cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A298222 The first of three consecutive squares the sum of which is equal to the sum of three consecutive primes.

Original entry on oeis.org

1444, 5776, 6400, 9604, 10816, 13924, 14400, 14884, 22500, 28900, 36100, 61504, 62500, 67600, 88804, 115600, 136900, 166464, 211600, 232324, 291600, 315844, 425104, 454276, 577600, 602176, 715716, 817216, 937024, 1016064, 1020100, 1290496, 1397124, 1507984
Offset: 1

Views

Author

Colin Barker, Jan 15 2018

Keywords

Examples

			1444 is in the sequence because 1444+1521+1600 (consecutive squares) = 4565 = 1511+1523+1531 (consecutive primes).

Links

Crossrefs

Cf. A000040, A000290, A054643, A298073, A298168, A298169, A298223.

Programs

  • PARI
    L=List(); forprime(p=2, 400000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(12*t-24, &sq) && (sq-6)%6==0, u=(sq-6)\6; listput(L, u^2))); Vec(L)

A298223 The first of three consecutive primes the sum of which is equal to the sum of three consecutive squares.

Original entry on oeis.org

1511, 5923, 6553, 9791, 11003, 14153, 14633, 15121, 22787, 29231, 36473, 61991, 62987, 68111, 89393, 116273, 137633, 167267, 212501, 233279, 292673, 316957, 426401, 455603, 579113, 603719, 717397, 819017, 938953, 1018057, 1022113, 1292737, 1399477, 1510427
Offset: 1

Views

Author

Colin Barker, Jan 15 2018

Keywords

Examples

			1511 is in the sequence because 1511+1523+1531 (consecutive primes) = 4565 = 1444+1521+1600 (consecutive squares).

Links

Crossrefs

Cf. A000040, A000290, A054643, A298073, A298168, A298169, A298222.

Programs

  • PARI
    L=List(); forprime(p=2, 400000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(12*t-24, &sq) && (sq-6)%6==0, u=(sq-6)\6; listput(L, p))); Vec(L)

A298250 The first of three consecutive pentagonal numbers the sum of which is equal to the sum of three consecutive primes.

Original entry on oeis.org

176, 35497, 45850, 68587, 87725, 229126, 488776, 705551, 827702, 1085876, 1127100, 1255380, 1732900, 1914785, 1972840, 2453122, 2737126, 2749297, 2818776, 3245026, 4598126, 5116190, 5522882, 6180335, 6658120, 6939126, 6958497, 7088327, 7114437, 7140595
Offset: 1

Views

Author

Colin Barker, Jan 15 2018

Keywords

Examples

			176 is in the sequence because 176+210+247 (consecutive pentagonal numbers) = 633 = 199+211+223 (consecutive primes).

Links

Crossrefs

Cf. A000040, A000326, A054643, A298073, A298168, A298169, A298222, A298223, A298251.

Programs

  • Maple
    N:= 10^8: # to get all terms where the sums <= N
Res:= NULL:
mmax:= floor((sqrt(8*N-23)-5)/6):
M:= [seq(seq(4*i+j,j=2..3),i=0..mmax/4)]:
M3:= map(m -> 9/2*m^2+15/2*m+6, M):
for i from 1 to nops(M) do
m:= M3[i];
  r:= ceil((m-8)/3);
  p1:= prevprime(r+1);
  p2:= nextprime(p1);
  p3:= nextprime(p2);
  while p1+p2+p3 > m do
    p3:= p2; p2:= p1; p1:= prevprime(p1);
  od:
  if p1+p2+p3 = m then
    Res:= Res, M[i]*(3*M[i]-1)/2;
  fi
od:
Res; # Robert Israel, Jan 16 2018
  • Mathematica
    Module[{prs3=Total/@Partition[Prime[Range[10^6]],3,1]},Select[ Partition[ PolygonalNumber[ 5,Range[ 5000]],3,1],MemberQ[ prs3,Total[#]]&]][[All,1]] (* Harvey P. Dale, Dec 25 2022 *)
  • PARI
    L=List(); forprime(p=2, 8000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(72*t-207, &sq) && (sq-15)%18==0, u=(sq-15)\18; listput(L, (3*u^2-u)/2))); Vec(L)

A298251 The first of three consecutive primes the sum of which is equal to the sum of three consecutive pentagonal numbers.

Original entry on oeis.org

199, 35951, 46351, 69221, 88427, 230291, 490481, 707573, 829883, 1088419, 1129693, 1258109, 1736101, 1918157, 1976243, 2456939, 2741159, 2753351, 2822881, 3249419, 4603351, 5121713, 5528623, 6186407, 6664429, 6945559, 6964949, 7094839, 7120963, 7147121
Offset: 1

Views

Author

Colin Barker, Jan 15 2018

Keywords

Examples

			199 is in the sequence because 199+211+223 (consecutive primes) = 633 = 176+210+247 (consecutive pentagonal numbers).

Links

Crossrefs

Cf. A000040, A000326, A054643, A298073, A298168, A298169, A298222, A298223, A298250.

