A053585 -id:A053585 - cp's OEIS frontend

A241919 If n is a prime power, p_i^e, a(n) = i, (with a(1)=0), otherwise difference (i-j) of the indices of the two largest distinct primes p_i, p_j, i > j in the prime factorization of n: a(n) = A061395(n) - A061395(A051119(n)).

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

Offset: 1

Antti Karttunen, May 13 2014

nonn

See A242411 and A241917 for other variants.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A241917, A242411, A051119, A061395, A122111.

Cf. A049084, A027748.

a241919 1 = 0
a241919 n = i - j where
            (i:j:_) = map a049084 $ reverse (1 : a027748_row n)
-- Reinhard Zumkeller, May 15 2014

from sympy import factorint, primefactors, primepi
def a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
def a053585(n):
    if n==1: return 1
    p = primefactors(n)[-1]
    return p**factorint(n)[p]
def a051119(n): return n/a053585(n)
def a(n): return a061395(n) - a061395(a051119(n)) # Indranil Ghosh, May 19 2017

(define (A241919 n) (- (A061395 n) (A061395 (A051119 n))))

a(n) = A061395(n) - A061395(A051119(n)).

A242411 If n is a prime power, p_i^e, a(n) = 0, otherwise difference (i-j) of the indices of the two largest distinct primes p_i, p_j, i > j in the prime factorization of n: a(n) = A061395(n) - A061395(A051119(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 1, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 2, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 1, 0, 10, 2, 0, 3, 3, 0, 6, 7, 1, 0, 1, 0, 11, 1, 7, 1, 4, 0, 2, 0, 12, 0, 2, 4, 13, 8, 4, 0

Offset: 1

Antti Karttunen, May 13 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000961 (positions of zeros).
Cf. A241917, A241919, A051119, A061395.
Cf. A049084, A027748.

a242411 1 = 0
a242411 n = i - j where
            (i:j:_) = map a049084 $ ps ++ [p]
            ps@(p:_) = reverse $ a027748_row n
-- Reinhard Zumkeller, May 15 2014

from sympy import factorint, primefactors, primepi
def a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])
def a053585(n):
    if n==1: return 1
    p = primefactors(n)[-1]
    return p**factorint(n)[p]
def a051119(n): return n/a053585(n)
def a(n): return 0 if n==1 or len(primefactors(n))==1 else a061395(n) - a061395(a051119(n)) # Indranil Ghosh, May 19 2017

(define (A242411 n) (if (= 1 (A001221 n)) 0 (- (A061395 n) (A061395 (A051119 n)))))

If A001221(n) = 1, then a(n) = 0, otherwise a(n) = A241919(n) = A061395(n) - A061395(A051119(n)).

A027854 Mutinous 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

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, 160, 165, 168, 175, 176, 180, 182, 189, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280

Offset: 1

Leroy Quet

nonn

Numbers n > 1 such that n/A053585(n) > A006530(n). - Michael De Vlieger, Jul 13 2017
If p = A006530(a(n)) then p * a(n) is in the sequence. E.g., as 12 is in the sequence with gpf(12) = A006530(12) = 3, 12*3^k is in the sequence for k > 0. Conjecture: if m is in the sequence then so is A003961(m). - David A. Corneth, Jul 13 2017
At present this and A027855 are complements in the set of integers >= 2. If a 1 were inserted at the start, then this and A027855 are complements in the set of positive integers. - Harry Richman, Sep 08 2019
The sequence is closed under multiplication (a semigroup). For, suppose x = p^i*m1, y = q^j*m2 are in the sequence, with p, q, p^i, p^j as given, with m1 > p and m2 > q, and suppose q >= p. If q = p then xy/q^(i+j) = m1*m2 > q. If q > p, then xy/q^j = p^i*m1*m2 > q (since q > p and p is greater than all primes in m1). - Richard Peterson, May 29 2022
There are subsequences that constitute subsemigroups: Consider as a subsequence all terms x such that x/p^k > a*p^b, with p,k as specified in the definition and a,b fixed real numbers greater than or equal to 1. Each pair (a,b) determines a subsequence that is also a subsemigroup of the original (1,1) semigroup that constitutes the whole sequence. The proof of closure is similar. To see that such proposed subsemigroups are nonempty, choose any prime p greater than 2 and multiply p by a sufficiently large power of 2. - Richard Peterson, May 29 2022
This sequence is a subsequence and subsemigroup of A289484. - Richard Peterson, Oct 29 2022

