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.

A245197 Numbers n where tau(n) and n-tau(n) are perfect squares, with tau(n) the number of divisors of n (A000005).

Original entry on oeis.org

1, 8, 85, 125, 365, 445, 533, 629, 965, 1685, 1800, 1853, 2340, 2605, 2813, 3029, 3973, 4765, 5045, 5220, 5629, 5933, 6245, 6893, 8285, 8653, 11029, 11453, 11885, 12773, 14165, 15133, 16645, 17165, 17460, 17693, 20453, 21029, 22205, 22805, 23413, 24653, 27229
Offset: 1

Views

Author

Reinhard Muehlfeld, Jul 13 2014

Keywords

Comments

Subsequence of A036436. - Michel Marcus, Jul 15 2014
Intersection of A036436 and A245388. - Robert Israel, Jul 20 2014

Links

Crossrefs

Cf. A000005 (tau(n)), A049820 (n - tau(n)), A036436 (n such that tau(n) is square), A245388 (n such that n - tau(n) is square).

Programs

  • Maple
    filter:= proc(n) local t;
  t:= numtheory:-tau(n);
  issqr(t) and issqr(n-t)
end proc;
select(filter, [$1..10^5]); # Robert Israel, Jul 20 2014
  • Mathematica
    okQ[n_] := With[{t = DivisorSigma[0, n]}, IntegerQ@Sqrt[t] && IntegerQ@Sqrt[n-t]];
Select[Range[10^5], okQ] (* Jean-François Alcover, Feb 08 2023 *)
  • PARI
    isok(n) = issquare(numdiv(n)) && issquare(n-numdiv(n)); \\ Michel Marcus, Jul 15 2014

A356833 Primes p such that the minimum number of divisors among the numbers between p and NextPrime(p) is a square.

Original entry on oeis.org

5, 13, 19, 31, 37, 43, 53, 61, 67, 73, 79, 83, 89, 103, 109, 127, 131, 139, 151, 157, 163, 173, 181, 193, 199, 211, 223, 233, 241, 251, 257, 263, 269, 271, 277, 293, 307, 311, 313, 317, 331, 337, 353, 367, 373, 379, 383, 389, 397, 401, 409, 421, 433, 443, 449, 457, 461, 463, 467, 479
Offset: 1

Views

Author

Claude H. R. Dequatre, Sep 16 2022

Keywords

Examples

			13 is a term because up to the next prime 17, tau(14) = 4, tau(15) = 4, tau(16) = 5, thus the smallest tau(k) is 4 and 4 is a square (2^2).
23 is prime but not a term because up to the next prime 29, tau(24) = 8, tau(25) = 3, tau(26) = 4, tau(27) = 4, tau(28) = 6, thus the smallest tau(k) = 3 and 3 is not a square.

Crossrefs

Cf. A000005, A000040, A000290, A061112, A036436.
Cf. A353284, A353285, A353286, A357170, A357175.

Programs

  • PARI
    isok(p)=issquare(vecmin(apply(numdiv, [p+1..nextprime(p+1)-1])));
forprime(p=3, 2000, if(isok(p), print1(p", ")))

A174331 n such that tau(Fibonacci(n)) is a perfect square.

Original entry on oeis.org

1, 2, 6, 8, 9, 10, 14, 18, 19, 20, 22, 26, 27, 28, 30, 31, 32, 34, 40, 41, 42, 52, 53, 55, 59, 61, 64, 66, 71, 73, 74, 77, 79, 85, 87, 89, 92, 93, 94, 95, 97, 99, 101, 107, 109, 113, 115, 116, 117, 120, 121, 123, 125, 127, 128, 129, 130, 133, 135, 138, 143, 146, 147, 149
Offset: 1

Views

Author

Michel Lagneau, Mar 15 2010

Keywords

Comments

tau = A000005 is the number of divisors of n.

Examples

			40 is in the sequence because tau(Fibonacci(40)) = tau(102334155) = 64 is square.

Links

Crossrefs

Cf. A000005, A000045, A000290, A036436, A063375.

