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.

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A320712 Indices of primes followed by a gap (distance to next larger prime) of 28.

Original entry on oeis.org

429, 462, 685, 781, 1116, 1231, 1274, 1288, 1327, 1392, 1585, 1708, 1710, 1891, 1944, 2065, 2154, 2367, 2417, 2606, 2663, 2729, 2980, 3012, 3069, 3227, 3519, 3653, 3990, 4018, 4168, 4196, 4595, 4603, 4618, 4797, 4856, 4867, 5123, 5191, 5294, 5375, 5432, 5476, 5498, 5593, 5627, 5688, 5703
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A124595.

Links

Crossrefs

Equals A000720 o A124595.
Indices of 28's in A001223.
Row 14 of A174349.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Programs

  • PARI
    A(N=100,g=28,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A124595(n)).
A320712 = { i > 0 | prime(i+1) = prime(i) + 28 }.

A320714 Indices of primes followed by a gap (distance to next larger prime) of 32.

Original entry on oeis.org

738, 1315, 3022, 3042, 3607, 4112, 4300, 4362, 4555, 4619, 4761, 4893, 5204, 5358, 5615, 5637, 6215, 6265, 6479, 6610, 6706, 6933, 7295, 7829, 7884, 8049, 8198, 8548, 9085, 9155, 9524, 9588, 9641, 9826, 9924, 10463, 10824, 11367, 11590, 11701, 11729, 11869, 12159, 12258, 12275, 12327
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A126784.

Links

Crossrefs

Equals A000720 o A126784.
Indices of 32's in A001223.
Row 16 of A174349.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Programs

  • Maple
    p:= 2: Res:= NULL: count:= 0:
for k from 1 while count < 100 do
  q:= nextprime(p);
  if q - p = 32 then
    count:= count+1;
    Res:= Res, k;
  fi;
  p:= q;
od:
Res; # Robert Israel, Oct 25 2018
  • Mathematica
    PrimePi/@Select[Partition[Prime[Range[15000]],2,1],#[[2]]-#[[1]]==32&][[;;,1]] (* Harvey P. Dale, Jun 19 2024 *)
  • PARI
    A(N=100,g=32,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A126784(n)).
A320714 = { i>0 | prime(i+1) = prime(i) + 32 }.

A320715 Indices of primes followed by a gap (distance to next larger prime) of 34.

Original entry on oeis.org

217, 1059, 1229, 1409, 1457, 1986, 2169, 2310, 2406, 3221, 3505, 3692, 3995, 4324, 4923, 5130, 5518, 6050, 6152, 6168, 6434, 7257, 7362, 7604, 7694, 7915, 8293, 8555, 8584, 8651, 8859, 9017, 9341, 9598, 9796, 9869, 10028, 10092, 10116, 10150, 10211, 10234, 10317, 10657, 10744, 10762
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134116.

Links

Crossrefs

Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134116.
Indices of 34's in A001223.
Row 17 of A174349.

Programs

  • Mathematica
    Position[Differences[Prime[Range[11000]]],34]//Flatten (* Harvey P. Dale, Jan 19 2021 *)
  • PARI
    A(N=100,g=34,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A134116(n)).
A320715 = { i>0 | prime(i+1) = prime(i) + 34 }.

A320716 Indices of primes followed by a gap (distance to next larger prime) of 36.

Original entry on oeis.org

1183, 1532, 1663, 1847, 2146, 2489, 2500, 2550, 2700, 2976, 3087, 3238, 3461, 4236, 4483, 4681, 4692, 4834, 4849, 4946, 5178, 5836, 6062, 6098, 6269, 6591, 6613, 6787, 6862, 6904, 7091, 7178, 7200, 7285, 7577, 7743, 8057, 8097, 8215, 8355, 8572, 8637, 8767, 8832, 8877, 9023, 9129, 9161
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134117.

Links

Crossrefs

Cf. A029707, A029709 (analog for gaps 2 & 4), A320701, A320702, ... A320720 (analog for gaps 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134117.
Indices of 36's in A001223.
Row 18 of A174349.

Programs

  • PARI
    A(N=100,g=36,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A134117(n)).
A320716 = { i>0 | prime(i+1) = prime(i) + 36 }.

A320717 Indices of primes followed by a gap (distance to next larger prime) of 38.

Original entry on oeis.org

3302, 4052, 4154, 4743, 5093, 5229, 5782, 5902, 6131, 6406, 6802, 7145, 7164, 7399, 7718, 7789, 8303, 8782, 9237, 9957, 10073, 10431, 10465, 10541, 10549, 10580, 10981, 11244, 11818, 11853, 12147, 12574, 13094, 13237, 13286, 13337, 13435, 13669, 13906, 14186, 14270, 14301, 14380, 14397
Offset: 1

Views

Author

M. F. Hasler, Oct 19 2018

Keywords

Comments

Indices of the primes listed in A134118.

Links

Crossrefs

Cf. A029707, A029709 (analog for gaps 2 & 4), A320701, A320702, ... A320720 (analog for gaps 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134118.
Indices of 38's in A001223.
Row 19 of A174349.

Programs

  • PARI
    A(N=100,g=38,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

Formula

a(n) = A000720(A134118(n)).

A350851 Cumulative sums of the first ceiling(n/2)+1 elements of rows 0 to n in Pascal's triangle.

Original entry on oeis.org

1, 3, 6, 13, 24, 50, 92, 191, 354, 736, 1374, 2860, 5370, 11182, 21090, 43909, 83112, 172958, 328340, 682862, 1299528, 2700820, 5150688, 10697070, 20437756, 42415272, 81170004, 168337168, 322613196, 668607412, 1283037084, 2657319103, 5105342946, 10567113352, 20323851054
Offset: 0

Views

Author

J. Stauduhar, Jan 18 2022

Keywords

Examples

			The first ceiling(n/2)+1 elements from the first four rows of Pascal's are:
     1
    1 1
   1 2
  1 3 3
So a(0)=1, a(1)=a(0)+1+1=3, a(2)=a(1)+1+2=6, a(3)=a(2)+1+3+3=13.

Crossrefs

Cf. A007318, A116496 (for n>=2, first differences).

Programs

  • Python
    seq=[];prev=[];total=0
for n in range(30):
  row=[1]
  last=int(n/2)
  for k in range(last):
    row.append(prev[k]+prev[k+1])
  if n%2==1:
    row.append(row[-1])
  prev=row
  total+=sum(row)
  seq.append(total)
print(seq)
Previous Showing 21-26 of 26 results.

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