A054643 -id:A054643 - cp's OEIS frontend

A298302 The first of three consecutive primes the sum of which is equal to the sum of three consecutive heptagonal numbers.

17, 967, 7477, 15877, 17093, 24337, 69467, 99689, 123983, 241333, 375773, 457307, 501077, 525983, 604411, 654587, 772001, 780347, 1007099, 1023037, 1124593, 1192651, 1206497, 1423921, 1488797, 1598791, 1610809, 1692071, 1809221, 2297759, 2538623, 3017849

Offset: 1

Colin Barker, Jan 16 2018

nonn

			17 is in the sequence because 17+19+23 (consecutive primes) = 59 = 7+18+34 (consecutive hexagonal numbers).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..100 from Colin Barker)

Cf. A000040, A000566, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298272, A298273, A298301.

L=List(); forprime(p=2, 4000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(120*t-519, &sq) && (sq-21)%30==0, u=(sq-21)\30; listput(L, p))); Vec(L)

A098726 Take three consecutive primes starting with the n-th prime. Calculate d(i,j) = abs(prime(i) - prime(j)), for all {i,j}, i.e., all possible differences. a(n) is the number of distinct differences (which can be either 3 or 2).

Original entry on oeis.org

3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3

Offset: 1

Labos Elemer, Oct 05 2004

nonn

a(n) = 2 iff the consecutive prime differences are equal.

It appears that a(n) = 2 for n in A064113. - Michel Marcus, Jul 27 2017

Cf. A080370, A054643, A064113.

k=3;t=Table[Abs[Prime[n+i]-Prime[n+j]], {i, 0, k-1}, {j, 0, k-1}]; u=Delete[Union[Flatten[t]], 1];a(n)=Length[u]

Name edited by Michel Marcus, Jul 27 2017

A298102 The first of five consecutive integers the sum of which is equal to the sum of five consecutive prime numbers.

Original entry on oeis.org

77, 279, 293, 327, 347, 353, 401, 437, 509, 641, 675, 683, 785, 803, 839, 885, 947, 961, 1169, 1177, 1193, 1239, 1325, 1337, 1395, 1433, 1461, 1501, 1545, 1639, 1683, 1715, 1731, 1777, 1809, 1915, 1955, 1989, 2031, 2059, 2139, 2145, 2345, 2387, 2393, 2431

Offset: 1

Colin Barker, Jan 12 2018

nonn

Also: Number m such that 5 * m + 10 is the sum of 5 consecutive primes. - David A. Corneth, Jan 12 2018

			77 is in the sequence because 77+78+79+80+81 = 395 = 71+73+79+83+89.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A054643, A298073, A298103.

p = {2, 3, 5, 7, 11}; lst = {}; While[p[[1]] < 3001, t = Plus @@ p; If[Mod[t, 10] == 5, AppendTo[lst, (t - 10)/5]]; p = Join[Rest@p, {NextPrime[p[[-1]]]}]]; lst (*  Robert G. Wilson v, Jan 14 2018 *)
Select[(#-10)/5&/@(Total/@Partition[Prime[Range[400]],5,1]),IntegerQ] (* Harvey P. Dale, Jun 22 2019 *)

L=List(); forprime(p=2, 2500, q=nextprime(p+1); r=nextprime(q+1); s=nextprime(r+1); t=nextprime(s+1); u=p+q+r+s+t; if((u-10)%5==0, listput(L, (u-10)\5))); Vec(L)

upto(n) = my(res = List(), pr = primes(5), s = vecsum(pr)); while(pr[5] < n, if(s == 5 * pr[3], listput(res, pr[1])); lp = nextprime(pr[5] + 1); s += (lp - pr[1]); for(i = 1, 4, pr[i] = pr[i+1]); pr[5] = lp); res \\ David A. Corneth, Jan 12 2018

New name by David A. Corneth, Jan 12 2018

A298103 The first of five consecutive prime numbers the sum of which is equal to the sum of five consecutive integers.

