 191, 287, 395, 615, 839, 1079, 1439, 1679, 1931, 2391, 3015, 3479, 3959, 4619, 5039, 5615, 6395, 7215, 8447, 9599, 10199, 10811, 11447, 12095, 14111, 16379, 17679, 18767, 20423, 22199, 23399, 25271, 26891, 28551, 30615, 32039

Offset: 1

Armand Turpel (armandt(AT)unforgettable.com)

a(n) is also the Frobenius number of the numerical semigroup generated by prime(n) and prime(n+1). - Victoria A Sapko (vsapko(AT)math.unl.edu), Feb 21 2001

Joshua Oliver, Table of n, a(n) for n = 1..2000 (first 1000 terms from Vincenzo Librandi)
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Frobenius numbers for k successive primes: this sequence (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), A138992 (k=6), A138993 (k=7), A138994 (k=8).

Cf. A001043, A006094.

[NthPrime(n)*NthPrime(n+1)-NthPrime(n)-NthPrime(n+1): n in [1..45]]; // Vincenzo Librandi, Dec 18 2012

f[n_] := FrobeniusNumber[{Prime[n], Prime[n + 1]}]; Array[f, 41] (* Robert G. Wilson v, Aug 04 2012 *)
Times@@#-Total[#]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Dec 27 2015 *)

a(n)=my(p=prime(n),q=nextprime(p+1)); p*q-p-q \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A006094(n) - A001043(n). - Michel Marcus, Mar 02 2019

A034964 Sums of five consecutive primes.

Original entry on oeis.org

28, 39, 53, 67, 83, 101, 119, 139, 161, 181, 199, 221, 243, 263, 287, 311, 331, 351, 373, 395, 421, 449, 473, 497, 517, 533, 559, 587, 617, 647, 683, 707, 733, 759, 787, 811, 839, 863, 891, 917, 941, 961, 991, 1023, 1057, 1089, 1123, 1151, 1169, 1193

Offset: 1

Patrick De Geest, Oct 15 1998

Except for the first term, all terms are odd. - Alonso del Arte, Dec 30 2011

			a(1) = prime(1+0) + prime(1+1) + prime(1+2) + prime(1+3) + prime(1+4) = 2 + 3 + 5 + 7 + 11 = 28.
a(2) = prime(2+0) + prime(2+1) + prime(2+2) + prime(2+3) + prime(2+4) = 3 + 5 + 7 + 11 + 13 = 39.

Owen O'Shea and Underwood Dudley, The Magic Numbers of the Professor, Mathematical Association of America (2007), p. 62

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A011974, A034707.

Cf. A131686 (sums of five consecutive squares of primes).

[&+[ NthPrime(n+k): k in [0..4] ]: n in [1..100] ]; // Vincenzo Librandi, Apr 03 2011

A034964:=n->add(ithprime(i), i=n..n+4): seq(A034964(n), n=1..50); # Wesley Ivan Hurt, Sep 14 2014

Plus@@@Partition[Prime[Range[100]],5,1] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

a(n) = sum(k=n, n+4, prime(k)); \\ Michel Marcus, Sep 03 2016

first(n) = {my(psum = 28, pr = List([2,3,5,7,11]), res = List([28])); for(i=2,n, psum -= pr[1]; listpop(pr, 1); listput(pr, nextprime(pr[4] + 1)); psum += pr[5]; listput(res, psum)); res} \\ David A. Corneth, Oct 14 2017

BB = primes_first_n(60)
L = []
for i in range(55):
    L.append(BB[i]+BB[i+1]+BB[i+2]+BB[i+3]+BB[i+4])
L # Zerinvary Lajos, May 14 2007

a(n) = Sum_{i=n..n+4} prime(i). - Wesley Ivan Hurt, Sep 14 2014

Offset changed to 1 by Joerg Arndt, Sep 04 2016

A074924 Numbers whose square is the sum of two successive primes.

