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.

A120736 Numbers k such that every prime p that divides d(k) (A000005) also divides k.

Original entry on oeis.org

1, 2, 6, 8, 9, 10, 12, 14, 18, 22, 24, 26, 30, 34, 36, 38, 40, 42, 46, 54, 56, 58, 60, 62, 66, 70, 72, 74, 78, 80, 82, 84, 86, 88, 90, 94, 96, 102, 104, 106, 108, 110, 114, 118, 120, 122, 126, 128, 130, 132, 134, 136, 138, 142, 146, 150, 152, 154, 156, 158, 166, 168
Offset: 1

Views

Author

Leroy Quet, Jun 29 2006

Keywords

Comments

Sequence is identical to A048751 except for terms 1 and 2 that are included here. - Michel Marcus, Jun 06 2014
Numbers k such that tau(k) = A000005(k) divides the product of the divisors of k (A007955). - Jaroslav Krizek, Sep 05 2017

Examples

			d(26) = 4. 2 is the only prime dividing 4. 2 divides 26, so 26 is in the sequence.

Links

Crossrefs

Cf. A000005, A120737.

Programs

  • Magma
    [n: n in [1..1000] | Denominator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]) eq 1];  // Jaroslav Krizek, Sep 05 2017
  • Maple
    isA120736 := proc(n) local d,p; d := numtheory[tau](n) ; p := 2 ; while p <= n do if ( d mod p ) = 0 then if ( n mod p ) <> 0 then RETURN(false) ; fi ; fi ; p := nextprime(p) ; od ; RETURN(true) ; end: for n from 1 to 200 do if isA120736(n) then printf("%d,",n) ; fi ; od ;
# R. J. Mathar, Aug 17 2006
  • Mathematica
    Select[Range@ 168, Divisible[Times @@ Divisors@ #, DivisorSigma[0, #]] &] (* Michael De Vlieger, Sep 05 2017 *)

Extensions

More terms from R. J. Mathar, Aug 17 2006

A137926 a(n) = the largest divisor of n that is coprime to A000005(n). (A000005(n) = the number of positive divisors of n.)

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 1, 1, 5, 11, 1, 13, 7, 15, 16, 17, 1, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 4, 37, 19, 39, 5, 41, 21, 43, 11, 5, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 5, 61, 31, 7, 64, 65, 33, 67, 17, 69, 35, 71, 1, 73, 37, 25, 19, 77, 39, 79
Offset: 1

Views

Author

Leroy Quet, Feb 23 2008

Keywords

Examples

			6 has 4 positive divisors. The divisors of 6 are 1,2,3,6. The divisors of 6 that are coprime to 4 are 1 and 3. 3 is the largest of these; so a(6) = 3.

Links

Crossrefs

Cf. A046642 (a(n)=n), A120737 (a(n)=1), A137927.

Programs

  • Maple
    f := proc (n) local D, t; D := numtheory:-divisors(n); t := nops(D); max(select(proc (d) options operator, arrow; igcd(d, t) = 1 end proc, D)) end proc:
map(f, [$1..100]); # Robert Israel, Feb 11 2018
  • Mathematica
    Table[Select[Divisors[n], GCD[ #, Length[Divisors[n]]] == 1 &][[ -1]], {n, 1, 80}] (* Stefan Steinerberger, Mar 09 2008 *)
  • PARI
    a(n) = {my(d = divisors(n)); vecmax(select(x->(gcd(x, #d) == 1), d));} \\ Michel Marcus, Feb 12 2018

Extensions

More terms from Stefan Steinerberger, Mar 09 2008

A302974 a(n) = numerator of tau(n)^n / n^tau(n).