Original entry on oeis.org

71, 271, 281, 313, 337, 347, 389, 431, 499, 631, 661, 673, 769, 787, 827, 877, 937, 947, 1153, 1163, 1181, 1229, 1307, 1319, 1373, 1427, 1451, 1489, 1531, 1621, 1667, 1699, 1721, 1759, 1789, 1901, 1933, 1979, 2017, 2039, 2131, 2137, 2339, 2381, 2383, 2417

Offset: 1

Colin Barker, Jan 12 2018

nonn

			71 is in the sequence because 71+73+79+83+89 = 395 = 77+78+79+80+81.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A054643, A298073, A298102.

L=List(); forprime(p=2, 2500, q=nextprime(p+1); r=nextprime(q+1); s=nextprime(r+1); t=nextprime(s+1); u=p+q+r+s+t; if((u-10)%5==0, listput(L, p))); Vec(L)

A298312 The first of three consecutive octagonal numbers the sum of which is equal to the sum of three consecutive primes.

Original entry on oeis.org

12160, 74576, 158240, 181056, 269400, 371008, 601216, 606600, 848008, 980408, 1242920, 2075008, 3292816, 3680776, 4477408, 4685000, 5627960, 7505008, 8263480, 9289280, 10397408, 10419760, 10735208, 10757920, 12726680, 13000008, 14200576, 15426936, 15700256

Offset: 1

Colin Barker, Jan 17 2018

nonn

			12160 is in the sequence because 12160+12545+12936 (consecutive octagonal numbers) = 37641 = 12541+12547+12553 (consecutive primes).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..70 from Colin Barker)

Cf. A000040, A000567, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298272, A298273, A298301, A298302, A298313.

L=List(); forprime(p=2, 20000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(36*t-180, &sq) && (sq-12)%18==0, u=(sq-12)\18; listput(L, 3*u^2-2*u))); Vec(L)

from _future_ import division
from sympy import prevprime, nextprime
A298312_list, n, m = [], 1, 30
while len(A298312_list) < 10000:
    k = prevprime(m//3)
    k2 = nextprime(k)
    if prevprime(k) + k + k2 == m or k + k2 + nextprime(k2) == m:
        A298312_list.append(n*(3*n-2))
    n += 1
    m += 18*n + 3 # Chai Wah Wu, Jan 22 2018

A298313 The first of three consecutive primes the sum of which is equal to the sum of three consecutive octagonal numbers.

Original entry on oeis.org

12541, 75521, 159617, 182519, 271181, 373091, 603901, 609289, 851197, 983819, 1246757, 2079997, 3299081, 3687421, 4484737, 4692497, 5636171, 7514477, 8273437, 9299831, 10408577, 10430921, 10746557, 10769281, 12739037, 13012487, 14213621, 15440531, 15713959

Offset: 1

Colin Barker, Jan 17 2018

nonn

			12541 is in the sequence because 12541+12547+12553 (consecutive primes) = 37641 = 12160+12545+12936 (consecutive octagonal numbers).

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..70 from Colin Barker)

Cf. A000040, A000567, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298272, A298273, A298301, A298302, A298312.

Module[{nn=5000,oct3},oct3=Total/@Partition[PolygonalNumber[8,Range[nn]],3,1];Select[ Partition[Prime[Range[PrimePi[Ceiling[Max[oct3]/3]]]],3,1],MemberQ[ oct3,Total[ #]]&]][[All,1]] (* Harvey P. Dale, Dec 03 2022 *)

L=List(); forprime(p=2, 20000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(36*t-180, &sq) && (sq-12)%18==0, u=(sq-12)\18; listput(L, p))); Vec(L)

from _future_ import division
from sympy import prevprime, nextprime
A298313_list, n, m = [], 1, 30
while len(A298313_list) < 10000:
    k = prevprime(m//3)
    k2 = prevprime(k)
    k3 = nextprime(k)
    if k2 + k + k3 == m:
        A298313_list.append(k2)
    elif k + k3 + nextprime(k3) == m:
        A298313_list.append(k)
    n += 1
    m += 18*n + 3 # Chai Wah Wu, Jan 22 2018

A358393 First of three consecutive primes p,q,r such that pq + pr - qr, pq - pr + qr and -pq + pr + q*r are all prime.