Original entry on oeis.org

6, 10, 12, 24, 42, 48, 62, 72, 84, 90, 110, 120, 122, 174, 204, 208, 220, 232, 240, 264, 306, 326, 336, 372, 386, 408, 410, 444, 454, 456, 468, 470, 474, 522, 546, 550, 594, 600, 630, 640, 642, 686, 740, 750, 762, 766, 788, 802, 852, 876, 882, 920, 936, 970

Offset: 1

Zak Seidov, Oct 02 2002

nonn

			6^2 = 17 + 19, 1610^2 = 1296041 + 1296059.

Zak Seidov, Table of n, a(n) for n = 1..22054 (all terms up to 10^6).

Square roots of squares in A001043.
Cf. A062703 (the squares), A061275 (lesser of the primes), A064397 (index of that prime).
Cf. A064397 (numbers n such that prime(n) + prime(n+1) is a square), A071220 (prime(n) + prime(n+1) is a cube), A074925 (n^3 is sum of 2 consecutive primes).

filter:= proc(n) local t; t:= n^2/2; prevprime(ceil(t)) + nextprime(floor(t)) = n^2 end proc:
select(filter, [$3..1000]); # Robert Israel, Nov 19 2024

Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[50000]],2,1]),IntegerQ] (* Harvey P. Dale, Oct 04 2014 *)
f@n_ := Sqrt@Select[(2*Range@n)^2, # == Plus @@ NextPrime[#/2, {-1, 1}] &]; f@485 (* Hans Rudolf Widmer, Nov 19 2024 *)

is(n)=if(n%2, return(0)); nextprime(n^2/2+1)+precprime(n^2/2)==n^2 \\ Charles R Greathouse IV, Apr 29 2015

select( {is_A074924(n)=!bittest(n=n^2,0) && precprime(n\2)+nextprime(n\/2)==n}, [1..999]) \\ M. F. Hasler, Jan 03 2020

A74924=[6]; apply( A074924(n)={while(n>#A74924, my(N=A74924[#A74924]); until( is_A074924(N+=2),);A74924=concat(A74924,N));A74924[n]}, [1..99]) \\ M. F. Hasler, Jan 03 2020

from itertools import count, islice
from sympy import nextprime, prevprime
def agen(): # generator of terms
    for k in count(4, step=2):
        kk = k*k
        if prevprime(kk//2+1) + nextprime(kk//2-1) == kk:
            yield k
print(list(islice(agen(), 54))) # Michael S. Branicky, May 24 2022

a(n) = sqrt(A062703(n)). - Zak Seidov, May 26 2013

a(n) = A000040(i) + A000040(i+1) with i = A064397(n) = A000720(A061275(n)). - M. F. Hasler, Jan 03 2020

Crossrefs section corrected and extended by M. F. Hasler, Jan 03 2020

A011974 2 followed by the numbers that are the sum of 2 successive primes.

Original entry on oeis.org

2, 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, 520

Offset: 1

N. J. A. Sloane

All the terms in the sequence, except for a(2), are even. - K. D. Bajpai, Aug 26 2014

			From _K. D. Bajpai_, Aug 26 2014: (Start)
a(6) = 24 is in the sequence because prime(5) + prime(6) = 11 + 13 = 24.
a(8) = 36 is in the sequence because prime(7) + prime(8) = 17 + 19 = 36.
(End)

Archimedeans Problems Drive, Eureka, 26 (1963), 12.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040.

Join[{2},Total/@Partition[Prime[Range[40]],2,1]] (* Harvey P. Dale, May 04 2013 *)

Essentially same as A001043.

The terms a(40) to a(56) from K. D. Bajpai, Aug 26 2014

A096342 Primes of the form p*q + p + q, where p and q are two successive primes.

