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.

A069782 Numbers k such that gcd(d(k^3), d(k)) = 2^w for some w.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107
Offset: 1

Views

Author

Labos Elemer, Apr 08 2002

Keywords

Comments

The first missing integer is 432 (see in A069781).

Examples

			Below 100000 only 314 integers are missing, collected in A069781.

Links

Crossrefs

Cf. A000005, A069780, A069781, A037992, A061701.

Programs

  • Mathematica
    f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]]; Do[s=f[n]; If[IntegerQ[Log[2, s]], Print[{n, s}]], {n, 1, 100000}]
  • PARI
    is(n)=my(f=factor(n)[, 2], g=gcd(prod(i=1, #f, 3*f[i]+1), prod(i=1, #f, f[i]+1))); g>>valuation(g, 2)==1 \\ Charles R Greathouse IV, Oct 16 2015

A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

Original entry on oeis.org

432, 576, 648, 1600, 2000, 2160, 2880, 2916, 3024, 3136, 3240, 4032, 4536, 4752, 4800, 5000, 5488, 5616, 6000, 6336, 7128, 7344, 7488, 7744, 8208, 8424, 9408, 9792, 9936, 10125, 10800, 10816, 10944, 11016, 11200, 12312, 12528, 13248, 13392
Offset: 1

Views

Author

Labos Elemer, Apr 08 2002

Keywords

Comments

The complement of this sequence in the positive integers A000027 is A069782. - M. F. Hasler, Jan 18 2015
The numbers of the form 4*3^(7*m - 1), m >= 1, are terms. - Marius A. Burtea, Oct 18 2019

Examples

			For n<100000, gcd[d(n^3),d[n]] = {5,7,10,14,20,28,40,80} which is obtained for n={20736,576,432,2880,54000,20160,2160,15120} respectively.

Links

Crossrefs

Cf. A000005, A061701, A037992, A069780, A069782.

Programs

  • Magma
    f:=func; [k:k in [1..14000]| not IsIntegral(Log(2,f(k)))]; // Marius A. Burtea, Oct 18 2019
  • Mathematica
    f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]] Do[s=f[n]; If[ !IntegerQ[Log[2, s]], Print[n]], {n, 1, 100000}]
Select[Range[14000],!IntegerQ[Log[2,GCD[DivisorSigma[0,#^3], DivisorSigma[ 0,#]]]]&] (* Harvey P. Dale, Mar 20 2018 *)
  • PARI
    is(n)=my(f=factor(n)[,2], g=gcd(prod(i=1,#f,3*f[i]+1), prod(i=1,#f,f[i]+1))); g!=1<Charles R Greathouse IV, Oct 16 2015

Formula

log_2(gcd(A000005(n^3), A000005(n))) is nonintegral.

A069783 a(n) = gcd(d(n!^3), d(n!)), where d() is the number of divisors function.

Original entry on oeis.org

1, 2, 4, 8, 16, 2, 4, 32, 32, 10, 20, 8, 16, 32, 448, 448, 1792, 32, 64, 80, 3200, 1280, 2560, 320, 448, 1792, 25088, 101920, 203840, 128, 256, 4096, 81920, 112640, 2048, 8960, 17920, 1024, 2048, 5120, 10240, 5734400, 11468800, 1003520, 250880, 8960, 17920
Offset: 1

Views

Author

Labos Elemer, Apr 08 2002

Keywords

Links

Crossrefs

Cf. A000005, A000142, A069780, A069781, A069782.

Programs

  • Mathematica
    Table[GCD[DivisorSigma[0, (n!)^3], DivisorSigma[0, n! ]], {n, 1, 100}]
  • PARI
    a(n) = my(e = factor(n!)[, 2]); gcd(vecprod(apply(x -> x + 1, e)), vecprod(apply(x -> 3*x + 1, e))) \\ Amiram Eldar, Mar 07 2025

Formula

a(n) = A069780(n!).

A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 18, 19, 30, 31, 32, 35, 38, 39
Offset: 1

Views

Author

Labos Elemer, Apr 08 2002

Keywords

Comments

From David A. Corneth, Jul 31 2017: (Start)
Theorem: There are no further terms.
Proof:
Let e_n(n, p) be the exponent of p in n!. The prime p has exponent e_(n, p) = n/p for sqrt(n) < p < n in n!. n/4 <= p < n/3, e_(n, p) = 3 so e_(n, p) * 3 = 9. and for n/5 <= p < n/4, e = 4. The gap g_n between prime(n) and prime(n+1) is about sqrt(n) * log(n). There is a gap of n/4 - n/5 = n/20 between n/5 and n/4. primepi(1000) = 168, so for n > 5*1000, the gap between n/5 and the next prime is about sqrt(168) * log(168) ~= 66. This is much less than n/20. No 40 <= m <= 15000 is in the sequence, which completes the proof. (End)

Crossrefs

Cf. A000005, A000142, A069780, A069781, A069782, A069783.

Programs

  • Mathematica
    Do[s=GCD[DivisorSigma[0, (n!)^3], DivisorSigma[0, n! ]]; If[IntegerQ[n/100], Print[{n}]]; If[IntegerQ[Log[2, s]], Print[n]], {n, 1, 10000}]
  • PARI
    val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {my(p1 = p2 = 1); forprime(p=2, n, v = val(n, p); p1 *= (v + 1); p2 *= (3*v + 1)); g = gcd(p1, p2); g==2^(valuation(g, 2))} \\ David A. Corneth, Jul 31 2017

Extensions

Keywords fini and full added by David A. Corneth, Jul 31 2017

A069785 a(n) = A061680(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 15, 15, 3, 5, 135, 135, 99, 99, 9, 63, 21, 21, 459, 459, 135, 19, 15, 15, 15, 21, 189, 189, 585, 585, 18225, 18225, 675, 15, 135, 891, 8505, 25515, 81, 81, 7695, 7695, 1575, 1575, 4725, 6615, 40635, 40635, 945, 1215, 3645, 3645, 151875, 151875
Offset: 1

Views

Author

Labos Elemer, Apr 09 2002

Keywords

Examples

			Observe cases when consecutive terms are equal: n={1,2,3,4,6,10,...,78,80,82,88,96}.

Links

Crossrefs

Cf. A000005, A000142, A061680, A061701, A048691, A069780, A069781, A069782, A069783, A069784.

Programs

  • Mathematica
    a[n_] := Module[{e = FactorInteger[n!][[;;, 2]]}, GCD[Times @@ (2*e+1), Times @@ (e+1)]]; Array[a, 100] (* Amiram Eldar, Dec 02 2023 *)
  • PARI
    a(n) = {my(e = factor(n!)[,2]); gcd(vecprod(apply(x -> 2*x+1, e)), vecprod(apply(x -> x+1, e)));} \\ Amiram Eldar, Dec 02 2023

Formula

a(n) = A061680(A000142(n)). - Amiram Eldar, Dec 02 2023
Showing 1-5 of 5 results.

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