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.

A131100 Indices of A002808 such that A081545(m) = Product_{k=m*(m-1)/2+1..m*(m+1)/2} A002808(a(k)).

Original entry on oeis.org

1, 1, 4, 1, 2, 3, 1, 2, 4, 5, 1, 2, 3, 5, 6, 1, 2, 3, 4, 5, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 8, 10, 1, 2, 3, 4, 5, 6, 7, 8, 13, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13
Offset: 1

Views

Author

M. F. Hasler, Jun 14 2007

Keywords

Examples

			a(2..3) = {1,4} since A081545(2) = A002808(1)*A002808(4) + 1;
a(7..10) = {1,2,4,5} since A081545(4) = A002808(1)*A002808(2)*A002808(4)*A002808(5) + 1.
Triangle begins:
1;
1, 4;
1, 2, 3;
1, 2, 4, 5;
1, 2, 3, 5, 6;
1, 2, 3, 4, 5, 7;
1, 2, 3, 4, 5, 6, 7;
...

Links

Crossrefs

Cf. A002808, A081545.

Extensions

Offset changed to 1 by and more terms from Jinyuan Wang, Dec 04 2020

A073918 Smallest prime which is 1 more than a product of n distinct primes: a(n) is a prime and a(n) - 1 is a squarefree number with n prime factors.

Original entry on oeis.org

2, 3, 7, 31, 211, 2311, 43891, 870871, 13123111, 300690391, 6915878971, 200560490131, 11406069164491, 386480064480511, 18826412648012971, 693386350578511591, 37508276737897976011, 3087649419126112110271, 183452981525059000664911, 11465419967969569966774411
Offset: 0

Views

Author

Amarnath Murthy, Aug 18 2002

Keywords

Comments

Apparently the same as record values of A055734: least k such that phi(k) has n distinct prime factors, where phi is Euler's totient function. If the Mathematica program is used for large n, then "fact" should be reduced to, say, 1.1 in order to increase the search speed. - T. D. Noe, Dec 17 2003

Examples

			a(0) = 1 + 1 = 2 (empty product of zero primes).
a(1) = 1 + 2 = 3.
a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3*5 = 31.
a(4) = 1 + 2*3*5*7 = 211.
a(5) = 1 + 2*3*5*7*11 = 1 + 11# = 2311.
a(6) = 1 + 2*3*5*7*11*19 = 43891, since 13# + 1 and 11#*17 + 1 = 17#/13 + 1 is not prime, and 17#/p + 1 is larger than a(6) for all p in {2, ..., 11}.
The index of the smallest prime which is not a factor of a(n)+1 is (1, 2, 3, 4, 5, 6, 6, 7, 7, 9, 10, 12, 11, 12, 13, 15, 16, 15, 16, 18, 19, 20, 21, 22, 22, 23, 25, 27, 26, 29, 29, ...) for n = 0, 1, 2, ... - _M. F. Hasler_, May 31 2018

Links

Crossrefs

Cf. A055734 (number of distinct prime factors of phi(n)).
Cf. A073917, A098026.
Cf. A000040 (primes), A002110 (primorial), A081545 (same with composite instead of primes).

Programs

  • Mathematica
    Generate[pIndex_, i_] := Module[{p2, t}, p2=pIndex; While[p2[[i]]++; Do[p2[[j]]=p2[[i]]+j-i, {j, i+1, Length[p2]}]; t=Times@@Prime[p2]; t
  • PARI
    A073918(n, b=0 /*best*/, p=1 /*product*/, f=[]/*factors*/)={ if( #f= f[n+1] ) || !b = A073918( n-1, b, p*f[n], f), f[n]= nextprime( f[n]+1 ) ); b } \\ then, e.g.: apply(A073918, [0..30]). - M. F. Hasler Jun 16 2007

Formula

From M. F. Hasler, Jun 16 2007 (Start):
Conjecture: For any m > 0 there is K > 0 such that for all k > K, a(k)-1 is divisible by the first m primes.
Corollary: For any m > 1 there is K > 0 such that for all k > K, a(k) = 1 (mod m).
Conjecture 2: Let K(m) be the smallest possible K satisfying the above Conjecture. Then K(m) ~ m, i.e., a(k) ~ A002110(k), only very few of the last factors will be a bit larger. (End)
Remark: the last "~" above was not intended to mean asymptotic equivalence. It appears that lim inf a(n)/A002110(n) = 1, but the lim sup might well be larger. It would be interesting to know whether it has a finite value. - M. F. Hasler, May 31 2018

Extensions

More terms from Vladeta Jovovic, Aug 20 2002
Edited by M. F. Hasler, May 31 2018

A081546 Smallest prime which is 1 more than the product of n (not necessarily distinct) composite numbers.

Original entry on oeis.org

2, 5, 17, 97, 257, 3457, 12289, 40961, 65537, 786433, 5308417, 14155777, 104857601, 167772161, 1811939329, 3221225473, 24159191041, 77309411329, 206158430209, 2061584302081, 2748779069441, 6597069766657, 39582418599937
Offset: 0

Views

Author

Amarnath Murthy, Apr 01 2003

Keywords

Examples

			a(0) = 1 + 1 = 2, a(3) = 4*4*6 + 1 = 97.

Crossrefs

Cf. A081545.

Extensions

More terms from Michel ten Voorde Jun 13 2003
Added missing term 12289 and a(13)-a(22) from Donovan Johnson, Nov 05 2009

A081547 Smallest composite number which is 1 more than the product of n (not necessarily distinct) prime numbers.

Original entry on oeis.org

4, 10, 9, 25, 33, 65, 129, 385, 513, 1025, 2049, 4097, 8193, 16385, 32769, 98305, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2003

Keywords

Comments

a(2*n+1) = 2^(2*n+1)+1, n>0. - Vladeta Jovovic, Apr 02 2003

Links

Crossrefs

Cf. A081545, A081546, A081548, A073919, A007583.

Programs

  • Mathematica
    cno[n_]:=Module[{a=2^n+1},If[PrimeQ[a],2^(n-1)*3+1,a]]; Join[{4,10}, Array[cno,30,3]] (* Harvey P. Dale, Mar 24 2012 *)
  • Python
    from sympy import isprime
def A081547(n): return 10 if n==2 else ((3<Chai Wah Wu, Sep 02 2024

Formula

For n>2, a(n) = 2^n+1 unless this is a Fermat prime (A019434), in which case a(n) = 2^(n-1)*3+1 (which is divisible by 5). - Dean Hickerson, Apr 05 2003

Extensions

More terms from Vladeta Jovovic, Apr 02 2003

A081548 Smallest composite number which is 1 more than the product of n distinct primes.

Original entry on oeis.org

4, 15, 106, 391, 3991, 30031, 510511, 9699691, 223092871, 6469693231, 255887521891, 7420738134811, 304250263527211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071, 117288381359406970983271, 7858321551080267055879091, 557940830126698960967415391
Offset: 1

Views

Author

Amarnath Murthy, Apr 01 2003

Keywords

Links

Crossrefs

Cf. A002110, A014545, A081545, A081546, A081547.

Formula

If n is in A014545, then a(n) > A002110(n)+1, otherwise a(n) = A002110(n)+1. - Chai Wah Wu, Sep 02 2024

Extensions

Corrected and extended by David Wasserman, Jun 08 2004
a(18)-a(20) from Chai Wah Wu, Sep 02 2024
Showing 1-5 of 5 results.

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