Original entry on oeis.org

11, 23, 47, 167, 251, 359, 479, 719, 1847, 2111, 2591, 3719, 6719, 7559, 8819, 10607, 12539, 14591, 19319, 27551, 29231, 31319, 51071, 53819, 68111, 97967, 149759, 155219, 172199, 177239, 195359, 199799, 234239, 273527, 305783, 314711, 339863

Offset: 1

Giovanni Teofilatto, Jun 29 2004

nonn

a(n) == 3 mod 4.
Primes arising in A126148. - Jonathan Vos Post, Mar 08 2007
Number of primes <10^n: 0, 3, 8, 15, 26, 49, 99, 220, 514, 1228, 2991, 7746, 20218, 54081, ..., . - Robert G. Wilson v

			a(4)=167 because 11*13 + 11 + 13=167.

Vincenzo Librandi and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Librandi)

Cf. A000040, A001043, A006094, A126148, A126199.

a = {}; Do[p = Prime[n]Prime[n + 1] + Prime[n] + Prime[n + 1]; If[ PrimeQ[p], AppendTo[a, p]], {n, 110}]; a (* Robert G. Wilson v, Jul 01 2004 *)
Select[Times@@#+Total[#]&/@Partition[Prime[Range[200]],2,1],PrimeQ] (* Harvey P. Dale, Nov 25 2018 *)

list(lim)=my(v=List(),p=2,t); forprime(q=3,, t=p*q+p+q; if (t>lim, return(Set(v))); if(isprime(t), listput(v,t)); p=q) \\ Charles R Greathouse IV, Sep 15 2015

More terms from Robert G. Wilson v, Jul 02 2004

A034965 Primes that are sum of five consecutive primes.

Original entry on oeis.org

53, 67, 83, 101, 139, 181, 199, 263, 311, 331, 373, 421, 449, 587, 617, 647, 683, 733, 787, 811, 839, 863, 941, 991, 1123, 1151, 1193, 1361, 1381, 1579, 1609, 1801, 1831, 1861, 1949, 1979, 2081, 2113, 2143, 2221, 2273, 2297, 2357, 2423, 2459, 2689, 2731

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			53 = 5 + 7 + 11 + 13 + 17.
373 = 67 + 71 + 73 + 79 + 83.

Harvey P. Dale and Syed Iddi Hasan, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A001043, A011974, A034707, A152468. Also Cf. A034964, of which this sequence is a subset.

ts_prod_n:=proc(n) local i,ans; ans:=[ ]: for i from 1 to n do if isprime(ithprime(i)+ithprime(i+1)+ithprime(i+2)+ithprime(i+3)+ithprime(i+4))= 'true' then ans:=[op(ans), ithprime(i)+ithprime(i+1)+ithprime(i+2)+ithprime(i+3)+ithprime(i+4) ]: fi od: end: ts_prod_n(701); # Jani Melik, May 05 2006

Select[Table[Plus@@Prime[Range[n, n + 4]], {n, 200}], PrimeQ] (* Alonso del Arte, Dec 30 2011 *)
Select[Total/@Partition[Prime[Range[200]],5,1],PrimeQ] (* Harvey P. Dale, May 24 2012 *)

Corrected example by Paul S. Coombes, Dec 29 2011

A062703 Squares that are the sum of two consecutive primes.

Original entry on oeis.org

36, 100, 144, 576, 1764, 2304, 3844, 5184, 7056, 8100, 12100, 14400, 14884, 30276, 41616, 43264, 48400, 53824, 57600, 69696, 93636, 106276, 112896, 138384, 148996, 166464, 168100, 197136, 206116, 207936, 219024, 220900, 224676, 272484, 298116, 302500, 352836

Offset: 1

Jason Earls, Jul 11 2001

			prime(7) + prime(8) = 17 + 19 = 36 = 6^2.

