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 1-8 of 8 results.

A249016 Indices of records in A055460 (number of primes dividing the squarefree part of n!).

Original entry on oeis.org

1, 2, 3, 5, 13, 17, 21, 23, 33, 42, 43, 56, 57, 75, 84, 99, 101, 105, 109, 119, 133, 139, 157, 162, 163, 182, 183, 207, 208, 219, 220, 255, 257, 263, 267, 303, 305, 307, 315, 340, 341, 343, 383, 385, 387, 397, 411, 420, 421, 423, 469, 483, 485, 489, 505, 519, 523, 547, 552, 553, 581, 602, 603, 609, 618
Offset: 1

Views

Author

Max Alekseyev, Oct 19 2014

Keywords

Links

Crossrefs

Cf. A249017 (values of records), A055204

Programs

  • PARI
    np=vector(10^6); b=-1; r=0; for(n=1, 10^6, f=factor(n); for(i=1, matsize(f)[1], if(f[i, 2]%2, r += (-1)^np[f[i, 1]]; np[f[i, 1]]=1-np[f[i, 1]]; )); if(r>b, b=r; print1(n, ", ")))

A249017 Records in A055460 (number of primes dividing the squarefree part of n!).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88, 91, 93, 95, 98, 99, 100
Offset: 1

Views

Author

Max Alekseyev, Oct 19 2014

Keywords

Links

Crossrefs

Cf. A055460, A055204, A249016 (indices of records).

Programs

  • PARI
    np=vector(10^6); b=-1; r=0; for(n=1,10^6, f=factor(n); for(i=1,matsize(f)[1], if(f[i,2]%2, r += (-1)^np[f[i,1]]; np[f[i,1]]=1-np[f[i,1]]; )); if(r>b,b=r;print1(b,", ")) )

A177333 Smallest factor in the factorization of n! over distinct terms of A050376.

Original entry on oeis.org

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

Views

Author

Vladimir Shevelev, May 06 2010

Keywords

Examples

			The factorization of 10! = 3628800 is 2^8*3^4*5^2*7^1, where 2^8 > 3^4 > 5^2 > 7, so a(10)=7 is the smallest of these 4 factors.

References

  • V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Links

Crossrefs

Cf. A050376, A177329, A055460, A176525, A001358, A176472, A176509, A064380, A050292.

Programs

  • Maple
    A177333 := proc(n) local a,p,pow2 ; a := n! ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := min(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177333(n),n=2..120) ; # R. J. Mathar, Jun 16 2010
  • Mathematica
    b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Min[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 105, 2] (* Amiram Eldar, Sep 17 2019 *)

Extensions

Corrected from a(10) on and extended beyond a(30) by R. J. Mathar, Jun 16 2010

A177334 Largest factor in the factorization of n! over distinct terms of A050376.

Original entry on oeis.org

2, 3, 4, 5, 16, 16, 16, 81, 256, 256, 256, 256, 256, 256, 256, 256, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 65536, 4294967296, 4294967296, 4294967296, 4294967296, 4294967296, 4294967296
Offset: 2

Views

Author

Vladimir Shevelev, May 06 2010

Keywords

Comments

Each number >=2 has a unique factorization over distinct terms of A050376.
This is obtained from the standard prime factor representation by splitting the exponents into a sum of powers of 2, and further factorization according to the nonzero term of this base-2 representation.
The largest factor of this representation of A000142(n) defines this sequence.

References

  • V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Links

Crossrefs

Cf. A050376, A177329, A055460, A177333, A176525, A001358, A176509, A064380, A050292.

Programs

  • Maple
    A177334 := proc(n) local a,p,pow2 ; a := 1 ; for p in ifactors(n!)[2] do pow2 := convert( op(2,p),base,2) ; for j from 1 to nops(pow2) do if op(j,pow2) <> 0 then a := max(a,op(1,p)^(2^(j-1))) ; end if; end do: end do: return a ; end proc:
seq(A177334(n),n=2..60) ; # R. J. Mathar, Jun 16 2010
  • Mathematica
    b[n_] :=2^(-1+Position[ Reverse@IntegerDigits[n, 2],?(#==1&)])//Flatten; a[n] := Module[{np = PrimePi[n]}, v=Table[0,{np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j,2,n}],  {k,1,np}]; Max[(Prime/@Range[np])^(b/@v) // Flatten]]; Array[a, 38, 2]  (* Amiram Eldar, Sep 17 2019 *)

Extensions

a(18) and a(19) corrected and sequence extended by R. J. Mathar, Jun 16 2010

A241139 Number of nonprimes in factorization of n! over distinct terms of A050376.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 7, 7, 4, 4, 5, 5, 6, 6, 8, 9, 10, 10, 9, 9, 11, 11, 12, 12, 10, 9, 8, 8, 9, 10, 11, 11, 12, 12, 11, 12, 14, 14, 16, 15, 15, 15, 13, 13, 14, 14, 14, 14, 16, 16, 16, 16, 17, 19, 21, 21, 18, 18, 19, 16, 14, 14, 16, 16, 17
Offset: 2

Views

Author

Vladimir Shevelev, Apr 16 2014

Keywords

Examples

			Factorization of 4! over distinct terms of A050376 is 4! = 2*3*4. This factorization contains only one A050376-nonprime. So a(4)=1.

References

  • V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Links

Crossrefs

Cf. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124.

