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 1-6 of 6 results.

A092853 Number of composites > mean composite (=A092802(n)) below 10^n.

Original entry on oeis.org

2, 37, 418, 4398, 45288, 461339, 4671939, 47150884, 474823446, 4774453663, 47957215384, 481331669604, 4828116680970, 48407394207052, 485163305702187
Offset: 1

Views

Author

Enoch Haga, Mar 07 2004

Keywords

Examples

			At 10^1 there are 4 composites: 4+6+8+9=27. The rounded mean is 27\4=7. there are 2 composites over 7: 8 and 9, so a(1)=2.

Crossrefs

Cf. A092802, A092853, A092854.

Formula

a(n) = A092871(n) - A092852(n) = A065894(n) - 1 - A092852(n). - Max Alekseyev, Aug 15 2013

Extensions

Terms a(9)-a(15) from Max Alekseyev, Aug 15 2013

A092852 Number of composites <= A092802(n).

Original entry on oeis.org

2, 36, 412, 4371, 45118, 460161, 4663480, 47087659, 474329018, 4770493824, 47924729801, 481060418376, 4825817782189, 48387664042144, 484992123875142, 4859631205206357, 48681601698828085, 487571138851821274, 4882443976989269954, 48884842829781286250, 489393391263430721900
Offset: 1

Views

Author

Enoch Haga, Mar 07 2004

Keywords

Examples

			Up to 10^1 there are 4 composites: 4 + 6 + 8 + 9 = 27. The rounded mean is A092802(1) = floor(27/4) = 7. There are 2 composites below 7: 4 and 6, so a(1) = 2.

Crossrefs

Cf. A000720, A062298, A065855, A092802, A092853, A092854.

Formula

a(n) = A065855(A092802(n)) = A062298(A092802(n)) - 1 = A092802(n) - A000720(A092802(n)) - 1.

Extensions

a(9)-a(15) from Max Alekseyev, Aug 14 2013
a(16)-a(21) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 05 2024

A092854 Difference between number of composites > and <= mean (=A092802(n)) below 10^n.

Original entry on oeis.org

0, 1, 6, 27, 170, 1178, 8459, 63225, 494428, 3959839, 32485583, 271251228, 2298898781, 19730164908, 171181827045
Offset: 1

Views

Author

Enoch Haga, Mar 07 2004

Keywords

Examples

			a(3)=6 because the count at 10^3 in A092853 is 418 and in A092852 it is 412. 418-412=6.

Crossrefs

Cf. A092802, A092852, A092853.

Formula

a(n) = A092853(n) - A092852(n).

Extensions

a(9)-a(15) from Max Alekseyev, Aug 15 2013

A092801 Standard deviation (rounded) of composites below 10^n.

Original entry on oeis.org

2, 28, 285, 2873, 28795, 288244, 2883807, 28846206, 288514821, 2885502969, 28857521613, 288593332699, 2886069270370
Offset: 1

Views

Author

Enoch Haga, Mar 06 2004

Keywords

Comments

As with the primes in A091716, each succeeding term seems a close multiple of that preceding.
Exact values: a(8)=sqrt(1055690147582754761176900206278/1268700426680407), a(9)=sqrt(2343463734895394416636292841043365/28152824997414728), a(10)=sqrt(758560349177461014920842710257059832362/91106022529587615169). [From Sean A. Irvine, Apr 07 2010]
Exact values from Sean A. Irvine are population standard deviations. - Hiroaki Yamanouchi, Sep 23 2014

Examples

			a(3) = 285 because this is the computed and rounded sample standard deviation of the composites below 10^3.

References

  • John E. Freund, Modern elementary statistics, 5th ed. (Prentice-Hall, 1979), pp. 42-47

Crossrefs

Cf. A092802, A091716.

Extensions

Corrected and extended by Sean A. Irvine, Apr 07 2010
a(8) corrected and a(11)-a(13) from Hiroaki Yamanouchi, Sep 23 2014

A092871 Number of composites < 10^n.

Original entry on oeis.org

0, 4, 73, 830, 8769, 90406, 921500, 9335419, 94238543, 949152464, 9544947487, 95881945185, 962392087980, 9653934463159, 96795058249196, 970155429577329, 9720761658966073, 97376442842345765, 975260045712259138, 9765942332723655391, 97779180397439081158
Offset: 0

Views

Author

Enoch Haga, Mar 08 2004

Keywords

Comments

The number 1 is omitted from the count as it is neither prime nor composite

Examples

			10^3 = 1000. 1000-2 = 998. a(3) = 830 because the 830 composites+168 primes must total 998.

Links

Crossrefs

Cf. A065894, A006880, A092801, A092802.

Programs

  • Mathematica
    Table[10^i-PrimePi[10^i]-2,{i,14}] (* Harvey P. Dale, Oct 01 2011 *) (* Mathematica's implementation of PrimePi does not work for 10^15 or above *)

Formula

For n>0, a(n) = A065894(n) - 1 = 10^n - 2 - A006880(n). - Max Alekseyev, Aug 15 2013

Extensions

Edited by Max Alekseyev, Aug 15 2013

A091716 Standard deviation (rounded) of primes below 10^n.

Original entry on oeis.org

2, 29, 298, 2962, 29412, 292821, 2921863, 29170821, 291324189, 2910238255, 29078387910, 290589147156, 2904276036695
Offset: 1

Views

Author

Enoch Haga, Mar 05 2004

Keywords

Comments

It appears that a good estimate for the standard deviation of primes below 10^(n+1) is about 10 times the term for 10^n.
Heuristically, if we use a model where each positive integer x has probability approximately 1/log(x) of being prime, we should expect the standard deviation of the primes below N to be approximately N/sqrt(12). - Robert Israel, Sep 23 2014

Examples

			a(6) = 292821 (rounded from 292820.634) because this is the computed and rounded sample standard deviation of the 78498 primes below 10^6.

References

  • John E. Freund, Modern elementary statistics, 5th ed. (Prentice-Hall, 1979), pp. 42-47

Crossrefs

Cf. A092800, A092801, A092802.

Programs

  • Maple
    seq(round(Statistics:-StandardDeviation(select(isprime, [$2 .. 10^n-1]))),n=1..7); # Robert Israel, Sep 23 2014

Extensions

a(9)-a(13) from Hiroaki Yamanouchi, Sep 23 2014
Showing 1-6 of 6 results.

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