Michael S. Branicky, Table of n, a(n) for n = 1..22054 (terms 1..100 from Harry J. Smith)

Squares in A001043. See A226524 for cubes.
Cf. A074924 (square roots), A061275 (lesser of the primes), A064397 (index of that prime).
Cf. A080665 (same with sum of three consecutive primes).

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{m = Floor[n/2]}, s = PrevPrim[m] + NextPrim[m]; If[s == n, True, False]]; Select[ Range[550], f[ #^2] &]^2
t := Table[Prime[n] + Prime[n + 1], {n, 15000}]; Select[t, IntegerQ[Sqrt[#]] &] (* Carlos Eduardo Olivieri, Feb 25 2015 *)

{for(n=1,100,(p=precprime(n^2/2))+nextprime(p+2) == n^2 && print1(n^2", "))} \\ Zak Seidov, Feb 17 2011

A062703(n)=A074924(n)^2 \\ M. F. Hasler, Jan 03 2020

from itertools import count, islice
from sympy import nextprime, prevprime
def agen(): # generator of terms
    for k in count(4, step=2):
        kk = k*k
        if prevprime(kk//2+1) + nextprime(kk//2-1) == kk:
            yield kk
print(list(islice(agen(), 37))) # Michael S. Branicky, May 24 2022

a(n) = A074924(n)^2.

a(n) = A000040(i) + A000040(i+1) with i = A064397(n) = A000720(A061275(n)). - M. F. Hasler, Jan 03 2020

Edited by Robert G. Wilson v, Mar 02 2003

Edited (crossrefs completed, obsolete PARI code deleted) by M. F. Hasler, Jan 03 2020

A072669 Primes of the form prime(x) + prime(x+1) - 1.

Original entry on oeis.org

7, 11, 17, 23, 29, 41, 59, 67, 83, 89, 127, 137, 151, 197, 239, 257, 307, 359, 383, 389, 409, 433, 449, 461, 479, 491, 547, 557, 563, 599, 617, 647, 683, 701, 739, 751, 761, 797, 809, 827, 839, 863, 881, 929, 977, 1063, 1087, 1103, 1163, 1229, 1249, 1283, 1289, 1319, 1373

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Aug 12 2002

Consider m such that prime(m) + prime(m+1) = prime(k) + 1 for some k; sequence gives prime(k).

A118072 is a subsequence, hence this sequence is infinite on Dickson's conjecture. - Charles R Greathouse IV, Apr 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001043, A092738, A072666, A072667.

f[n_] := Prime[n] + Prime[n + 1] - 1; f[ # ] & /@ Select[ Range[120], PrimeQ[ f[ # ]] &] (* Robert G. Wilson v, Apr 14 2004 *)
Select[Total[#]-1&/@Partition[Prime[Range[200]],2,1],PrimeQ] (* Harvey P. Dale, Aug 06 2012 *)

p=2;forprime(q=3,1e6,if(isprime(p+q-1),print1(p+q-1", "));p=q) \\ Charles R Greathouse IV, Apr 18 2013

Definition reworded by Jorge Coveiro, Apr 12 2004

Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A127337 Numbers that are the sum of 10 consecutive primes.

Original entry on oeis.org

129, 158, 192, 228, 264, 300, 340, 382, 424, 468, 510, 552, 594, 636, 682, 732, 780, 824, 870, 912, 954, 1008, 1060, 1114, 1164, 1216, 1266, 1320, 1376, 1434, 1494, 1546, 1596, 1650, 1704, 1752, 1800, 1854, 1914, 1974, 2030, 2084, 2142, 2192, 2250, 2310, 2374

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of coefficient of x^9 of the polynomial Product_{j=0..9} (x - prime(n+j)) of degree 10; the roots of this polynomial are prime(n), ..., prime(n+9).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127338, A127339.

