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.

A242016 Smallest prime factor of A135972(n), the n-th composite Mersenne number.

Original entry on oeis.org

3, 3, 3, 7, 3, 23, 3, 3, 7, 3, 3, 3, 7, 3, 47, 3, 31, 3, 7, 3, 233, 3, 3, 7, 3, 31, 3, 223, 3, 7, 3, 13367, 3, 431, 3, 7, 3, 2351, 3, 127, 3, 7, 3, 6361, 3, 23, 3, 7, 3, 179951, 3, 3, 7, 3, 31, 3, 193707721, 3, 7, 3
Offset: 1

Views

Author

Felix Fröhlich, Aug 11 2014

Keywords

Links

Crossrefs

Cf. A135972, A242017. Subsequence of A056608.

Programs

  • PARI
    for(n=2, 1e2, if(!ispseudoprime(2^n-1), p=factor(2^n-1)[1, 1]; print1(p, ", ")))

Formula

a(n) = A020639(A135972(n)).

A135981 Number of distinct prime factors of A135972(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 3, 3, 4, 4, 5, 3, 4, 2, 6, 3, 3, 3, 6, 3, 6, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 6, 3, 7, 6, 4, 3, 9, 2, 7, 5, 7, 3, 6, 6, 8, 4, 6, 2, 11, 3, 6, 7, 3, 8, 2, 7, 4, 9, 3, 12, 3, 5, 7, 7, 4, 7, 3, 9, 6, 5, 2, 12, 3, 5, 6, 10, 11, 5, 9, 3, 6, 5, 12, 2, 5, 8, 12
Offset: 2

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Examples

			A135972(3) = 15 = 3*5 which has a(3)=2 distinct prime factors.

Links

Crossrefs

Cf. A046051, A046053, A022307, A135972.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^n - 1], c = FactorInteger[2^n - 1]; d = Length[c]; AppendTo[k, d]], {n, 1, 100}]; k

Formula

a(n) = A001221(A135972(n)) .

Extensions

Offset set to 2, definition shortened - R. J. Mathar, Oct 01 2009

A353786 Number of distinct nonprime numbers of the form 2^k - 1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, May 12 2022

Keywords

Examples

			Divisors of 255 are [1, 3, 5, 15, 17, 51, 85, 255], of these of the form 2^k - 1 (A000225) are 1, 3, 15 and 255, but only three of them are counted (because 3 is a prime), therefore a(255) = 3.

Links

Crossrefs

Cf. A000225, A000265, A147645, A154402.
Cf. A065442, A135972, A173898.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, !PrimeQ[#] && # + 1 == 2^IntegerExponent[# + 1, 2] &]; Array[a, 120] (* Amiram Eldar, May 12 2022 *)
  • PARI
    A353786(n) = { my(m=1,s=0); while(m<=n, s += (!isprime(m))*!(n%m); m += (m+1)); (s); };

Formula

a(n) = A154402(n) - A147645(n).
a(n) = a(2*n) = a(A000265(n)).
For all primes p, a(p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A135972(n) = A065442 - A173898 = 1.0902409734... . - Amiram Eldar, Dec 31 2023

A329534 Irregular triangle read by rows: for n >= 1 row n lists the k from [1, 2, ... , n] such that A002378(k-1) = (k-1)*k == 0 (mod n).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 3, 4, 6, 1, 7, 1, 8, 1, 9, 1, 5, 6, 10, 1, 11, 1, 4, 9, 12, 1, 13, 1, 7, 8, 14, 1, 6, 10, 15, 1, 16, 1, 17, 1, 9, 10, 18, 1, 19, 1, 5, 16, 20, 1, 7, 15, 21, 1, 11, 12, 22, 1, 23, 1, 9, 16, 24, 1, 25
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 15 2019

Keywords

Comments

n-th row length gives 1 for n = 1, and 2^A001221(n) for n >= 2 , that is A034444(n). [Proof: Unique lifting theorem (e.g., Apostol, 5.30 (a), p.121) for this congruence, and only two solutions 1 and p for primes p. See also the Yuval Dekel, Sep 21 2003, comment in A034444. - Wolfdieter Lang, Feb 05 2020]

Examples

			The irregular triangle T(n,k) begins
n\k  1  2  3  4 ...
1:   1
2:   1  2
3:   1  3
4:   1  4
5:   1  5
6:   1  3  4  6
7:   1  7
8:   1  8
9:   1  9
10:  1  5  6 10
11:  1 11
12:  1  4  9 12
13:  1 13
14:  1  7  8 14
15:  1  6 10 15
16:  1 16
17:  1 17
18:  1  9 10 18
19:  1 19
20:  1  5 16 20
...

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.

Crossrefs

Cf. A000010, A000225, A000688, A000961, A001221, A006881, A006530, A007875, A020639, A024619, A034444, A077610, A135972, A309307.

Programs

  • Magma
    [[k: k in [1..n] | k^2 mod n eq k]: n in [1..38]];
  • Mathematica
    Table[Select[Range@ n, Mod[-n + # (# - 1), n] == 0 &], {n, 25}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)
  • PARI
    row(n) = select(x->(Mod(x, n) == Mod(x, n)^2), [1..n]); \\ Michel Marcus, Nov 19 2019

Extensions

Edited by Wolfdieter Lang, Feb 05 2020

A357604 Number of prime powers in the sequence of the floor of n/k for k <= n, A010766.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 5, 6, 6, 8, 8, 7, 8, 7, 8, 10, 11, 9, 10, 11, 12, 12, 13, 11, 12, 14, 14, 15, 16, 14, 15, 16, 17, 16, 17, 16, 17, 18, 18, 20, 21, 19, 21, 21, 21, 22, 23, 22, 23, 23, 24, 26, 27, 22, 23, 24, 25, 28, 28, 28, 29, 29, 30, 30, 31, 27
Offset: 1

Views

Author

Randell Heyman, Oct 06 2022

Keywords

Comments

Conjecture: a(n+1) - a(n) has all its record values at n = A135972(k) and the record values of a(n-1) - a(n) are a subsequence of A025487. This was verified for n = 1..20000. - Thomas Scheuerle, Oct 06 2022

Examples

			For n=8 we have floor(8/1) = 8 = 2^3, a prime power; floor(8/2) = 4 = 2^2, a prime power; floor(8/3) = floor(8/4) = 2 = 2^1, a prime power. Each remaining term of the sequence is 1, which is not a prime power, so a(8) = 4.

Links

Crossrefs

Cf. A000961, A025487, A135972, A010766.

Programs

  • MATLAB
    function a = A357604( max_n )
    for n = 1:max_n
        s = floor(n./[1:n]); c = 0;
        for m = 1:n-1
            f = factor(s(m));
            if s(m) > 1 && length(unique(f)) == 1
                c = c+1;
            end
        end
        a(n) = c;
    end
end % Thomas Scheuerle, Oct 06 2022
  • PARI
    a(n) = sum(k=1,n, isprimepower(n\k)!=0); \\ Thomas Scheuerle, Oct 07 2022

Formula

a(n) = c*n + O(n^(1/2)), where c is the sum of 1/(q*(q+1)) over all prime powers q.

Extensions

a(12)-a(72) from Thomas Scheuerle, Oct 06 2022
Showing 1-5 of 5 results.

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