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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A257375 Numbers n such that n, n+4, n+6, n+10, n+16, n+18, n+24, n+28, n+30, n+34, n+40, n+46, n+48, n+54, n+58, n+60 and n+66 are all prime.

Original entry on oeis.org

13, 47624415490498763963983, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 163027495131423420474913
Offset: 1

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Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, this sequence, A257376, A257377.

A257376 Numbers n such that n, n+6, n+8, n+12, n+18, n+20, n+26, n+32, n+36, n+38, n+42, n+48, n+50, n+56, n+60, n+62 and n+66 are all prime.

Original entry on oeis.org

1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 124211857692162527019731
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, this sequence, A257377.

Extensions

a(1) corrected by Tim Johannes Ohrtmann, Dec 17 2015

A257377 Numbers n such that n, n+2, n+6, n+12, n+14, n+20, n+24, n+26, n+30, n+36, n+42, n+44, n+50, n+54, n+56, n+62 and n+66 are all prime.

Original entry on oeis.org

17, 37630850994954402655487, 53947453971035573715707, 174856263959258260646207, 176964638100452596444067, 207068890313310815346497, 247620555224812786876877, 322237784423505559739147
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Apr 21 2015

Keywords

Links

Crossrefs

Initial members of all of the first prime k-tuplets:
twin primes: A001359.
prime triples: A007529 out of A022004, A022005.
prime quadruplets: A007530.
prime 5-tuples: A086140 out of A022007, A022006.
prime sextuplets: A022008.
prime septuplets: A257124 out of A022009, A022010.
prime octuplets: A065706 out of A022011, A022012, A022013.
prime nonuplets: A257125 out of A022547, A022548, A022545, A022546.
prime decaplets: A257127 out of A027569, A027570.
prime 11-tuplets: A257129 out of A213646, A213647.
prime 12-tuplets: A257131 out of A213601, A213645.
prime 13-tuplets: A257135 out of A214947, A257137, A257138, A257139, A257140, A257141.
prime 14-tuplets: A257166 out of A257167, A257168.
prime 15-tuplets: A257169 out of A257304, A257305, A257306, A257307.
prime 16-tuplets: A257308 out of A257369, A257370.
prime 17-tuplets: A257373 out of A257374, A257375, A257376, this sequence.

A210439 The minimal Skewes number for prime n-tuplets.

Original entry on oeis.org

1369391, 337867, 1172531, 21432401, 251331775687, 7572964186421, 1203255673037261
Offset: 2

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Author

Alexei Kourbatov, Jan 20 2013

Keywords

Comments

More formally: the least prime in the prime n-tuplet at which for the first time pi_n(p) > C_n*Li_n(p). Here pi_n(p) is the n-tuplet counting function; C_n is the Hardy-Littlewood constant, and Li_n(x) is the integral from 2 to x of (1/(log t)^n) dt.
If, for a given n, there is more than one type of n-tuplets, then a(n) is determined by the n-tuplet type for which the first sign change of pi_n - C_n*Li_n occurs earlier than for the other type(s).
For the special case n=1, the term a(1) is the Skewes number, i.e., the first prime p for which pi(p) > Li(p). The term a(1) is not included in the sequence because it is not precisely known.

Examples

			Initially, for twin primes we have pi_2(p) < C_2 Li_2(p). The inequality is reversed for the first time for the 10744th pair of twin primes (1369391,1369393), therefore a(2) = 1369391.
Similarly, for prime triples (p,p+4,p+6), pi_3(p) < C_3 Li_3(p) until the 652nd triple (337867,337871,337873) where the inequality is reversed for the first time. Thus a(3)=337867. (The reversal for the other type of triples (p,p+2,p+6) occurs much later, so triples (p,p+2,p+6) do not contribute a term to this sequence.)
From _Hugo Pfoertner_, Aug 26 2021, Oct 24 2021: (Start)
a(8) corresponds to the 134292-th 8-tuple of the form p + [0, 2, 6, 8, 12, 18, 20, 26], found using a program provided by _Norman Luhn_. This type of 8-tuple is the one that leads to the earliest crossing of the corresponding comparison value (see linked illustration), while the other two possible configurations (enumerated in A022012 and A022013 or in A346997 and A346998) are still far from crossing their respective applicable comparison values. The other two possible 8-tuples, which lead to the crossing that occurs later, determine the terms A332493(8) and A348053(8), dependent on the criterion applied to decide what is "later". (End)