			From _Michael De Vlieger_, Jul 13 2017: (Start)
12 is a term since 12/A053585(12) = 12/3 = 4, A006530(12) = 3, and 4 > 3.
30 is a term since 30/A053585(30) = 30/5 = 6, A006530(30) = 5, and 6 > 5.
(End)

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A006530, A027855, A053585.

Select[Range@ 280, Function[n, (n/Apply[Power, Last@ #]) > #[[-1, 1]] &@ FactorInteger[n]]] (* Michael De Vlieger, Jul 13 2017 *)

isok(n) = {my(f = factor(n)); my(maxf = #f~); my(p = f[maxf, 1]); my(pk = f[maxf, 2]); (n/p^pk) > p;} \\ Michel Marcus, Jan 16 2014

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)*primefactors(n)[-1]]) # Indranil Ghosh, Jul 13 2017

Extended by Ray Chandler, Nov 17 2008

Offset changed to 1 by Michel Marcus, Jan 16 2014

A242415 Reverse the deltas of indices of distinct primes in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12, 13, 35, 10, 16, 17, 18, 19, 45, 21, 77, 23, 24, 25, 143, 27, 175, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 49, 75, 187, 1573, 53, 54, 33, 875, 247, 667, 59, 60, 61, 899, 63, 64, 65

Offset: 1

Antti Karttunen, May 24 2014

nonn

This self-inverse permutation (involution) of natural numbers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)]. It also preserves the exponent of the largest prime: A053585(a(n)) = A053585(n).
From the above it follows, that this fixes prime powers (A000961), among other numbers.
Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which reverses the order of horizontal line segments of the "steps" in Young (or Ferrers) diagram of a partition, but keeps the order of vertical line segments intact. Please see the last example in the example section and compare also to the comments given in A242419.

			For n = 10 = 2*5 = p_1 * p_3, we get p_(3-1) * p_3 = 3 * 5 = 15, thus a(10) = 15.
For n = 20 = 2*2*5 = p_1^2 * p_3^1, we get p_(3-1)^2 * p_3^1 = 3^2 * 5 = 45, thus a(20) = 45.
For n = 84 = 2*2*3*7 = p_1^2 * p_2 * p_4, when we reverse the deltas of indices, but keep the exponents same, we get p_(4-2)^2 * p_(4-1) * p_4 = p_2^2 * p_3 * p_4 = 3^2 * 5 * 7 = 315, thus a(84) = 315.
For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
Reversing the order of horizontal line segment lengths (1,2,2) to (2,2,1), but keeping the order of vertical line segment lengths as (3,2,1), we get a new Young diagram
   _ _
  |   |
  |   |
  |   |_ _
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,2,4,4,5), encoded in A112798 by p_2^3 * p_4^2 * p_5^1 = 3^3 * 7^2 * 11 = 14553, thus a(2200) = 14553.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A242413.
Cf. A112798, A122111, A153212, A000040, A241919, A067029, A242378, A051119, A225891.
{A000027, A069799, A242415, A242419} form a 4-group.

If n = p_a^e_a * p_b^e_b * ... * p_h^e_h * p_i^e_i * p_j^e_j * p_k^e_k, where p_a < ... < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_a .. e_k are their nonzero exponents, then a(n) = p_{k-j}^e_a * p_{k-i}^e_b  * p_{k-h}^e_c * ... * p_{k-a}^e_j * p_k^e_k.
As a recurrence: a(1) = 1, and for n>1, a(n) = (A000040(A241919(n))^A067029(n)) * A242378(A241919(n), a(A051119(A225891(n)))).
By composing/conjugating related permutations:
a(n) = A069799(A242419(n)) = A242419(A069799(n)).
a(n) = A122111(A069799(A122111(n))) = A153212(A069799(A153212(n))).

