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 1-7 of 7 results.

A063047 Minimum m where (c_n)^m is mutinous (i.e., part of sequence A027854), where c_n is the n-th positive integer not a prime power.

Original entry on oeis.org

2, 3, 1, 3, 2, 2, 2, 2, 4, 1, 4, 2, 1, 3, 5, 2, 1, 5, 3, 1, 2, 2, 1, 5, 1, 3, 3, 2, 2, 2, 1, 3, 5, 1, 5, 1, 2, 2, 3, 3, 1, 1, 6, 2, 3, 2, 2, 1, 6, 1, 2, 6, 4, 2, 1, 2, 3, 4, 6, 2, 1, 3, 2, 2, 2, 2, 1, 6, 1, 2, 4, 1, 2, 2, 3, 2, 6, 2, 1, 6, 4, 3, 1, 4, 2, 1, 2, 7, 1, 2, 2, 1, 4, 7, 2, 1, 3, 7, 2, 3, 1, 2, 2, 1, 3
Offset: 1

Views

Author

Leroy Quet, Aug 03 2001

Keywords

Comments

Prime powers (p^k, k = nonnegative integer) raised to a power are never mutinous.

Examples

			a(1) = 2 because the first non-prime-power is 6; and 6^2 = 36, but not 6^1, is mutinous.

Crossrefs

Cf. A027854, A024619.

Formula

m = ceiling[log(p)/(log(c_n) - k log(p))], where p is the largest prime to divide c_n and p^k is the highest power of p to divide c_n.

Extensions

Definition clarified by Jonathan Sondow, May 18 2014

A289484 Semigroup of numbers under multiplication, distinct from mutinous sequence.

Original entry on oeis.org

12, 24, 30, 36, 40, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 96, 105, 108, 112, 120, 126, 132, 135, 140, 144, 150, 154, 156, 160, 165, 168, 175, 176, 180, 182, 189, 192, 195, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 231, 234, 240, 252, 260, 264, 270, 273, 275, 276, 280
Offset: 1

Views

Author

Richard Locke Peterson, Jul 06 2017

Keywords

Comments

Definition: Set S(S numbers) of all numbers n whose prime factorization contains at least one initial product greater than a later prime. That is, write n as n=p1^e1*p2^e2*...pm^em, with the ei>0 and p1
Examples: 156=2^2*3*13 is in S, since 2^2>3. Another example is 200=2^3*5^2, since 8>5, so 200 is in S.
Counterexamples: 20=2^2*5 and 42=2*3*7 are not S numbers because 2^2<5 and 2*3<7.
Properties: No primes or prime powers are in S, nor are any numbers pq with p and q prime. S is closed under multiplication, so it is a semigroup. In fact, any positive multiple of an S number is also an S number.
Subset of generators: The numbers 12,30,40,45,56,63,.., belong to an infinite subset of S that could be called "S primes" because no proper factor of an S prime is an S number, and because every S number is a positive multiple of at least one of the S primes.
Algebra: If one adjoins the numbers 0 and all the negatives of numbers in S into S and call the result S#, then S# remains a semigroup and is the set union of infinitely many principal ideals:S#=(12)U(30)U(40)U(45)U...U.(note presence of S primes). But S# itself is not an ideal, because it is not closed under addition.
Density: In the integers from 1 to 500, about 19% are in S. Using Wolfram Alpha, about 63% of the integers from 10^40+1 to 10^40+62 were found to be S numbers.

Links

Crossrefs

Cf. A027854, mutinous numbers. Contained in S, differs in that for a mutinous number it must be the greatest prime in the factorization that is exceeded by the initial product, while in S, the prime that is exceeded can be any of the primes later than the initial primes. For example 156 is an S number but not a mutinous number.

Programs

  • Maple
    isA289484 := proc(n)
        local pset,p,pprodidx,pprod,nu ;
        pset := sort(convert(numtheory[factorset](n),list)) ;
        pprod := 1;
        for pprodidx from 1 to nops(pset)-1 do
                p := pset[pprodidx] ;
                nu := padic[ordp](n,p) ;
                pprod := pprod*p^nu ;
                if pprod > pset[pprodidx+1] then
                        return true;
                end if;
        end do:
        return false ;
end proc:
for n from 1 to 300 do
        if isA289484(n) then
                printf("%d,",n) ;
        end if;
end do: # R. J. Mathar, Oct 20 2017
  • Mathematica
    Select[Range@ 280, Function[f, AnyTrue[Range[Length@ f - 1], Times @@ Map[#1^#2 & @@ # &, #1] > #2[[1, 1]] & @@ TakeDrop[f, #] &]]@ FactorInteger@ # &] (* Michael De Vlieger, Aug 17 2017 *)
  • PARI
    is(n)=my(f=factor(n),t=1); for(i=1,#f~, if(t>f[i,1], return(1)); t*=f[i,1]^f[i,2]); 0 \\ Charles R Greathouse IV, Jul 10 2017
  • PARI
    has(f)=my(t=1); for(i=1,#f~, if(t>f[i,1], return(1)); t*=f[i,1]^f[i,2]); 0
list(lim)=my(m=Map(),v=List()); forfactored(n=12,lim\=1, if(mapisdefined(m,n), next); if(has(n[2]), forstep(k=n[1],lim,n[1], mapput(m,k,0)))); for(n=12,lim, if(mapisdefined(m,n), listput(v,n))); m=0; Vec(v) \\ Charles R Greathouse IV, Jul 12 2017