[&+[ NthPrime(n+k): k in [0..9] ]: n in [1..90] ]; // Vincenzo Librandi, Apr 03 2011

A127337 := proc(n)
    local i ;
    add(ithprime(n+i),i=0..9) ;
end proc:
seq(A127337(n),n=1..30) ; # R. J. Mathar, Apr 24 2023

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 9}]], {x, 1, 50}]; a
Table[Plus@@Prime[Range[n, n + 9]], {n, 50}] (* Alonso del Arte, Feb 15 2011 *)
ListConvolve[ConstantArray[1, 10], Prime[Range[50]]]
Total/@Partition[Prime[Range[60]],10,1] (* Harvey P. Dale, Jan 31 2013 *)

{m=46;k=10;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=46;k=10;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

from sympy import prime
def a(n): return sum(prime(n + i) for i in range(10))
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Dec 09 2021

# faster version for generating initial segment of sequence
from sympy import nextprime
def aupton(terms):
    alst, plst = [], [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
    for n in range(terms):
        alst.append(sum(plst))
        plst = plst[1:] + [nextprime(plst[-1])]
    return alst
print(aupton(47)) # Michael S. Branicky, Dec 09 2021

a(n) = A127336(n)+A000040(n+9). - R. J. Mathar, Apr 24 2023

Edited by Klaus Brockhaus, Jan 13 2007

A007443 Binomial transform of primes.

Original entry on oeis.org

2, 5, 13, 33, 83, 205, 495, 1169, 2707, 6169, 13889, 30993, 68701, 151469, 332349, 725837, 1577751, 3413221, 7349029, 15751187, 33616925, 71475193, 151466705, 320072415, 674721797, 1419327223, 2979993519, 6245693407, 13068049163

Offset: 1

N. J. A. Sloane

Equals row sums of triangle A164738. Example: a(4) = 33 = sum of terms in row 4 of triangle A164738: (2, 3, 5, 3, 5, 7, 5, 3). - Gary W. Adamson, Aug 23 2009

It might have been more natural to define this sequence with offset 0, which would also make the formula simpler. Then a(n) would be the first term of the sequence obtained from the primes by applying n times the operation "take sums of successive terms", Ts(k) = s(k)+s(k+1). - M. F. Hasler, Jun 02 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..3000 (terms 1..1000 from Vincenzo Librandi)
Vaclav Kotesovec, Plot of a(n) / (2^n * n * log(n)) for n = 2..20000
N. J. A. Sloane, Transforms

Cf. A164738.
Cf. A001043, A096277, A096278, A096279. See A287915 for indices of primes.
First differences give A178167.

a:=n->add(binomial(n-1,k-1)*ithprime(k),k=1..n): seq(a(n),n=1..30); # Muniru A Asiru, Oct 23 2018

A007443[n_]:=Sum[Binomial[n-1,k-1]Prime[k],{k,n}];Array[A007443,50] (* or *)
Module[{nmax=50,b},b=Prime[Range[nmax]];Join[{2},Table[First[b=ListConvolve[{1,1},b]],nmax-1]]] (* Paolo Xausa, Oct 31 2023 *)

A007443(n)=sum(k=1,n,binomial(n-1,k-1)*prime(k)) \\ M. F. Hasler, Jun 02 2017

a(n) = Sum_{k=1..n} binomial(n-1,k-1)*prime(k). - M. F. Hasler, Jun 02 2017

G.f.: Sum_{k>=1} prime(k)*x^k/(1 - x)^k. - Ilya Gutkovskiy, Apr 21 2019

More terms from Vladimir Joseph Stephan Orlovsky, May 21 2010

cp's OEIS Frontend

A037165 a(n) = prime(n)*prime(n+1) - prime(n) - prime(n+1).

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A034964 Sums of five consecutive primes.

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A074924 Numbers whose square is the sum of two successive primes.

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A011974 2 followed by the numbers that are the sum of 2 successive primes.

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A096342 Primes of the form p*q + p + q, where p and q are two successive primes.

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Mathematica

PARI

Extensions

A034965 Primes that are sum of five consecutive primes.

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