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

Leroy Quet

nonn

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

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A006530, A027854, A051119.

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

Select[Range@100, #1^(#2 + 1) & @@ FactorInteger[#][[-1]] > # &] (* Ivan Neretin, Jul 09 2015 *)

is(n) = my(f = factor(n)); c = n\f[#f~, 1]^f[#f~, 2]; c < f[#f~, 1] \\ David A. Corneth, Aug 19 2019

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

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

A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 1, 1, 3, 1, 1, 0, 3, 0, 1, 2, 0, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 1, 2, 1, 2, 0, 4, 0, 1, 0, 3, 1, 1, 1, 3, 0, 3, 1, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 2, 0, 3, 2

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The prime factors of 300 are {2,2,3,5,5} of which {2,2,3} are less than the largest, so a(300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 0's are A000961. Positions of 1's are A325230. Positions of terms > 1 are A307517.

Cf. A000094, A000245, A001222, A052126, A053585, A061395, A062977, A064989, A071178, A105441, A108951, A256617.

Table[PrimeOmega[n/Power@@FactorInteger[n][[-1]]],{n,100}]

A071178(n) = if(1==n, 0, factor(n)[omega(n), 2]);
A325226(n) = (bigomega(n) - A071178(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A001222(n/A053585(n)).

a(n) = A001222(n) - A071178(n) = A062977(A108951(n)). - Antti Karttunen, Nov 17 2019

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A085231 Numbers k in whose canonical factorization the power of the smallest prime factor is greater than the power of the greatest prime factor.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 63, 80, 96, 112, 120, 135, 144, 160, 168, 175, 176, 189, 192, 208, 224, 240, 275, 280, 288, 297, 315, 320, 325, 336, 351, 352, 360, 384, 405, 416, 425, 448, 459, 475, 480, 504, 513, 528, 539, 544, 560, 567, 575, 576, 608, 621, 624

Offset: 1

Reinhard Zumkeller, Jun 22 2003

p*a(n) is a term for all primes p with A020639(a(n)) < p < A006530(a(n)).

			The canonical factorization of 240 is 2^4 * 3 * 5. 2^4 = 16 > 5, therefore 240 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006530, A020639, A028233, A053585, A085232, A363122.
A085233 is a subsequence.
Subsequence of A102749.

pfgQ[n_]:=Module[{fe=#[[1]]^#[[2]]&/@FactorInteger[n]},fe[[1]]>fe[[-1]]]; Select[Range[700],pfgQ] (* Harvey P. Dale, Dec 11 2017 *)

A028233(a(n)) > A053585(a(n)).

Edited by Peter Munn, Jun 01 2025

A085232 In canonical prime factorization: power of smallest prime factor is less than power of greatest prime factor.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

A028233(a(n)) < A053585(a(n));
p*a(n) is a term for all primes p with A020639(a(n))A006530(a(n));
a(n)=A057714(n-1) for n<28: a(28)=60, A057714(28-1)=62.

			60 = 2^2 * 3 * 5 with 2^2=4 < 5, therefore 60 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A085231.

spfQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]>1&&fi[[1,1]]^fi[[1,2]] < fi[[-1,1]]^fi[[-1,2]]]; Select[Range[120],spfQ] (* Harvey P. Dale, Jul 30 2018 *)

A336363 Number of iterations of map k -> k*sigma(p^e)/p^e needed to reach a power of 2, where p is the largest prime factor of k and e is its exponent, when starting from k = n. a(n) = -1 if number of the form 2^k is never reached.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 2, 2, 1, 2, 1, 4, 0, 4, 3, 3, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 4, 1, 0, 4, 4, 3, 3, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 2, 1, 6, 2, 4, 2, 4, 3, 5, 1, 5, 5, 5, 4, 2, 1, 4, 0, 4, 4, 5, 4, 4, 3, 4, 3, 5, 4, 3, 3, 3, 3, 3, 2, 6, 3, 3, 2, 5, 3, 5, 2, 5, 4, 7, 2, 2, 2, 3, 1, 7, 6, 4, 2, 5, 4, 3, 2, 5

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Informally: starting from k=n, keep on replacing p^e, the maximal power of the largest prime factor in k, with (1 + p + p^2 + ... + p^e), until a power of 2 is reached. Sequence counts the steps needed.