A027855 Antimutinous numbers: n>1 such that n/p^k < p, where p is the largest prime dividing n and p^k is the highest power of p dividing n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87
Offset: 1

Views

Author

Leroy Quet

Keywords

Comments

Numbers which can be expressed as m*p^k, for p prime and m < p and k > 0. List contains n if A006530(n) > A051119(n). - Harry Richman, Aug 19 2019

Links

Crossrefs

Cf. A006530, A027854, A051119.

Programs

  • Maple
    isA027855 := proc(n) local p,k,pk; if n <= 1 then false; else p := A006530(n) ; pk := p ; while n mod ( pk*p) = 0 do pk := pk*p ; od: if n< p*pk then true ; else false ; fi ; fi ; end proc:
for n from 2 to 120 do if isA027855(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Dec 02 2007
  • Mathematica
    Select[Range@100, #1^(#2 + 1) & @@ FactorInteger[#][[-1]] > # &] (* Ivan Neretin, Jul 09 2015 *)
  • PARI
    is(n) = my(f = factor(n)); c = n\f[#f~, 1]^f[#f~, 2]; c < f[#f~, 1] \\ David A. Corneth, Aug 19 2019
  • Python
    from sympy import factorint, primefactors
def a053585(n):
    if n==1: return 1
    p = primefactors(n)[-1]
    return p**factorint(n)[p]
print([n for n in range(2, 301) if n//a053585(n)Indranil Ghosh, Jul 13 2017

Extensions

More terms from R. J. Mathar, Dec 02 2007

A056077 Indices n of terms of sequence A001142, Product_{k=0..n} binomial(n,k), that are divisible by all primes <= n.

Original entry on oeis.org

1, 2, 4, 6, 10, 11, 12, 16, 18, 22, 23, 28, 29, 30, 35, 36, 39, 40, 42, 44, 46, 47, 52, 55, 58, 59, 60, 62, 66, 69, 70, 71, 72, 78, 79, 82, 83, 88, 89, 95, 96, 100, 102, 104, 106, 107, 108, 111, 112, 119, 125, 126, 130, 131, 134, 136, 138, 139, 143, 148, 149, 150, 153
Offset: 1

Views

Author

Leroy Quet, Jul 26 2000

Keywords

Comments

a(n) + 1 is either a prime or a "mutinous number" (A027854).

Examples

			11 is included because Product_{k=0..11} binomial(11, k) is divisible by 2, 3, 5, 7 and 11.

Links

Crossrefs

Cf. A001142, A056606, A027854, A002110.

Programs

  • Maple
    isA056077 := proc(n) local radh; radh := proc(n) option remember;
mul(k, k = numtheory:-factorset(mul(k^k/factorial(k), k=0..n))) end;
type(radh(n)/radh(n-1), integer) end: # isA056077(0) = true.
select(isA056077, [$1..153]); # Peter Luschny, Dec 21 2019
  • Mathematica
    With[{s = Select[Range@ 154, Function[n, (n/Apply[Power, Last@ #]) > #[[-1, 1]] &@ FactorInteger[n]]]}, -1 + Union[s, Prime@ Range@ PrimePi@ Max@ s]] (* Michael De Vlieger, Sep 23 2017 *)

Formula

Let h(m) = Product(PrimeDivisors(Product_{k=0..m} k^k/k!)). If h(m-1) divides h(m) then m is in this sequence. # Peter Luschny, Dec 21 2019

Extensions

Extended by Ray Chandler, Nov 17 2008

A132982 The non-prime-power "antimutinous" numbers. (Antimutinous numbers are those integers m, m >1, where m/p^k < p, p = largest prime divisor of m, p^k = largest power of p that divides m.)