Original entry on oeis.org

261977, 496163, 1943101, 2204273, 2502827, 2632627, 2822381, 2878543, 3291593, 3431891, 4122043, 4269679, 5205671, 5224361, 5565139, 6248881, 6600989, 6881291, 7568963, 8181317, 8251277, 8377777, 9005561, 9644911, 10226233, 11096753, 11767801, 12252271, 13197361, 13574489, 13730263, 14064901

Offset: 1

J. M. Bergot and Robert Israel, Nov 13 2022

nonn

			a(1) = 261977 is a term because 261977, 261983 and 262007 are consecutive primes with 261977*261983 + 261977*262007 - 261983*262007 = 68631948349,
261977*261983 - 261977*262007 + 261983*262007 = 68635092433, and
-261977*261983 + 261977*262007 + 261983*262007 = 68647667329 prime.

Robert Israel, Table of n, a(n) for n = 1..2000

Contained in A054643.

q:= 2: r:= 3:
R:= NULL: count:= 0:
while count < 40 do
  p:= q; q:= r; r:= nextprime(r);
  s:= p*(q+r)+q*r;
  if  isprime(s-2*p*q) and isprime(s-2*p*r) and isprime(s-2*q*r) then       R:= R, p; count:= count+1;
  fi
od:
R;

f[p_, q_, r_] := PrimeQ[p*q + p*r - q*r] && PrimeQ[p*q - p*r + q*r] && PrimeQ[-p*q + p*r + q*r]; Select[Partition[Prime[Range[10^6]], 3, 1], f @@ # &][[;; , 1]] (* Amiram Eldar, Nov 13 2022 *)

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        pq, pr, qr = p*q, p*r, q*r
        if all(isprime(t) for t in [pq+pr-qr, pq-pr+qr, -pq+pr+qr]): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 15))) # Michael S. Branicky, Nov 13 2022

A366414 Primes p such that p and the four previous primes are in arithmetic progression.

Original entry on oeis.org

9843139, 37772549, 53868769, 71427877, 78364669, 79080697, 98150141, 99591553, 104437009, 106457629, 111267539, 121174931, 121174961, 168236239, 199450219, 203909011, 207068923, 216618307, 230952979, 234058991, 235524901, 253412437, 263651281, 268843153, 294485483, 296239907

Offset: 1

Harvey P. Dale, Oct 09 2023

			9843019, 9843049, 9843079, 9843109, 9843139 are the 5 consecutive primes starting from A059044(1) and ending at a(1).

Cf. A054643, A122535, A006562, A181424, A054803, A059044, A054800, A122535.

Select[Partition[Prime[Range[10^7]],5,1],Length[Union[Differences[#]]]==1&][[;;,5]]

cp's OEIS Frontend

A298302 The first of three consecutive primes the sum of which is equal to the sum of three consecutive heptagonal numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A098726 Take three consecutive primes starting with the n-th prime. Calculate d(i,j) = abs(prime(i) - prime(j)), for all {i,j}, i.e., all possible differences. a(n) is the number of distinct differences (which can be either 3 or 2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A298102 The first of five consecutive integers the sum of which is equal to the sum of five consecutive prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A298103 The first of five consecutive prime numbers the sum of which is equal to the sum of five consecutive integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A298312 The first of three consecutive octagonal numbers the sum of which is equal to the sum of three consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

A298313 The first of three consecutive primes the sum of which is equal to the sum of three consecutive octagonal numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A358393 First of three consecutive primes p,q,r such that p*q + p*r - q*r, p*q - p*r + q*r and -p*q + p*r + q*r are all prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A366414 Primes p such that p and the four previous primes are in arithmetic progression.

Views

Author

Keywords

Examples

Crossrefs

Programs

A358393 First of three consecutive primes p,q,r such that pq + pr - qr, pq - pr + qr and -pq + pr + q*r are all prime.