Programs

  • Mathematica
    Select[Range[150], IntegerQ[Sqrt[DivisorSigma[0,Fibonacci[#]]]] &]
  • PARI
    is(n) = issquare(numdiv(fibonacci(n))) \\ Felix Fröhlich, Sep 08 2019

Formula

{n: A063375(n) in A000290}. - R. J. Mathar, Jul 09 2012

A190266 Numbers k such that tau(k-1) = (tau(k))^2 = tau(k+1), where tau(k) = A000005(k) (number of divisors of k).

Original entry on oeis.org

7, 1241, 1673, 1751, 1769, 2471, 2839, 3161, 3305, 3497, 3711, 4135, 4265, 4279, 4471, 4711, 5191, 5433, 5561, 6017, 6041, 6103, 6313, 6809, 6953, 7031, 7241, 7463, 7671, 8023, 8057, 8345, 8791, 8889, 9079, 10167, 10793, 10841, 11111, 11209, 11391, 11751, 12297, 12729
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 06 2011

Keywords

Examples

			a(1)=7 because tau(6) = (tau(7))^2 = tau(8) = 4;
a(2)=1241 because tau(1240) = (tau(1241))^2 = tau(1242) = 16.

Links

Crossrefs

Cf. A000005, A074757, A090502. Subsequence of A036436.

Programs

  • Mathematica
    Transpose[Select[Partition[Range[15000], 3, 1], DivisorSigma[0, #[[2]]]^2 == DivisorSigma[0, First[#]] == DivisorSigma[0, Last[#]]&]][[1]] + 1 (* Amiram Eldar, Jul 17 2019 after Harvey P. Dale at A175116 *)
  • PARI
    isA190266(n)=my(t=numdiv(n-1)); issquare(t) & t==numdiv(n+1) & t==numdiv(n)^2 \\ Charles R Greathouse IV, May 14 2011

Formula

A000005(a(n)-1) = (A000005(a(n)))^2 = A000005(a(n)+1).

Extensions

Data corrected by Amiram Eldar, Jul 17 2019

A245199 Numbers n where phi(n) and tau(n) are perfect squares.

Original entry on oeis.org

1, 8, 10, 34, 57, 74, 85, 125, 185, 202, 219, 394, 451, 456, 489, 505, 514, 546, 570, 629, 640, 679, 680, 802, 985, 1000, 1026, 1057, 1154, 1285, 1354, 1365, 1387, 1417, 1480, 1717, 1752, 1938, 2005, 2016, 2047, 2176, 2190, 2340, 2457, 2509, 2565, 2594, 2649
Offset: 1

Views

Author

Reinhard Muehlfeld, Jul 13 2014

Keywords

Comments

Numbers n such that A000005(n) and A000010(n) are perfect squares.
Intersection of A036436 and A039770. - Michel Marcus, Jul 15 2014

Examples

			8 is in the sequence because phi(8) = 4, tau(8) = 4, and 4 is a perfect square.
12 is not in the sequence because tau(12) = 6 is not a square.

Links

Crossrefs

Cf. A000005, A000010, A036436, A039770.

Programs

  • Maple
    filter:= proc(n) uses numtheory; issqr(phi(n)) and issqr(tau(n)) end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 27 2014
  • Mathematica
    fQ[n_] := IntegerQ@ Sqrt@ EulerPhi[n] && IntegerQ@ Sqrt@ DivisorSigma[0, n]; Select[ Range@ 3000, fQ] (* Robert G. Wilson v, Jul 21 2014 *)
Select[Range[3000],AllTrue[{Sqrt[EulerPhi[#]],Sqrt[DivisorSigma[0, #]]}, IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 01 2018 *)
  • PARI
    isok(n) = issquare(numdiv(n)) && issquare(eulerphi(n)); \\ Michel Marcus, Jul 15 2014
  • Python
    from sympy import totient, divisor_count
from gmpy2 import is_square
[n for n in range(1,10**4) if is_square(int(divisor_count(n))) and is_square(int(totient(n)))] # Chai Wah Wu, Aug 04 2014
Showing 1-5 of 5 results.

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