Links

Crossrefs

Cf. A052435 (round(li(n)-pi(n)), where li is the logarithmic integral and pi(x) is the prime counting function).
Cf. A332493, A333586, A333587, A348053.
Cf A022011, A022012, A022013, A346996, A346997, A346998 (related to 8-tuplets).

Programs

  • PARI
    \\ See Alexei Kourbatov link.

Extensions

a(7) from Hugo Pfoertner, May 09 2020
a(8) from Hugo Pfoertner, Aug 26 2021

A346997 a(n) = A022012(10^n).

Original entry on oeis.org

17, 134764997, 9844128377, 345828727877, 9637575539147, 223528482767957, 4652382265065167, 89306626080020957
Offset: 0

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Author

Hugo Pfoertner, Aug 11 2021

Keywords

Comments

The terms are the (10^n)-th initial members of the prime octuplets of the form (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26). Terms a(5) and a(6) were found using a program provided by Norman Luhn during an effort to find A210439(8) and A332493(8).
Asymptotically for n -> infinity, C_HL*Integral_{x=2..a(n)} 1/log(x)^8 dx = 10^n, where C_HL = 475.36521172411318772... is the specific Hardy-Littlewood constant for this prime constellation. The predicted approximate values using this relationship would be a(6) = 4.629899*10^15 and a(7) = 8.9223552*10^16.

Links

Crossrefs

Cf. A022011, A022012, A022013, A210439, A332493, A346996, A346998.

Extensions

a(7) from Norman Luhn, Sep 13 2021

A346998 a(n) = A022013(10^n).

Original entry on oeis.org

88713, 302542763, 46328924003, 1409639621633, 37685138975573, 824339812580723, 16514635234360163, 308319877282402613
Offset: 0

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Author

Hugo Pfoertner, Aug 12 2021

Keywords

Comments

The terms are the (10^n)-th initial members of the prime octuplets of the form (p, p+6, p+8, p+14, p+18, p+20, p+24, p+26). Terms a(5) and a(6) were found using a program provided by Norman Luhn during an effort to find A210439(8) and A332493(8).
Since this prime constellation leads to the same Hardy-Littlewood constant as for A022011, the expected asymptotic behavior is also the same as in A346996 for large n. See the comment there for more information. Accordingly, the comparison value for a(6) is 1.647755*10^16 and the prediction for a(7) is 3.0824636*10^17.

Links

Crossrefs

Cf. A022011, A022012, A022013, A210439, A332493, A346996, A346997.

Extensions

a(7) from Norman Luhn and Hugo Pfoertner, Sep 13 2021

A186634 Irregular triangle, read by rows, giving dense patterns of n primes.

Original entry on oeis.org

0, 2, 0, 2, 6, 0, 4, 6, 0, 2, 6, 8, 0, 2, 6, 8, 12, 0, 4, 6, 10, 12, 0, 4, 6, 10, 12, 16, 0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20, 0, 2, 6, 8, 12, 18, 20, 26, 0, 2, 6, 12, 14, 20, 24, 26, 0, 6, 8, 14, 18, 20, 24, 26, 0, 2, 6, 8, 12, 18, 20, 26, 30, 0, 2, 6, 12, 14, 20, 24, 26, 30, 0, 4, 6, 10, 16, 18, 24, 28, 30, 0, 4, 10, 12, 18, 22, 24, 28, 30, 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 0, 2, 6, 12, 14, 20, 24, 26, 30, 32
Offset: 2

Views

Author

T. D. Noe, Feb 24 2011

Keywords

Comments

The first pattern (0,2) is for twin primes (p,p+2). Row n contains A083409(n) patterns, each one consisting of 0 followed by n-1 terms. In each row the patterns are in lexicographic order.
These numbers (in a slightly different order) appear in Table 1 of the paper by Tony Forbes. Sequence A186702 gives the least prime starting a given pattern.