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 62, 65, 66, 68, 69, 74, 75, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130
Offset: 1

Views

Author

Leroy Quet, Nov 19 2007

Keywords

Examples

			30 = 2*3*5. 2*3 is > 5, so 30 is not in the sequence because 30 is mutinous (see A027854).
27 = 3^3. 27/3^3 is < 3, so 27 is antimutinous. But 27 is a power of a prime, so 27 is not in the sequence.
However, 20 = 2^2 * 5^1. And 20/5^1 is < 5, so 20 is antimutinous. Also, 20 is not a power of a prime. So 20 is in the sequence.

Links

Crossrefs

Cf. A027854, A027855.

Programs

  • Mathematica
    a = {}; For[n = 2, n < 200, n++, If[ !PrimeQ[n], If[ ! Length[FactorInteger[n]] == 1, b = FactorInteger[n]; m = 0; For[j = 1, j < Length[b] + 1, j++, If[b[[j, 1]]^b[[j, 2]] > m, m = b[[j, 1]]^b[[j, 2]]]; If[n/m < FactorInteger[m][[1, 1]], AppendTo[a, n]]]]]]; a (* Stefan Steinerberger, Nov 20 2007 *)

Extensions

More terms from Stefan Steinerberger, Nov 20 2007

A132858 Composite "antimutinous" numbers. An antimutinous number is an integer m > 1 where m/p^k < p, where p is the largest prime divisor of m and p^k is the largest power of p dividing m.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 18, 20, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 42, 44, 46, 49, 50, 51, 52, 54, 55, 57, 58, 62, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118
Offset: 1

Views

Author

Leroy Quet, Nov 21 2007

Keywords

Comments

{a(k)-1} is the complement of sequence A056077. In other words, {a(k)} contains precisely those positive integers m where A001142(m-1) (= product{k=1 to m-1} k^(2k-m)) is not divisible by all primes <= m-1.

Links

Crossrefs

Cf. A027855, A132982, A027854, A056077, A001142.

Programs

  • Mathematica
    antiQ[n_] := Module[{f = FactorInteger[n], p, k}, p = f[[-1, 1]]; k = f[[-1, 2]]; n/p^k < p]; Select[Range[118], CompositeQ[#] && antiQ[#] &] (* Amiram Eldar, Feb 24 2020 *)

Extensions

Extended by Ray Chandler, Nov 17 2008

A319404 a(n) is the period of the periodic k-sequence q_k=lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)).

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 60, 105, 280, 504, 2520, 27720, 27720, 51480, 72072, 45045, 720720, 1361360, 12252240, 46558512, 33256080, 21162960, 232792560, 5354228880, 1070845776, 2059318800, 2974571600, 11473347600, 80313433200, 2329089562800, 2329089562800, 4512611027925
Offset: 1

Views

Author

René Gy, Sep 18 2018

Keywords

Comments

For n>0, k>=0, the k-sequence q_k=lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)) is a periodic integer sequence with period a(n).
a(n) is a divisor of A003418(n-1) and a multiple of A003418(n)/n.
a(n) = A003418(n-1) if n is a member of A027854 (a mutinous number), otherwise a(n) = A003418(n)/q^v where q^v is the highest prime power which divides n.
a(n) = A003418(n-1) iff n is a mutinous number or n is a prime number.
a(n) = A003418(n) iff n is a mutinous number.
lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)) is a divisor of lcm(1,2,...,n)/n, therefore a(n) is also the period of the periodic k-sequence r_k= binomial(k+n,n)*lcm(1,2,...,n)/lcm(k+1,k+2,...,k+n).
Let g be the smallest multiple of A003418(n)/n such that r_g=r_0=1 and r_{g+1}=r_1=gcd(m+1,A003418(n)), then a(n)=g.
a(n+j) is a multiple of binomial(n+j-1,j).
All these statements require proofs.

Examples

			For n = 5, a(5) = 12 since from k>=0, we have lcm(k+1,k+2,k+3,k+4,k+5)/5/binomial(k+5,5) =  12,2,4,3,4,2,12,1,4,6,4,1,12,2,4,3,4,2,12,1,4,6,4,1,12,2,4,3,4,2,12,1,4,6,4,1,12,..., etc. a periodic sequence of period 12.

Crossrefs

Cf. A003418, A027854.

Programs

  • Mathematica
    ll2[n0_, m0_] :=
Module[{f, g, i, n = n0, m = m0}, g = 1;
  If[1 <= m <= n, Do[f = LCM[g, n - i]; g = f, {i, 0, m - 1}], f = 1];f]
list3 = {1};
Do[i = 0; ll = ll2[m, m]/m; b = {1, ll };a = {0, 0 };
  While[ a != b, i = i + ll;
   a = { ll2[m + i - 1, m]/(m*Binomial[m + i - 1, m]), ll2[m + i, m]/(
     m*Binomial[m + i, m])}]; AppendTo[list3, i], {m, 2, 50}]; Print[list3]
Showing 1-7 of 7 results.

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