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 1-5 of 5 results.

A364500 a(n) = gcd(n, A005940(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 1, 10, 1, 12, 1, 2, 3, 16, 1, 2, 1, 20, 7, 2, 1, 24, 1, 2, 3, 4, 1, 6, 1, 32, 1, 2, 1, 4, 1, 2, 3, 40, 1, 14, 1, 4, 5, 2, 1, 48, 1, 2, 3, 4, 1, 6, 5, 8, 1, 2, 1, 12, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 15, 4, 11, 6, 1, 80, 1, 2, 1, 28, 5, 2, 3, 8, 1, 10, 7, 4, 1, 2, 5, 96, 1, 2, 33, 4
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2023

Keywords

Links

Crossrefs

Cf. A005940, A364499, A364501, A364502.
Cf. also A324198, A339969, A364255.

Programs

  • Mathematica
    nn = 100; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)
  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364500(n) = gcd(n, A005940(n));
  • PARI
    A364500(n) = { my(orgn=n,p=2,rl=0,z=1); n--; while(n, if(!(n%2), p=nextprime(1+p), rl++; if(1==(n%4), z *= p^min(rl,valuation(orgn,p)); rl=0)); n>>=1); (z); };

Formula

a(n) = gcd(n, A364499(n)) = gcd(A005940(n), A364499(n)).
a(n) = n / A364501(n) = A005940(n) / A364502(n).

A364502 a(n) = A005940(n) / gcd(n, A005940(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 9, 1, 7, 1, 15, 1, 25, 9, 9, 1, 11, 7, 21, 1, 5, 15, 45, 1, 49, 25, 25, 9, 125, 9, 81, 1, 13, 11, 33, 7, 55, 21, 21, 1, 77, 5, 105, 15, 35, 45, 135, 1, 121, 49, 49, 25, 245, 25, 45, 9, 343, 125, 375, 9, 625, 81, 27, 1, 17, 13, 39, 11, 65, 33, 99, 7, 91, 55, 11, 21, 25, 21, 189, 1, 143, 77, 231, 5
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2023

Keywords

Comments

Denominator of n / A005940(n).

Links

Crossrefs

Cf. A005940, A364500, A364501 (numerators), A364546 (positions of 1's).
Cf. also A364492.

Programs

  • Mathematica
    nn = 84; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[a[#]/GCD[a[#], #] &, nn] (* Michael De Vlieger, Jul 28 2023 *)
  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A364502(n) = { my(u=A005940(n)); (u / gcd(n, u)); };

A364545 Odd numbers k such that k divides A005940(k).

Original entry on oeis.org

1, 3, 5, 125, 245, 375, 715, 845, 847, 1215, 2873, 11583, 12635, 21879, 24255, 31213, 33495, 36125, 42875, 48125, 48841, 71269, 100793, 102245, 104907, 157035, 173641, 191607, 206045, 240787, 244205, 251459, 302575, 313937, 351509, 359513, 375687, 384475, 388531, 417605, 419957, 444889, 468999, 521703, 586177, 635375
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2023

Keywords

Crossrefs

Odd terms in A364544.
Cf. A005940, A364501.
Cf. also A364495, A364547.

Programs

  • Mathematica
    nn = 2^20; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[1, nn, 2], Divisible[a[#], #] &] (* Michael De Vlieger, Jul 28 2023 *)
  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364545(n) = ((n%2)&&!(A005940(n)%n));

A364544 Numbers k such that k divides A005940(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 125, 128, 160, 192, 245, 250, 256, 320, 375, 384, 490, 500, 512, 640, 715, 750, 768, 845, 847, 980, 1000, 1024, 1215, 1280, 1430, 1500, 1536, 1690, 1694, 1960, 2000, 2048, 2430, 2560, 2860, 2873, 3000, 3072, 3380, 3388, 3920, 4000, 4096, 4860, 5120
Offset: 1

Views

Author

Antti Karttunen, Jul 28 2023

Keywords

Comments

If k is a term, then also 2*k is present in this sequence, and vice versa.
A029747 is included as a subsequence, because it gives the known fixed points of map n -> A005940(n).

Crossrefs

Positions of 1's in A364501.
Subsequence of A364542.
Subsequences: A029747, A364545 (odd terms).
Cf. A005940.
Cf. also A364494, A364546.

Programs

  • Mathematica
    nn = 5120; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Select[Range[nn], Divisible[a[#], #] &] (* Michael De Vlieger, Jul 28 2023 *)
  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364544(n) = !(A005940(n)%n);

A365464 a(n) = n / gcd(n, A356867(n)).

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 7, 1, 1, 10, 11, 4, 13, 7, 5, 8, 17, 1, 19, 5, 7, 22, 23, 1, 1, 13, 1, 28, 29, 10, 31, 8, 11, 17, 5, 4, 37, 19, 13, 8, 41, 7, 43, 11, 5, 46, 47, 8, 49, 1, 17, 13, 53, 1, 5, 28, 19, 58, 59, 5, 61, 31, 7, 64, 13, 22, 67, 17, 23, 7, 71, 1, 73, 37, 1, 76, 77, 13, 79, 5, 1, 82, 83, 28, 17, 43, 29, 44, 89, 10
Offset: 1

Views

Author

Antti Karttunen, Sep 15 2023

Keywords

Comments

Numerator of n / A356867(n).

Links

Crossrefs

Cf. A356867, A365462, A365463, A365465 (denominators).
Cf. also A364501.

Programs

  • PARI
    up_to = 19683;
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v356867 = A356867list(up_to);
A356867(n) = v356867[n];
A365464(n) = n/gcd(n, A356867(n));

Formula

a(n) = n / A365463(n).
Showing 1-5 of 5 results.

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