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 1-3 of 3 results.

A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).

Original entry on oeis.org

2, 4, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070
Offset: 1

Views

Author

Lekraj Beedassy, Aug 10 2006

Keywords

Comments

If n > 2, then a(n) = product of n-1 consecutive distinct prime divisors. E.g. a(5)=210, the product of 4 consecutive and distinct prime divisors, 2,3,5,7. - Enoch Haga, Dec 08 2007
From Bill McEachen, Jul 10 2022: (Start)
Rather than have code merely generating the conjectured values, one can compare values of sequence terms at the same position n. Specifically, locate new maximums where (p,p+even) are both prime, where even=2,4,6,8,... and the datum set is taken with even=4. A new maximum implies a new jumping champion.
Doing this produces the terms 2,4,6,30,210,2310,30030,.... Looking at the plot of a(n) ratio for gap=2/gap=6, the value changes VERY slowly, and is 2.14 after 50 million terms (one can see the trend via Plot 2 of A001359 vs A023201 (3rd option seqA/seqB vs n). The ratio for gap=4/gap=2 ~ 1, implying they are equally frequent. (End)

Links

Crossrefs

Cf. A087103, A087104, A001223, A000230, A001632, A038664, A086977-A086980, A085237, A005250, A053686, A054587, A093737-A093753, A093972-A093984.

Programs

  • Mathematica
    2,4,Table[Product[Prime[k],{k,1,n-1}],{n,3,30}]
  • PARI
    print1("2, 4");t=2;forprime(p=3,97,print1(", ",t*=p)) \\ Charles R Greathouse IV, Jun 11 2011

Formula

Consists of 4 and the primorials (A002110).
a(1) = 2, a(2) = 4, a(3) = 6, a(n+1)/a(n) = Prime[n] for n>2.

Extensions

Corrected and extended by Alexander Adamchuk, Aug 11 2006
Definition corrected and clarified by Jonathan Sondow, Aug 16 2011

A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.

Original entry on oeis.org

0, 7, 44, 299, 1940, 13549, 99987, 768752, 6089791, 49392723, 408550278, 3435528229, 29289695650, 252672394234, 2201981901415, 19360330918473, 171550299264139, 1530609037414453
Offset: 1

Views

Author

Enoch Haga, Apr 15 2004

Keywords

Comments

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the former, whereas A080841 considers the latter. - N. J. A. Sloane, Mar 07 2021

Examples

			a(2) = 7 because there are 7 prime gaps of 6 below 10^2.

Links

Crossrefs

Cf. A007508, A080841, A093737, A093739.
See also A023201, A031924.

Programs

  • Mathematica
    Accumulate@ Array[Count[Differences@ Prime@ Range[PrimePi[10^(# - 1) + 1], PrimePi[10^# - 1]], 6] &, 8] (* Michael De Vlieger, Apr 09 2021 *)
  • UBASIC
    20 N=1:dim T(34); 30 A=nxtprm(N); 40 N=A; 50 B=nxtprm(N); 60 D=B-A; 70 for x=2 to 34 step 2; 80 if D=X and B<10^2+1 then T(X)=T(X)+1; 90 next X; 100 if B>10^2+1 then 140; 110 B=A; 120 N=N+1; 130 goto 30; 140 for x=2 to 34 step 2; 150 print T(X);, 160 next (This program simultaneously finds values from 2 to 34 -- if gap=2 add 1-- adjust lines 80 and 100 for desired 10^n)

Extensions

a(10)-a(13) from Washington Bomfim, Jun 22 2012
a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

A213930 Table of frequencies of gaps of size 2d between consecutive primes below 10^n, n >= 1; d = 1,2,...,A213949(n).

Original entry on oeis.org

2, 8, 7, 7, 1, 35, 40, 44, 15, 16, 7, 7, 0, 1, 1, 205, 202, 299, 101, 119, 105, 54, 33, 40, 15, 16, 15, 3, 5, 11, 1, 2, 1, 1224, 1215, 1940, 773, 916, 964, 484, 339, 514, 238, 223, 206, 88, 98, 146, 32, 33, 54, 19, 28, 19, 5, 4, 3, 5
Offset: 1

Views

Author

Washington Bomfim, Jun 24 2012

Keywords

Comments

Sum of elements in line n is Pi(10^n)-2. Column d is the sequence of the numbers of gaps of size 2d between consecutive primes up to 10^n. For example, column 1 is A007508, and column 2 is A093737. Column 3 corresponds to the jumping champion 6. Column 15 corresponds to the next champion 30. It is interesting that local maximums appear in the beginning of this column, 11 in line 4, and 146 in line 5.

Examples

			Table begins
   2
   8    7    7   1
  35   40   44  15  16   7   7   0   1   1
  205  202  299 101 119 105  54  33  40  15  16  15  3  5  11  1  2  1
1224 1215 1940 773 916 964 484 339 514 238 223 206 88 98 146 32 33 54 19 28...

Links

Crossrefs

Cf. A038460, A000720, A007508, A093737, A213949 (row lengths).

Programs

  • Mathematica
    Table[t2 = Sort[Tally[Table[Prime[k + 1] - Prime[k], {k, 2, PrimePi[10^n] - 1}]]]; maxDiff = t2[[-1, 1]]/2; t3 = Table[0, {k, maxDiff}];Do[t3[[t2[[i, 1]]/2]] = t2[[i, 2]], {i, Length[t2]}]; t3, {n, 5}] (* T. D. Noe, Jun 25 2012 *)
Showing 1-3 of 3 results.

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

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