A141809 -id:A141809 - cp's OEIS frontend

A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.

0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 3, 1, 1, 0, 0, 4, 2, 1, 0, 0, 0, 4, 2, 1, 1, 0, 0, 0, 7, 2, 1, 1, 0, 0, 0, 0, 7, 4, 1, 1, 0, 0, 0, 0, 0, 8, 4, 2, 1, 0, 0, 0, 0, 0, 0, 8, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 10, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 11, 5, 2, 2, 1, 1, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Jul 07 2003

T(n,1) = A011371(n); T(n,2) = A054861(n) for n>1;
T(n,k) = number of occurrences of prime(k) as factor in numbers <= n (with repetitions);
Sum{T(n,k): 1<=k<=n} = A022559(n);
T(n, A000720(n)) = 1; T(n,k) = 0, A000720(n)T(n,k) = A115627(n,k) for n > 1 and k=1..A000720(n). - Reinhard Zumkeller, Nov 01 2013

			0;
1,0;
1,1,0;
3,1,0,0;
3,1,1,0,0;
4,2,1,0,0,0;
4,2,1,1,0,0,0;
7,2,1,1,0,0,0,0;
7,4,1,1,0,0,0,0,0;
8,4,2,1,0,0,0,0,0,0;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A141809, A115627, A000142.

a085604 n k = a085604_tabl !! (n-2) !! (k-1)
a085604_row 1 = [0]
a085604_row n = a115627_row n ++ (take $ a062298 $ fromIntegral n) [0,0..]
a085604_tabl = map a085604_row [1..]
-- Reinhard Zumkeller, Nov 01 2013

T[n_, k_] := Module[{p = Prime[k], jm}, jm = Floor[Log[p, n]]; Sum[Quotient[n, p^j], {j, 1, jm}]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

A067240 If n = Product_{i} p_i^e_i, a(n) = Sum_{i} (p_i - 1)*p_i^(e_i - 1).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 4, 6, 5, 10, 4, 12, 7, 6, 8, 16, 7, 18, 6, 8, 11, 22, 6, 20, 13, 18, 8, 28, 7, 30, 16, 12, 17, 10, 8, 36, 19, 14, 8, 40, 9, 42, 12, 10, 23, 46, 10, 42, 21, 18, 14, 52, 19, 14, 10, 20, 29, 58, 8, 60, 31, 12, 32, 16, 13, 66, 18, 24, 11, 70, 10, 72, 37, 22, 20, 16, 15, 78, 12, 54, 41, 82, 10, 20, 43, 30, 14, 88, 11, 18, 24, 32, 47, 22, 18, 96, 43, 16, 22

Offset: 1

N. J. A. Sloane, Mar 10 2002

nonn

Additive sequence with a(p^e) = phi(p^e) = (p-1)*p^(e-1). - Charles R Greathouse IV, Sep 10 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186. (T on p. 181.)

Cf. A000010, A141809.

a067240 1 = 0
a067240 n = sum $ map a000010 $ a141809_row $ toInteger n
-- Reinhard Zumkeller, Jun 13 2012

with(numtheory); A067240 := proc(n) local e,j; e := ifactors(n)[2]: add((e[j][1]-1)*e[j][1]^(e[j][2]-1),j=1..nops(e)); end;

a[n_] := Total[ EulerPhi[ Power @@ #] & /@ FactorInteger[n]]; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 22 2012, after T. D. Noe *)

A067240(n)=
{
    local(f=factor(n), r=0, p, e);
    for (i=1, matsize(f)[1],
        p=f[i,1];  e=f[i,2];
        r += (p-1)*p^(e-1);
    );
    return(r);
} /* Joerg Arndt, Jun 10 2011 */

a(n)=my(f=factor(n)); sum(i=1,#f~, (f[i,1]-1)*f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Sep 10 2015

from sympy import factorint
a = lambda n: sum((p**(e-1))*(p-1) for p,e in factorint(n).items() if e > 0) # Darío Clavijo, Feb 15 2024

For n > 1: a(n) = Sum_{i} phi(p_i^e_i). - T. D. Noe, Jul 10 2003

A141810 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the sizes of the prime-powers. Row 1 = (1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 3, 4, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 3, 8, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 5, 8, 41, 2, 3, 7, 43, 4, 11, 5, 9, 2, 23, 47, 3, 16, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 7, 8, 3

Offset: 1

Leroy Quet, Jul 07 2008

In other words, except for row 1, row n contains the unitary prime power divisors of n. - Franklin T. Adams-Watters, May 05 2011

			60 has the prime factorization of 2^2 * 3^1 * 5^1. So row 60 is, arranged by the sizes of the prime-powers, (3,4,5).