Original entry on oeis.org

1, 1, 8, 81, 32, 256, 128, 16, 27, 65536, 2048, 729, 8192, 16777216, 1073741824, 152587890625, 131072, 2985984, 524288, 892616806656, 4398046511104, 1099511627776, 8388608, 281474976710656, 847288609443, 281474976710656, 18014398509481984, 1499253470328324096
Offset: 1

Views

Author

Jaroslav Krizek, Apr 16 2018

Keywords

Comments

tau(n) = the number of the divisors of n (A000005).
a(n) <= A302975(n) only for numbers n = 1, 2 and 3.

Examples

			For n = 6; tau(6)^6 / 6^tau(6) = 4^6 / 6^4 = 256 / 81; a(6) = 256.

Links

Crossrefs

Cf. A000005, A120737, A302975, A302976.

Programs

  • Magma
    [Numerator((NumberOfDivisors(n)^n) / (n^NumberOfDivisors(n))): n in[1..100]];
  • Mathematica
    Numerator[#[[2]]^#[[1]]/#[[1]]^#[[2]]]&/@Table[{n,DivisorSigma[0,n]},{n,30}] (* Harvey P. Dale, May 29 2023 *)

A302975 a(n) = denominator of tau(n)^n / n^tau(n).

Original entry on oeis.org

1, 1, 9, 64, 25, 81, 49, 1, 1, 625, 121, 1, 169, 2401, 50625, 1048576, 289, 1, 361, 15625, 194481, 14641, 529, 6561, 15625, 28561, 531441, 117649, 841, 2562890625, 961, 1, 1185921, 83521, 1500625, 262144, 1369, 130321, 2313441, 390625, 1681, 37822859361, 1849
Offset: 1

Views

Author

Jaroslav Krizek, Apr 16 2018

Keywords

Comments

tau(n) = the number of the divisors of n (A000005).
Conjecture: all terms are squares.
a(n) >= A302974(n) only for numbers n = 1, 2 and 3.

Examples

			For n = 6; tau(6)^6 / 6^tau(6) = 4^6 / 6^4 = 256 / 81; a(6) = 81.

Links

Crossrefs

Cf. A000005, A120737, A302974, A302976.

Programs

  • Magma
    [Denominator((NumberOfDivisors(n)^n) / (n^NumberOfDivisors(n))): n in[1..100]];
  • Mathematica
    Denominator[#[[2]]^#[[1]]/#[[1]]^#[[2]]]&/@Table[{n,DivisorSigma[0,n]},{n,50}] (* Harvey P. Dale, Sep 15 2019 *)

Formula

a(p) = p^2 for p = prime.
a(A120737(n)) = 1.

A302976 a(n) = tau(n)^n mod n^tau(n).

Original entry on oeis.org

0, 0, 8, 17, 7, 208, 30, 0, 0, 8576, 112, 0, 80, 22864, 36199, 159681, 155, 0, 116, 40062976, 83791, 142928, 255, 26138902528, 68, 302656, 362152, 454885376, 60, 544999124224, 374, 0, 226279, 629152, 399674, 27234498115233, 76, 956704, 956539, 3361080344576
Offset: 1

Views

Author

Jaroslav Krizek, Apr 16 2018

Keywords

Comments

tau(n) = the number of the divisors of n (A000005).
tau(n)^n > n^tau(n) for all n > 3.

Examples

			For n = 8; a(8) = 0 because tau(8)^8 mod 8^tau(8) = 4^8 mod 8^4 = 65536 mod 4096 = 0.

Crossrefs

Cf. A000005, A120737, A302974, A302975.

Programs

  • Magma
    [(NumberOfDivisors(n)^n) mod (n^NumberOfDivisors(n)): n in[1..100]];
  • Mathematica
    PowerMod[#[[2]],#[[1]],#[[1]]^#[[2]]]&/@Table[{n,DivisorSigma[0,n]},{n,40}] (* Harvey P. Dale, Jan 08 2023 *)

Formula

a(n) = A000005(n)^n mod n^A000005(n) = A302974(n) mod A302975(n).
a(A120737(n)) = 0.
Showing 1-5 of 5 results.

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