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.

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A278159 Run length transform of primorials, A002110.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 6, 30, 2, 4, 4, 12, 6, 12, 30, 210, 2, 4, 4, 12, 4, 8, 12, 60, 6, 12, 12, 36, 30, 60, 210, 2310, 2, 4, 4, 12, 4, 8, 12, 60, 4, 8, 8, 24, 12, 24, 60, 420, 6, 12, 12, 36, 12, 24, 36, 180, 30, 60, 60, 180, 210, 420, 2310, 30030, 2, 4, 4, 12, 4, 8, 12, 60, 4, 8, 8, 24, 12, 24, 60, 420, 4, 8, 8, 24, 8, 16, 24, 120, 12, 24, 24, 72, 60, 120, 420
Offset: 0

Views

Author

Antti Karttunen, Nov 16 2016

Keywords

Comments

Like every run length transform this sequence satisfies for all i, j: A278222(i) = A278222(j) => a(i) = a(j).

Examples

			For n=7, "111" in binary, there is a run of 1-bits of length 3, thus a(7) = product of A002110(3), = A002110(3) = 30.
For n=39, "10111" in binary, there are two runs, of lengths 1 and 3, thus a(39) = A002110(1) * A002110(3) = 2*30 = 60.

Links

Crossrefs

Cf. A002110, A005940, A124859, A227349, A246660, A278161, A278222.

Programs

  • Mathematica
    f[n_] := Product[Prime[k], {k, 1, n}]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 94}] (* Jean-François Alcover, Jul 11 2017 *)
  • Python
    from math import prod
from re import split
from sympy import primorial
def RLT(n,f):
    """ run length transform of a function f """
    return prod(f(len(d)) for d in split('0+', bin(n)[2:]) if d != '') if n > 0 else 1
def A278159(n): return RLT(n,primorial) # Chai Wah Wu, Feb 04 2022
  • Scheme
    (define (A278159 n) (fold-left (lambda (a r) (* a (A002110 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))
;; See A227349 for the required other functions.

Formula

a(n) = A124859(A005940(1+n)).

A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2
Offset: 2

Views

Author

Matthew Vandermast, Apr 28 2014

Keywords

Comments

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

Examples

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)

Crossrefs

Row lengths are A051903(n); row sums are A001222(n).
Cf. A217171.
These partitions are ranked by A238745.
For prime indices A296150 instead of exponents we get A321649, rev A321650.
A000700 counts self-conjugate partitions, ranked by A088902.
A003963 gives product of prime indices, conjugate A329382.
A008480 gives number of permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221.
A352486-A352490 are sets related to the fixed points of A122111.
Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Programs

  • Mathematica
    Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Formula

Row n is identical to row A124859(n) of table A212171.

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54
Offset: 1

Views

Author

José María Grau Ribas, Jan 04 2021

Keywords

Comments

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

Examples

			a(2^s) = 3 for all s>0.

Links

Crossrefs

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

Programs

  • Maple
    f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021
  • Mathematica
    fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]
  • PARI
    A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

Formula

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A354826 Dirichlet inverse of A238745.

Original entry on oeis.org

1, -2, -2, 0, -2, 5, -2, 0, 0, 5, -2, -2, -2, 5, 5, 0, -2, -2, -2, -2, 5, 5, -2, 0, 0, 5, 0, -2, -2, -17, -2, 0, 5, 5, 5, 8, -2, 5, 5, 0, -2, -17, -2, -2, -2, 5, -2, 0, 0, -2, 5, -2, -2, 0, 5, 0, 5, 5, -2, 16, -2, 5, -2, 0, 5, -17, -2, -2, 5, -17, -2, -8, -2, 5, -2, -2, 5, -17, -2, 0, 0, 5, -2, 16, 5, 5, 5, 0, -2, 16
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2022

Keywords

Links

Crossrefs

Cf. A238745.
Cf. also A354349, A354359.

Programs

  • PARI
    A124859(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1] = prod(j=1, f[k, 2], prime(j)); f[k, 2] = 1); factorback(f); }; \\ From A124859
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A238745(n) = A181819(A124859(n));
memoA354826 = Map();
A354826(n) = if(1==n,1,my(v); if(mapisdefined(memoA354826,n,&v), v, v = -sumdiv(n,d,if(dA238745(n/d)*A354826(d),0)); mapput(memoA354826,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA238745(n/d) * a(d).
Previous Showing 11-14 of 14 results.

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