Cf. A141809, A027748.

A080736 Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler's function A000010.

Original entry on oeis.org

1, 0, 2, 2, 4, 0, 6, 4, 6, 0, 10, 4, 12, 0, 8, 8, 16, 0, 18, 8, 12, 0, 22, 8, 20, 0, 18, 12, 28, 0, 30, 16, 20, 0, 24, 12, 36, 0, 24, 16, 40, 0, 42, 20, 24, 0, 46, 16, 42, 0, 32, 24, 52, 0, 40, 24, 36, 0, 58, 16, 60, 0, 36, 32, 48, 0, 66, 32, 44, 0, 70, 24, 72, 0, 40, 36, 60, 0, 78, 32

Offset: 1

N. J. A. Sloane, Mar 08 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A080737.

Cf. A000010, A141809.

a080736 n = if n `mod` 4 == 2 then 0 else a000010 n
-- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012

a[n_] := If[Mod[n, 4] == 2, 0, EulerPhi[n]]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)

{for(n=1,81,f=factor(n); print1(if(n==1,1,if(f[1,1]==2&&f[1,2]==1,0,prod(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])))),","))}

a(A016825(n)) = 0; a(A000040(n)) = A000040(n) - 1. - Reinhard Zumkeller, Jun 11 2012
a(n) = if n mod 4 = 2 then 0 else A000010(n). - Reinhard Zumkeller, Jun 13 2012
From Amiram Eldar, Nov 02 2023: (Start)
Multiplicative with a(2) = 0, a(2^e) = 2^(e-1) for e >= 2, and a(p^e) = (p-1)*p^(e-1) for an odd prime p.
Dirichlet g.f.: (1 - 2^(1-s) + 1/(2^s-1)) * zeta(s-1) / zeta(s).
Sum_{k=1..n} a(k) ~ (5/(2*Pi^2)) * n^2. (End)

More terms and PARI code from Klaus Brockhaus, Mar 10 2003

A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 60, 63, 80, 84, 90, 96, 112, 120, 126, 132, 135, 144, 156, 160, 168, 175, 176, 180, 189, 192, 204, 208, 224, 228, 240, 252, 264, 270, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 350, 351, 352, 360, 372, 378, 384, 405

Offset: 1

Leroy Quet, Jul 18 2008

Previous name was: Let p^b(n,p) be the largest power of the prime p that divides n. The integer n is included if the list of p^b(n,p)'s, where each p is a distinct prime divisor of n, arranged by size of each p^b(n,p) is not in the same order as the list of p^b(n,p)'s arranged by size of each prime p.
This sequence contains no squarefree integers.
90 is the smallest integer in this sequence but not in sequence A126855.
The number of terms < 10^n: 0, 12, 151, 1575, 16154, 161630, 1617052, ..., . - Robert G. Wilson v, Aug 31 2008
If k is in the sequence, then all powers of k are in the sequence. - Mike Jones, Jun 16 2022
If k is in the sequence then k*A020639(k)^m is in the sequence for m >= 0. - David A. Corneth, Jun 16 2022
Conjecture: There are infinitely many terms k such that k+1 is also a term. - Mike Jones, Jun 18 2022

			The prime factorization of 90 is, when arranged by size of the distinct primes, 2^1 * 3^2 * 5^1. Since 3^2 is > 5^1, even though 5 > 3, 90 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A383397.

Cf. A020639, A141809, A141810, A126855.

fQ[n_] := Block[{f = First@# ^ Last@# & /@ FactorInteger@n}, f != Sort@f]; Select[ Range@ 407, fQ@# &] (* Robert G. Wilson v, Aug 31 2008 *)

is(n) = { my(f = factor(n)); for(i = 1, #f~-1, if(f[i,1]^f[i,2] > f[i+1,1]^f[i+1,2], return(1) ) ); 0 } \\ David A. Corneth, Jun 16 2022

More terms from Robert G. Wilson v, Aug 31 2008

Simpler name from Mike Jones, Jun 15 2022

A333235 a(n) is the product of indices of unitary prime power divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 6, 9, 5, 8, 10, 11, 7, 12, 12, 10, 8, 13, 12, 14, 9, 15, 15, 16, 8, 17, 18, 16, 11, 20, 21, 19, 12, 18, 24, 20, 10, 21, 24, 28, 13, 22, 20, 23, 14, 22, 27, 24, 15, 32, 30, 24, 16, 25, 24, 26, 17, 35, 27, 36, 16, 28, 33, 26, 20

Offset: 1

Ilya Gutkovskiy, Mar 12 2020

Equivalently: replace each prime power p^e in the prime factorization of n by its index in A246655. - M. F. Hasler, Jun 16 2021

			a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.

