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.

A309245 Least number k > 0 which is not a divisor of n such that k^2 + n is a nonsquare powerful number (A102834).

Original entry on oeis.org

682, 5, 37, 11, 1879706, 463, 11, 10, 2046, 341881, 31, 74, 70, 5519, 793, 22, 1952785824219551870, 57, 559, 338, 4580728614212333152148, 503259461, 45, 926, 190, 109, 36, 62, 436, 832836278711, 63, 88, 2451448196948930, 7037029, 36, 33
Offset: 1

Views

Author

Amiram Eldar, Jul 18 2019

Keywords

Comments

De Koninck et al. calculated the first 50 terms of this sequence.

Examples

			a(1) = 682 since 682^2 + 1 = 465125 = 5^3 * 61^2 is a nonsquare powerful number and is the smallest k > 0 such that k^2 + 1 is not a nonsquare powerful number.

Links

Crossrefs

Cf. A001694, A102834, A140746, A309246.

Programs

  • Mathematica
    powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; powerfulNonsquare[n_] := !IntegerQ[Sqrt[n]] && powerfulQ[n]; a[n_] := Module[{k=1}, While[Divisible[n, k] || !powerfulNonsquare[k^2 + n], k++]; k]; Table[a[n], {n, 1, 16}]
  • PARI
    is_a102834(n) = ispowerful(n) && !issquare(n) \\ after Charles R Greathouse IV in A102834
a(n) = for(k=1, oo, if(n%k!=0 && is_a102834(k^2+n), return(k))) \\ Felix Fröhlich, Jul 19 2019

A090699 Decimal expansion of the Erdos-Szekeres constant zeta(3/2)/zeta(3).

Original entry on oeis.org

2, 1, 7, 3, 2, 5, 4, 3, 1, 2, 5, 1, 9, 5, 5, 4, 1, 3, 8, 2, 3, 7, 0, 8, 9, 8, 4, 0, 4, 3, 8, 2, 2, 3, 7, 2, 2, 9, 0, 6, 7, 1, 1, 3, 2, 9, 1, 3, 1, 6, 6, 0, 8, 5, 6, 7, 4, 9, 1, 7, 5, 7, 5, 8, 9, 6, 7, 0, 5, 9, 6, 6, 1, 7, 2, 6, 6, 4, 4, 4, 6, 8, 2, 0, 3, 7, 8, 5, 7, 2, 7, 8, 3, 8, 3, 1, 7, 6, 5, 1, 0, 2, 6, 6, 4
Offset: 1

Views

Author

Benoit Cloitre, Jan 14 2004

Keywords

Comments

Let N(x) denotes the number of 2-full integers not exceeding x. Then lim_{x->oo} N(x)/sqrt(x) = zeta(3/2)/zeta(3). Also related to Niven's constant.

Examples

			zeta(3/2)/zeta(3) = 2.17325431251955413823708984...

References

  • S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 112-114.

Links

Crossrefs

Cf. A001694 (powerful numbers), A102834 (nonsquare powerful numbers).
Cf. A033150, A051904.

Programs

  • Mathematica
    RealDigits[N[Zeta[3/2]/Zeta[3],150]][[1]] (* T. D. Noe, May 03 2006 *)
  • PARI
    zeta(3/2)/zeta(3) \\ Michel Marcus, Oct 06 2017

Formula

Product_{p prime} (1+1/p^(3/2)) = zeta(3/2)/zeta(3). - T. D. Noe, May 03 2006
Equals lim_{n->oo} (Sum_{k=1..n} A051904(k) - n)/sqrt(n) (Niven, 1969). - Amiram Eldar, Jul 11 2020

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 16 2007

A309246 Least number k which is not a divisor of n such that k^2 - n is a nonsquare powerful number.

Original entry on oeis.org

3, 11427, 15503069909027, 6, 73, 62531004125, 85227106679780, 20, 15, 71457130044805582612325294634331, 56, 47, 16, 33017, 1138, 68, 23, 19, 762488, 146, 1552808, 47, 6234, 32, 45, 2537, 51700, 54, 426, 83, 34, 40, 3601, 948281, 531783519104, 42, 73, 16493
Offset: 1

Views

Author

Amiram Eldar, Jul 18 2019

Keywords

Comments

De Koninck et al. calculated the first 50 terms of this sequence.

Links

Crossrefs

Cf. A001694, A102834, A309245.

Programs

Formula

a(2) = 11427 since 11427^2 - 2 = 130576327 = 7^3 * 617^2 is a nonsquare powerful number, and k^2 - 2 is not a nonsquare powerful number for all k < 11427.

A376171 Powerful numbers whose prime factorization has an odd maximum exponent.

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 216, 243, 288, 343, 392, 500, 512, 675, 800, 864, 968, 972, 1000, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1800, 1944, 2048, 2187, 2197, 2312, 2592, 2700, 2744, 2888, 3087, 3125, 3200, 3267, 3375, 3456, 3528, 3872, 3888, 4000
Offset: 1

Views

Author

Amiram Eldar, Sep 13 2024

Keywords

Comments

Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.
Powerful numbers k such that A051903(k) is odd.
Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.
The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.

Links

Crossrefs

Complement of A376170 within A001694.
Intersection of A001694 and A376142.
Subsequence of A102834.
Subsequences: A030078, A050997, A079395, A092759, A138031, A179665, A335988 \ {1}.
Cf. A051903, A082695, A376092.

Programs

  • Mathematica
    seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]
  • PARI
    is(k) = {my(f = factor(k), e = f[,2]); #e && ispowerful(f) && vecmax(e) % 2;}

Formula

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (-1)^k * s(k) = 0.29116340833243888282..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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