			For n = 15 = 3*5, we obtain the following path, when starting from k = n, and when we always replace the maximal power of the largest prime factor, p^e of k with sigma(p^e) = (1 + p + p^2 + ... + p^e) in the prime factorization k: 3^1 * 5^1 -> 3*(5+1) = 18 = 2^1 * 3^2 -> 2 * (1+3+9) = 26 = 2 * 13 -> 2 * (13+1) = 28 = 2^2 * 7 -> 4*(7+1) = 2^5, thus it took four iterations to reach a power of two, and a(15) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A000265, A000593, A209229, A053585.

Cf. also A331410, A336361, A336362.

A053585(n) = if(1==n, 1, my(f=factor(n)); f[#f~, 1]^f[#f~, 2]);
A336363(n) = if(!bitand(n,n-1),0,my(pe=A053585(n)); 1+A336363((n/pe)*sigma(pe)));

If A209229(n) = 1 [when n is a power of 2], a(n) = 0, otherwise a(n) = 1 + a(sigma(A053585(n))*(n/A053585(n))).

a(n) = a(2n) = a(A000265(n)).

A379094 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 156, 168, 180, 204, 228, 240, 252, 264, 270, 276, 280, 300, 312, 315, 336, 348, 350, 360, 372, 378, 408, 420, 440, 444, 456, 480, 492, 495, 504, 516, 520, 525, 528, 540, 550, 552, 560, 564, 585, 588, 594, 600, 616, 624, 630, 636, 650

Offset: 1

Peter Luschny, Dec 17 2024

nonn

A379097 is a subsequence.
From Michael De Vlieger, Dec 18 2024: (Start)
Proper subset of A126706.
Smallest powerful number is a(314) = 2700. (End)

			60 is a term because the factors in the canonical prime factorization are [4, 3, 5], a list that is neither increasing nor decreasing.
Primorials (A002110) are not terms of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A053585, A057714, A085232, A140831, A379097.

with(ArrayTools):
fact := n -> local p; [seq(p[1]^p[2], p in ifactors(n)[2])]:
isA379094 := proc(n) local f; f := fact(n);
is(not IsMonotonic(f, direction=decreasing, strict=false) and not IsMonotonic(f, direction=increasing, strict=false)) end:
select(isA379094, [seq(1..650)]);

Select[Range[650], Function[f, NoneTrue[{Sort[f], ReverseSort[f]}, # == f &]][Power @@@ FactorInteger[#]] &] (* Michael De Vlieger, Dec 18 2024 *)

is_a379094(n) = my(C=apply(x->x[1]^x[2], Vec(factor(n)~))); vecsort(C)!=C && vecsort(C,,4)!=C \\ Hugo Pfoertner, Dec 18 2024

cp's OEIS Frontend

A241919 If n is a prime power, p_i^e, a(n) = i, (with a(1)=0), otherwise difference (i-j) of the indices of the two largest distinct primes p_i, p_j, i > j in the prime factorization of n: a(n) = A061395(n) - A061395(A051119(n)).

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A242411 If n is a prime power, p_i^e, a(n) = 0, otherwise difference (i-j) of the indices of the two largest distinct primes p_i, p_j, i > j in the prime factorization of n: a(n) = A061395(n) - A061395(A051119(n)).

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A027854 Mutinous 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.

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A242415 Reverse the deltas of indices of distinct primes in the prime factorization of n.

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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.

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A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

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A085231 Numbers k in whose canonical factorization the power of the smallest prime factor is greater than the power of the greatest prime factor.

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