Cf. A322981 (the index of n = p^e in A246655).

N:= 1000: # for a(1)..a(N)
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k,k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
end proc:
map(f, [$1..N]); # Robert Israel, Feb 11 2021

PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Jun 16 2021

If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).
a(prime(n)) = A027883(n).
a(2^n) = A182908(n).
a(A246655(n)) = n.

A304233 If n = Product (p_j^k_j) then a(n) = min{p_j^k_j}*max{p_j^k_j}.

Original entry on oeis.org

1, 4, 9, 16, 25, 6, 49, 64, 81, 10, 121, 12, 169, 14, 15, 256, 289, 18, 361, 20, 21, 22, 529, 24, 625, 26, 729, 28, 841, 10, 961, 1024, 33, 34, 35, 36, 1369, 38, 39, 40, 1681, 14, 1849, 44, 45, 46, 2209, 48, 2401, 50, 51, 52, 2809, 54, 55, 56, 57, 58, 3481, 15, 3721, 62, 63, 4096, 65

Offset: 1

Ilya Gutkovskiy, May 08 2018

nonn

			a(60) = 15 because 60 = 2^2*3*5, min{2^2,3,5} = 3, max{2^2,3,5} = 5 and 3*5 = 15.

Ilya Gutkovskiy, Logarithmic scatter plot of a(n) up to n=30000
Eric Weisstein's World of Mathematics, Prime Factorization

Cf. A000977 (numbers n such that a(n) < n), A002110, A007774 (fixed points), A034684, A034699, A066048, A100484, A141809.

a[n_] := Min[#[[1]]^#[[2]] & /@FactorInteger[n]] Max[#[[1]]^#[[2]] & /@FactorInteger[n]]; Table[a[n], {n, 65}]

a(n) = A034684(n)*A034699(n).
a(p^k) = p^(2*k) where p is a prime.
a(A002110(k)) = A100484(k).

A059401 Numbers that contain as proper substrings every maximal prime power dividing them.

Original entry on oeis.org

1197, 2510, 2570, 5210, 5230, 5290, 12590, 14673, 15230, 20530, 21530, 22510, 23510, 23570, 24590, 25030, 25210, 25310, 25390, 25430, 25490, 25510, 25570, 25790, 25910, 25930, 26570, 26590, 27530, 28510, 28570, 29530, 29570, 32510

Offset: 1

Erich Friedman, Jan 29 2001

Most terms end in 0, since 2*5*prime will work if prime contains 2 and 5 as substrings. The other terms are listed in A059402.

There must be at least two maximal prime powers dividing each term. - Harvey P. Dale, Dec 04 2016

			1197 = 9 * 7 * 19 and all of these are substrings.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A141809, A010055.
Subsequence of A024619.
A059402 is a subsequence.
Equivalent sequence for primes instead of maximal prime powers: A050694.

import Data.List (isInfixOf)
a059401 n = a059401_list !! (n-1)
a059401_list = filter (\x -> a010055 x == 0 &&
               all (`isInfixOf` show x) (map show $ a141809_row x)) [1..]
-- Reinhard Zumkeller, Dec 16 2013

psmppQ[n_]:=Module[{pp=#[[1]]^#[[2]]&/@FactorInteger[n], idn= IntegerDigits[ n]}, Length[pp]>1&&And@@Table[ SequenceCount[ idn, IntegerDigits[pp[[i]]]]>0,{i,Length[pp]}]]; Select[Range[ 33000], psmppQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 04 2016 *)

Offset corrected by Reinhard Zumkeller, Dec 16 2013

Edited by Peter Munn, Sep 01 2022

Previous Showing 11-20 of 21 results. Next

cp's OEIS Frontend

A210208 Triangle read by rows in which row n lists the divisors of n that are prime powers, A000961.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A085604 T(n,k) = highest power of prime(k) dividing n!, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A067240 If n = Product_{i} p_i^e_i, a(n) = Sum_{i} (p_i - 1)*p_i^(e_i - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

A141810 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the sizes of the prime-powers. Row 1 = (1).

Views

Author

Keywords

Comments

Examples

Crossrefs

A080736 Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler's function A000010.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs