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.

A263297 The greater of bigomega(n) and maximal prime index in the prime factorization of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 2, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 3, 4, 5, 9, 4, 3, 6, 3, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 4, 13, 4, 14, 5, 3, 9, 15, 5, 4, 3, 7, 6, 16, 4, 5, 4, 8, 10, 17, 4, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 5, 21, 12, 3, 8, 5, 6, 22, 5
Offset: 1

Views

Author

Alexei Kourbatov, Oct 13 2015

Keywords

Comments

Also: minimal m such that n is the product of at most m primes not exceeding prime(m). (Here the primes do not need to be distinct; cf. A263323.)
By convention, a(1)=0, as 1 is the empty product.
Those n with a(n) <= k fill a k-simplex whose 1-sides span from 0 to k.
For a similar construction with distinct primes (hypercube), see A263323.
Each nonnegative integer occurs finitely often; in particular:
- Terms a(n) <= k occur A000984(k) = (2*k)!/(k!)^2 times.
- The term a(n) = 0 occurs exactly once.
- The term a(n) = k > 0 occurs exactly A051924(k) = (3*k-2)*C(k-1) times, where C(k)=A000108(k) are Catalan numbers.

Examples

			a(6)=2 because 6 is the product of 2 primes (2*3), each not exceeding prime(2)=3.
a(8)=3 because 8 is the product of 3 primes (2*2*2), each not exceeding prime(3)=5.
a(11)=5 because 11 is prime(5).

Links

Crossrefs

Cf. A000108, A000984, A001222, A051924, A061395, A263323, A325225, A331296 (ordinal transform), A331297.

Programs

  • Maple
    seq(`if`(n=1,0,max(pi(max(factorset(n))),bigomega(n))),n=1..80); # Peter Luschny, Oct 15 2015
  • Mathematica
    f[n_] := Max[ PrimePi[ Max @@ First /@ FactorInteger@n], Plus @@ Last /@ FactorInteger@n]; Array[f, 80]
  • PARI
    a(n)=if(n<2, return(0)); my(f=factor(n)); max(vecsum(f[,2]), primepi(f[#f~,1])) \\ Charles R Greathouse IV, Oct 13 2015

Formula

a(n) = max(A001222(n), A061395(n)).
a(n) <= pi(n), with equality when n is 1 or prime.

A326846 Length times maximum of the integer partition with Heinz number n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the minimal rectangle containing the Young digram of the integer partition with Heinz number n.

Links

Crossrefs

Cf. A047993, A056239, A061395, A106529, A112798.
Cf. A316413, A326836, A326837, A326844, A326845, A326848, A326849.
Cf. also A331297.

Programs

  • Mathematica
    Table[PrimeOmega[n]*PrimePi[FactorInteger[n][[-1,1]]],{n,100}]
  • PARI
    A326846(n) = if(1==n, 0, bigomega(n)*primepi(vecmax(factor(n)[, 1]))); \\ Antti Karttunen, Jan 18 2020

Formula

a(n) = A001222(n) * A061395(n).

Extensions

More terms from Antti Karttunen, Jan 18 2020

A331298 Lexicographically earliest infinite sequence such that a(i) = a(j) => A001222(i) = A001222(j) and A061395(i) = A061395(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 6, 9, 10, 11, 12, 13, 9, 14, 15, 11, 16, 17, 13, 18, 19, 20, 9, 21, 11, 22, 23, 17, 24, 25, 18, 26, 13, 20, 27, 28, 21, 29, 30, 22, 31, 32, 17, 33, 34, 35, 13, 17, 26, 36, 37, 20, 18, 38, 28, 39, 40, 29, 41, 42, 22, 43, 21, 32, 44, 45, 33, 22, 46, 35, 47, 48, 17, 49, 18, 36, 50, 51, 20, 52, 53, 38, 26, 54, 39, 55, 56, 29, 21, 57, 42, 58, 28, 59, 60, 22, 32, 29
Offset: 1

Views

Author

Antti Karttunen, Jan 18 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A001222(n), A061395(n)].
For all i, j:
A318891(i) = A318891(j) => a(i) = a(j),
a(i) = a(j) => A331297(i) = A331297(j) => A326846(i) = A326846(j),
a(i) = a(j) => A331281(i) = A331281(j),
a(i) = a(j) => A331282(i) = A331282(j).

Links

Crossrefs

Cf. A001222, A061395, A318891, A326846, A331281, A331282.
Cf. also A331297, A331299.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
Aux331298(n) = [bigomega(n),A061395(n)];
v331298 = rgs_transform(vector(up_to, n, Aux331298(n)));
A331298(n) = v331298[n];

A331295 Number of values of k, 1 <= k <= n, with f(k) = f(n), where f(n) = [A001222(n), A061395(n)].

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 19 2020

Keywords

Comments

Ordinal transform of A331298, or equally, of the ordered pair [A001222(n), A061395(n)].

Links

Crossrefs

Cf. A001222, A061395, A078899, A331296, A331297, A331298.

Programs

  • Mathematica
    f[n_] := f[n] = {PrimeOmega[n], PrimePi[FactorInteger[n]][[-1, 1]]};
a[n_] := Count[Array[f, n], f[n]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022 *)
  • PARI
    up_to = 1001;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
Aux331298(n) = [bigomega(n), A061395(n)];
v331295 = ordinal_transform(vector(up_to, n, Aux331298(n)));
A331295(n) = v331295[n];
Showing 1-4 of 4 results.

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