Examples

			The irregular triangle begins:
0, 2
0, 2, 6, 0, 4, 6
0, 2, 6, 8
0, 2, 6, 8, 12, 0, 4, 6, 10, 12
0, 4, 6, 10, 12, 16
0, 2, 6, 8, 12, 18, 20, 0, 2, 8, 12, 14, 18, 20

Links

Crossrefs

Cf. A020497, A008407, A186702, and sequences created by these patterns: A001359, A022004, A022005, A007530, A022006, A022007, A022008, A022009, A022010, A022011, A022012, A022013, A022545, A022546, A022547, A022548, A027569, A027570.

A182393 Numbers n such that 210*n + {11,13,17,19,23,29,31,37} are 8 consecutive primes.

Original entry on oeis.org

0, 75048, 122183, 445838, 868588, 1078331, 3152249, 4337790, 4962337, 5101537, 5572485, 6638215, 6948906, 8155956, 8298280, 9217084, 9752564, 11416369, 13331645, 13539754, 17782872, 19480161, 25473918, 25614474, 26299945, 27593165, 28335777, 28906807, 29231650
Offset: 1

Views

Author

Zak Seidov, Apr 27 2012

Keywords

Comments

Subsequence of A182387: a(2) = 75048 = A182387(5) = A182282(7), a(3) = 122183 = A182387(8) = A182282(29).

Links

Crossrefs

Cf. A014561, A022011, A022012, A022013, A182279, A182282, A182387.

Formula

a(n) = (A022011(n) - 11)/210. - Hugo Pfoertner, Nov 18 2022

A347849 Primes at lower end of record gaps between prime octuplets given by A347848.

Original entry on oeis.org

11, 25658441, 93625991, 226449521, 2843348351, 4090833821, 13421076281, 18856092371, 26092031081, 54270148391, 66449431661, 111422173391, 124168028051, 280837571081, 875319936761, 1247050623431, 3589081520021, 6363702282011, 7479508339601, 10804857261041, 15199582184861
Offset: 1

Views

Author

Hugo Pfoertner and Norman Luhn, Sep 16 2021

Keywords

Examples

			See A347848.

Links

Crossrefs

Cf. A022011, A347848.

Formula

a(n) == 11 (mod 210).

A350828 Number of prime octuplets with initial member (A065706) between 10^(n-1) and 10^n.

Original entry on oeis.org

0, 2, 0, 1, 1, 3, 3, 9, 28, 136, 541, 2936
Offset: 1

Views

Author

M. F. Hasler, Mar 01 2022

Keywords

Comments

"Between 10^(n-1) and 10^n" is equivalent to saying "with n (decimal) digits".
A prime octuplet is a sequence of 8 consecutive primes (p1, ..., p8) of minimal diameter p8 - p1 = 26.
Terms a(1)-a(12) computed from b-file a(1..18123) for A065706. Using Luhn's database, cf. LINKS, one can get 3 more terms.
So far, the last term of all the octuplets has the same number of digits as the initial term.

Examples

			a(1) = a(3) = 0 because there is no single-digit nor a 3-digit prime initial member of a prime octuplet.
a(2) = 2 because 11 and 17 are the only 2-digit members of A065706, i.e., primes to start a prime octuplet.
a(4) = a(5) = 1 because 1277 (resp. 88793) is the only prime with 4 (resp. 5) digits to start a prime octuplet.
Then there are a(6) = 3 six-digit primes, 113147, 284723 and 855713, which start a prime octuplet.

Links

Crossrefs

Cf. A065706 (initial members p of prime octuplets (p, ..., p+26)), A022011, A022012, A022013 (idem, specifically for each of the three possible patterns).
Cf. A350825, A350826, A350827: similar for quintuplets, sextuplets and septuplets.

Programs

  • PARI
    (D(v)=v[^1]-v[^-1])( [setsearch(A065706,10^n,1) | n<-[0..12]] ) \\ where A065706 is a vector of at least 3660 terms of that sequence.
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