Programs

  • Mathematica
    b[n_] := 2^(-1 + Position[Reverse@IntegerDigits[n, 2], ?(# == 1 &)]) // Flatten; a[n] := Module[{np = PrimePi[n]}, v = Table[0, {np}]; Do[p = Prime[k]; Do[v[[k]] += IntegerExponent[j, p], {j, 2, n}], {k, 1, np}]; Length[Select[(b /@ v) // Flatten, # > 1 &]]]; Array[a, 73, 2]  (* Amiram Eldar, Sep 17 2019 *)
  • PARI
    a(n)={my(f=factor(n!)[,2]); sum(i=1, #f~, hammingweight(f[i]>>1))} \\ Andrew Howroyd, Sep 17 2019

Formula

a(n) = A177329(n) - A055460(n).

Extensions

More terms from Peter J. C. Moses, Apr 17 2014

A348841 Number of primes with even exponents >= 2 in the prime power factorization of n!, for n >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 2, 1, 1, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 4, 2, 2, 2, 4, 4, 4, 3, 4, 4, 5, 5, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 5, 7, 7, 8, 8, 7, 5, 6, 6, 8, 6, 4, 2, 4, 4, 6, 6, 6, 7, 7, 5, 6, 6, 7, 7, 6, 6, 7, 7, 7, 6, 7, 7, 10, 10, 9
Offset: 1

Views

Author

Wolfdieter Lang, Nov 02 2021

Keywords

Comments

The restriction to positive exponents in the prime factor factorization is used to avoid the ambiguity due to p^0 = 1 for any prime. Then a(n) = A000720(n) - A055460(n), for n >= 1.

Examples

			n = 12: 12! = 479001600 = 2^10 * 5^2  *  3^5 * 7^1 * 11^1, hence a(12) = 2,  A055460(12) = 3 and A000720(12) = 5. This latter equation holds because 2, 3, 5, 7, 11 are the primes not exceeding 12.

Links

Crossrefs

Cf. A000720, A055460.
Cf. A000142, A162641.

Programs

  • Mathematica
    Table[Length@Select[FactorInteger[n!],EvenQ@Last@#&],{n,80}] (* Giorgos Kalogeropoulos, Nov 02 2021 *)
  • PARI
    a(n) = my(f=factor(n!)); #select(x->(! (x%2)), f[,2]); \\ Michel Marcus, Nov 03 2021
  • PARI
    a(n) = my(res = 0); forprime(p = 2, n\2, res+=(val(n, p)%2==0)); res
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Nov 03 2021

Formula

a(n) = A000720(n) - A055460(n), for n >= 1.
a(n) = A162641(A000142(n)). - Michel Marcus, Nov 03 2021

A348842 Number of Juniper Green games with n cards.

Original entry on oeis.org

0, 1, 1, 6, 10, 35, 47, 147, 216, 452, 512, 3055, 3365, 5602, 12160, 35951, 37959, 147889, 154998, 703094, 1178850, 1467813
Offset: 1

Views

Author

Wolfdieter Lang, Dec 23 2021

Keywords

Comments

For the rules of this two person game with cards labeled from 1 to n, for n >= 1, called JG(n), see the Ian Stewart links.
It is reported (see the FEEDBACK and the German version), that E. P. Wigner used this game in some lecture in the thirties. There the prime factorization of n! into prime powers, with the number of odd or even (>= 2) exponents, seems to have played a role (see A055460(n) and A348841(n) for the number of primes with these exponents in the factorization of n!, respectively).
The repertoire of card numbers for JG(n) that can be chosen if the latest removed card had label k is shown in A348390. Of course, only those card numbers not yet removed in earlier moves qualify. E.g., n = 4, k = 2: repertoire 1, 4.
The total number of games JG(n), for n >= 2, if the first removed card has label K = 2*k, for k = 1, 2, ... ,floor(n/2), is given in A348843.
For the irregular table which gives in row n the odd and even number of moves in the a(n) JG(n) games see A348844. This gives the number of times Alice (the first mover), respectively Bob wins.

Links

  • Ian Stewart, Juniper Green, Scientific American No. 3 March 1997, pp. 118-120.
  • Ian Stewart, FEEDBACK, Scientific American No. 5 November 1997 p. 112.
  • The author's email is given, juniper green Online game with 100 cards.

Crossrefs

Cf. A055460, A348390, A348843, A348844.

Formula

a(n) = Sum_{k=1..floor(n/2)} A348843(n, k) = Sum_{k=1..2*floor(n/2)} A348844(n, k), for n >= 2.

A055077 Multiplicity of Max{gcd(d, n!/d)} when d runs over the divisors of n!.

Original entry on oeis.org

1, 2, 4, 4, 8, 2, 4, 8, 8, 2, 4, 8, 16, 16, 16, 16, 32, 16, 32, 16, 64, 64, 128, 32, 32, 32, 64, 32, 64, 32, 64, 128, 512, 128, 128, 128, 256, 256, 256, 256, 512, 1024, 2048, 1024, 512, 128, 256, 128, 128, 256, 1024, 512, 1024, 256, 1024, 4096, 16384, 4096, 8192
Offset: 1

Views

Author

Labos Elemer, Jun 13 2000

Keywords

Comments

All terms are powers of 2.

Examples

			n=5, 5!=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The maximum is 2 and it occurs 8 times, so a(5)=8.

Crossrefs

Cf. A000188.

Programs

  • Mathematica
    Table[Count[#, Max@ #] &@ Map[GCD[#, m/#] &, Divisors[m]], {m, Range[30]!}] (* Michael De Vlieger, Oct 06 2019 *)

Formula

It appears that a(n) = 2^A055460(n). - Joerg Arndt, Oct 06 2019
Showing 1-8 of 8 results.

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

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