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 1-4 of 4 results.

A136022 First differences of A136021.

Original entry on oeis.org

19, 1028, 63326, 4417267, 336418691, 27095099730, 2264740360646, 194526458511192, 17050085068702458, 1517704371492095702, 136758640772944755122, 12445444666179480674989, 1141852084948606732236114, 105483948921834512942168239
Offset: 1

Views

Author

Enoch Haga, Dec 10 2007, Dec 12 2007

Keywords

Examples

			a(1)=19 because at 10^0 the prime divisor sum in A136021 is zero, so 19-0=19. From the second sum in A136021, 1047, subtract 19, so that a(2) = 1028.

Crossrefs

Cf. A136021.

Formula

a(n) = A136021(n)-A136021(n-1).

Extensions

a(6) from R. J. Mathar, Dec 12 2007
Edited by R. J. Mathar, Apr 17 2009
a(7)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jun 29 2014

A136023 The largest prime factor among all composites <= 10^n.

Original entry on oeis.org

5, 47, 499, 4999, 49999, 499979, 4999999, 49999991, 499999993, 4999999937, 49999999967, 499999999979, 4999999999937, 49999999999981, 499999999999999, 4999999999999997, 49999999999999993, 499999999999999931, 4999999999999999963, 49999999999999999951
Offset: 1

Views

Author

Enoch Haga, Dec 12 2007

Keywords

Comments

This is the largest single divisor contributing to A136021(n).
If 4 concatenated with n-1 nines is prime, it will be the largest factor. This candidate does not work for n=6 because 499999 is not prime.

Examples

			a(3)=499 because it is the largest prime factor of all k<=10^3; the largest k in that interval with this factor is k=2*499 = 998.

Links

Crossrefs

Cf. A136021, A052369.

Programs

  • Maple
    5, seq(prevprime(5*10^n),n=1..100); # Robert Israel, Jul 21 2014

Extensions

Edited and more terms added by R. J. Mathar, Apr 17 2009

A136024 Largest prime factor of odd composites less than 10^n.

Original entry on oeis.org

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333313, 3333333323, 33333333329, 333333333323, 3333333333301, 33333333333323, 333333333333307, 3333333333333301, 33333333333333323
Offset: 1

Views

Author

Enoch Haga, Dec 12 2007

Keywords

Comments

Last instance of the largest prime factor of odd N <= 10^n-1 associated with A136021.
This sequence is not the same as A051200. E.g., A051200(9)=333333331 is not prime and is different from a(9)=333333313. However, if A051200(n) is prime, then a(n)=A051200(n).

Examples

			a(1)=31 because it is the largest factor of odd N <= 10^2-1. The value of odd N where this factor first occurs is 3*31 = 93.

Crossrefs

Cf. A051200, A136021.

Programs

  • Mathematica
    a[n_]:=NextPrime[10^n/3,-1]; Array[a,17] (* Stefano Spezia, Aug 31 2025 *)
  • PARI
    a(n)=precprime(10^n\3)

Formula

a(n) = precprime(10^n/3) = A007917((10^n-1)/3). - Max Alekseyev, Sep 29 2015

Extensions

Clarified and extended by Charles R Greathouse IV, Oct 11 2009

A136025 Sum of distinct proper prime divisors of odd integers below 10^n.

Original entry on oeis.org

3, 373, 24307, 1691682, 127867801, 10233538789, 850896280551, 72812857079241, 6363727756215813, 565232434009370012, 50843507342073211151, 4620323131256374760046, 423405369424475640435621, 39074878176445767411791424
Offset: 1

Views

Author

Enoch Haga, Dec 12 2007

Keywords

Comments

Through 10^5 about 37.5% of total sums for all integers N comprise sums of odd N and the remaining 62.5% of even N.

Examples

			a(0)=3 because the only odd N <=10^1-1 having a prime factor is 9 and its factor is 3 and sum is 3.

Crossrefs

Cf. A136021, A136024, A143851.

Programs

  • Maple
    A105221 := proc(n) local a,ifs,p; ifs := ifactors(n)[2] ; a := 0 ; for p in ifs do if op(1,p) <> 1 and op(1,p) <> n then a := a+op(1,p) ; fi ; od: RETURN(a) ; end: A136025 := proc(n) local a,k ; a := 0 ; for k from 5 to 10^n-1 by 2 do a := a+A105221(k) ; od: RETURN(a) ; end: for n from 1 do print(A136025(n)); od: # R. J. Mathar, Jan 29 2008

Formula

a(n) = sum_{k=1,2,...,A093143(n)} A105221(2k-1). - R. J. Mathar, Jan 29 2008
a(n) = sum_{prime p, 3<=p<10^n} p*floor((10^n-p)/(2p)). - Max Alekseyev, Jan 30 2012

Extensions

a(6) from R. J. Mathar, Jan 29 2008
a(7)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jul 06 2014
Showing 1-4 of 4 results.

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