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.

A073262 Partial sums of first m composite numbers arising in A053781.

Original entry on oeis.org

4, 10, 18, 63, 132, 3692, 8025, 9200, 13205, 16275, 287382, 1674342, 13852716, 261109560, 509099424, 53226407187, 339194408709, 63673280091408, 230160748303410, 9613333448250222, 14247374595414090, 446710602178803172
Offset: 1

Views

Author

Labos Elemer, Jul 22 2002

Keywords

Comments

Required to extend A053781 without recomputation.

Crossrefs

Cf. A053781, A073263.

Programs

  • Mathematica
    c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] t=Table[0, {256}]; s=0; k=0; Do[s=s+c[n]; If[IntegerQ[s/n], k=k+1; t[[k]]=s; Print[s]], {n, 1, 12000000}] t

Formula

a(n)/A053781(n) is an integer.

Extensions

a(17)-a(19) from Klaus Brockhaus, Aug 12 2009
a(20)-a(22) from Ray Chandler, Jul 19 2010

A073263 Integer quotients of partial sum of first n composite and n (see A053781).

Original entry on oeis.org

4, 5, 6, 9, 12, 52, 75, 80, 95, 105, 422, 1002, 2841, 12165, 16944, 170997, 430041, 5844984, 11095683, 71438394, 86938085, 485470043, 3341061731, 11203279551, 25053055391, 29329179630, 150058180590, 278953774325, 771115150044
Offset: 1

Views

Author

Labos Elemer, Jul 22 2002

Keywords

Crossrefs

Cf. A002808, A053781, A073255-A073262.

Programs

  • Mathematica
    s=0; Do[s=s+FixedPoint[n+PrimePi[ # ]+1&, n]; If[IntegerQ[s/n], k=k+1; t[[k]]=s/n; Print[s/n]], {n, 1, 12000000}]

Formula

a(n)=A073262(n)/A053781(n)

Extensions

a(17) - a(19) from Klaus Brockhaus, Aug 12 2009
a(20)-a(26) from Ray Chandler, Jul 19 2010
a(27)-a(29) from Donovan Johnson, May 23 2011

A129749 Numbers k that divide the sum of the first k nonprimes.

Original entry on oeis.org

1, 8, 32, 44, 577, 5066, 5669, 8615, 39787, 59689, 109752, 146328, 15451380, 22173220, 28558717, 332573533, 1837410366, 3289933379, 19053646133, 370648112641
Offset: 1

Views

Author

J. M. Bergot, May 14 2007

Keywords

Comments

A variant of A045345 and A053781.
a(21) > 10^12. - Donovan Johnson, May 20 2010

Examples

			The sum of the first 44 nonprimes is 1452 = 33*44.

Crossrefs

Cf. A045345 (n divides sum of first n primes), A053781 (n divides sum of first n composites), A018252 (nonprimes), A051349 (sum of first n nonprimes).

Programs

  • Maple
    np:=proc(j) if isprime(j)=false then j else fi end: NP:=[seq(np(j),j=1..50000)]: a:=proc(n) if type(add(NP[j],j=1..n)/n,integer)=true then n else fi end: seq(a(n),n=1..nops(NP)); # Emeric Deutsch, May 16 2007
  • Mathematica
    Module[{nn=150000,np},np=Accumulate[Select[Range[nn],!PrimeQ[#]&]];Select[Thread[ {np,Range[Length[np]]}],Mod[#[[1]],#[[2]]]==0&]][[;;,2]] (* The program generates the first 11 terms of the sequence. To generate more, increase the nn constant. *) (* Harvey P. Dale, Jan 04 2024 *)

Extensions

Edited and a(5) to a(18) added by Klaus Brockhaus, May 17 2007
a(19) from Donovan Johnson, Sep 19 2009
a(20) from Donovan Johnson, May 20 2010

A179859 Numbers k that divide the sum of the first k noncomposites.

Original entry on oeis.org

1, 3, 7, 225, 487, 735, 50047, 142835, 170209, 249655, 316585343, 374788043, 2460457827, 2803329305, 6860334657, 65397031525, 78658228039
Offset: 1

Views

Author

Ray Chandler, Jul 29 2010

Keywords

Comments

A variant of A045345 (primes), A053781 (composites) and A129749 (nonprimes).

Examples

			The sum of the first 7 noncomposites is 42 = 6*7, so 7 is in the sequence.

Crossrefs

Cf. A008578 (noncomposites), A014284 (sum of first n noncomposites).
Cf. A045345, A053781, A129749, A179860, A179861.

Programs

  • PARI
    lista(kmax) = {my(m = 1, s = 1); print1(1, ", "); forprime(k = 1, kmax, m++; s += k; if(!(s % m), print1(m, ", ")));} \\ Amiram Eldar, May 24 2024

Formula

Numbers k such that k | A014284(k).
a(n) = A179861(n) / A179860(n).
a(n+1) = A158682(n) + 1.

Extensions

a(16)-a(17) from Amiram Eldar, May 24 2024
Showing 1-4 of 4 results.

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