Programs

  • Maple
    N:= 10^8: # to get all terms where the sums <= N
Res:= NULL:
mmax:= floor((sqrt(8*N-23)-5)/6):
M3:= map(t->9/2*t^2+15/2*t+6, [seq(seq(4*i+j,j=2..3),i=0..mmax/4)]):
for m in M3 do
  r:= ceil((m-8)/3);
  p1:= prevprime(r+1);
  p2:= nextprime(p1);
  p3:= nextprime(p2);
  while p1+p2+p3 > m do
    p3:= p2; p2:= p1; p1:= prevprime(p1);
  od:
  if p1+p2+p3 = m then
    Res:= Res, p1
  fi
od:
Res; # Robert Israel, Jan 16 2018
  • Mathematica
    Module[{nn=50000,pn},pn=Total/@Partition[PolygonalNumber[5,Range[ Ceiling[ (1+Sqrt[1+24 Prime[nn]])/6]]],3,1];Select[Partition[ Prime[ Range[ nn]],3,1],MemberQ[pn,Total[#]]&]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 12 2020 *)
  • PARI
    L=List(); forprime(p=2, 8000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(72*t-207, &sq) && (sq-15)%18==0, u=(sq-15)\18; listput(L, p))); Vec(L)

A298272 The first of three consecutive hexagonal numbers the sum of which is equal to the sum of three consecutive primes.

Original entry on oeis.org

6, 6216, 7626, 9180, 16836, 19900, 22366, 29646, 76636, 89676, 93096, 114960, 116886, 118828, 322806, 365940, 397386, 422740, 437580, 471906, 499500, 574056, 595686, 626640, 690900, 743590, 984906, 1041846, 1148370, 1209790, 1260078, 1357128, 1450956
Offset: 1

Views

Author

Colin Barker, Jan 16 2018

Keywords

Examples

			6 is in the sequence because 6+15+28 (consecutive hexagonal numbers) = 49 = 13+17+19 (consecutive primes).

Links

Crossrefs

Cf. A000040, A000384, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298273.

Programs

  • Maple
    N:= 100: # to get a(1)..a(100)
count:= 0:
mmax:= floor((sqrt(24*N-87)-9)/12):
for i from 1 while count < N do
  mi:= 2*i;
  m:= 6*mi^2+9*mi+7;
  r:= ceil((m-8)/3);
  p1:= prevprime(r+1);
  p2:= nextprime(p1);
  p3:= nextprime(p2);
  while p1+p2+p3 > m do
    p3:= p2; p2:= p1; p1:= prevprime(p1);
  od:
  if p1+p2+p3 = m then
    count:= count+1;
    A[count]:= mi*(2*mi-1);
  fi
od:
seq(A[i], i=1..count); # Robert Israel, Jan 16 2018
  • PARI
    L=List(); forprime(p=2, 2000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(24*t-87, &sq) && (sq-9)%12==0, u=(sq-9)\12; listput(L, u*(2*u-1)))); Vec(L)

A298273 The first of three consecutive primes the sum of which is equal to the sum of three consecutive hexagonal numbers.

Original entry on oeis.org

13, 6427, 7873, 9439, 17203, 20287, 22783, 30133, 77417, 90523, 93949, 115903, 117841, 119797, 324403, 367649, 399163, 424573, 439441, 473839, 501493, 576193, 597859, 628861, 693223, 746023, 987697, 1044733, 1151399, 1212889, 1263247, 1360417, 1454351
Offset: 1

Views

Author

Colin Barker, Jan 16 2018

Keywords

Examples

			13 is in the sequence because 13+17+19 (consecutive primes) = 49 = 6+15+28 (consecutive hexagonal numbers).

Links

Crossrefs

Cf. A000040, A000384, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298272.

Programs

  • Maple
    N:= 100: # to get a(1)..a(100)
count:= 0:
mmax:= floor((sqrt(24*N-87)-9)/12):
for i from 1 while count < N do
  mi:= 2*i;
  m:= 6*mi^2+9*mi+7;
  r:= ceil((m-8)/3);
  p1:= prevprime(r+1);
  p2:= nextprime(p1);
  p3:= nextprime(p2);
  while p1+p2+p3 > m do
    p3:= p2; p2:= p1; p1:= prevprime(p1);
  od:
  if p1+p2+p3 = m then
    count:= count+1;
  A[count]:= p1;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Jan 16 2018
  • PARI
    L=List(); forprime(p=2, 2000000, q=nextprime(p+1); r=nextprime(q+1); t=p+q+r; if(issquare(24*t-87, &sq) && (sq-9)%12==0, u=(sq-9)\12; listput(L, p))); Vec(L)

A298301 The first of three consecutive heptagonal numbers the sum of which is equal to the sum of three consecutive primes.

Original entry on oeis.org

7, 874, 7209, 15484, 16687, 23863, 68641, 98704, 122877, 239785, 373842, 455182, 498852, 523723, 601966, 652036, 769230, 777573, 1003939, 1019844, 1121245, 1189215, 1203049, 1420159, 1484946, 1594804, 1606807, 1687977, 1804975, 2292973, 2533612, 3012363
Offset: 1

Views

Author

Colin Barker, Jan 16 2018

Keywords

Examples

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

Links

Crossrefs

Cf. A000040, A000566, A054643, A298073, A298168, A298169, A298222, A298223, A298250, A298251, A298272, A298273, A298302.

Programs

  • PARI
    L=List(); forprime(p=2, 2000000, 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, (5*u^2-3*u)/2))); Vec(L)

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

Original entry on oeis.org

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

Views

Author

Colin Barker, Jan 16 2018

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • PARI
    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)

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

Views

Author

Colin Barker, Jan 12 2018

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A054643, A298073, A298103.

Programs

  • Mathematica
    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 *)
  • PARI
    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)
  • PARI
    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

Extensions

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

Views

Author

Colin Barker, Jan 12 2018

Keywords

Examples

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

Links

Crossrefs

Cf. A054643, A298073, A298102.

Programs

  • PARI
    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)
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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