Hauke Löffler - cp's OEIS frontend

A325943 a(n) = floor(n / omega(n)) where omega = A001221.

2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 6, 13, 7, 7, 16, 17, 9, 19, 10, 10, 11, 23, 12, 25, 13, 27, 14, 29, 10, 31, 32, 16, 17, 17, 18, 37, 19, 19, 20, 41, 14, 43, 22, 22, 23, 47, 24, 49, 25, 25, 26, 53, 27, 27, 28, 28, 29, 59, 20, 61, 31, 31, 64, 32, 22, 67, 34, 34, 23

Offset: 2

Hauke Löffler, Sep 09 2019

nonn

			a(2) = 2; 2 has one distinct prime divisor {2}, so a(2) = floor(2/1) = 2.
a(10) = 5; 10 has two distinct prime divisors {2,5}, so a(10) = floor(10/2) = 5.
a(15) = 7; 15 has two distinct prime divisors {3,5}, so a(15) = floor(15/2) = 7.

Hauke Löffler, Table of n, a(n) for n = 2..10001

[ n // (len(prime_divisors(n))) for n in range(2, 20) ]

A325938 a(n) = omega(n)^tau(n), where omega=A001221 and tau=A000005.

Original entry on oeis.org

0, 1, 1, 1, 1, 16, 1, 1, 1, 16, 1, 64, 1, 16, 16, 1, 1, 64, 1, 64, 16, 16, 1, 256, 1, 16, 1, 64, 1, 6561, 1, 1, 16, 16, 16, 512, 1, 16, 16, 256, 1, 6561, 1, 64, 64, 16, 1, 1024, 1, 64, 16, 64, 1, 256, 16, 256, 16, 16, 1, 531441, 1, 16, 64, 1, 16, 6561, 1, 64

Offset: 1

Hauke Löffler, Sep 09 2019

nonn

			a(5) = 1; 5 has one distinct prime divisor {5} and two divisors {1,5}, so a(5) = 1^2 = 1.
a(6) = 16; 6 has two distinct prime divisors {2,3} and four divisors {1,2,3,6}, so a(6) = 2^4 = 16.

Hauke Löffler, Table of n, a(n) for n = 1..10000

Cf. A000005(tau), A001221(omega), A110088, A248577.

a(n) = {omega(n)^numdiv(n)} \\ Andrew Howroyd, Sep 09 2019

[ len(prime_divisors(x))^(len(divisors(x))) for x in range(1,20) ]

a(n) = A001221(n) ^ A000005(n).

A309548 Numbers k such that sigma(k)! - 1 is prime, where sigma is A000203.

Original entry on oeis.org

2, 3, 4, 5, 6, 11, 13, 21, 29, 31, 37, 170, 180, 214, 234, 265, 362, 369, 10734, 14318, 19679, 19876, 39636, 48784, 62517, 76225, 77277, 83629, 85519, 90649, 92287

Offset: 1

Hauke Löffler, Aug 07 2019

			2 is a term because sigma(2) = 3. 3! - 1 = 5, a prime.
6 is a term because sigma(6) = 12. 12! - 1 = 479001599, a prime.

Cf. A000040, A000203, A002982, A055490, A184388.

isok(n) = isprime(sigma(n)!-1); \\ Michel Marcus, Aug 07 2019

[n for n in range(1,150) if is_prime(factorial(sigma(n))-1)]

a(12)-a(24) from Daniel Suteu, Aug 07 2019

a(25)-a(31) from Amiram Eldar, May 14 2023

A309432 Number of distinct digits in decimal representation of n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 4, 4, 3, 3, 4, 4, 2, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 4, 4, 4, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 3, 4

Offset: 0

Hauke Löffler, Aug 01 2019

			a(0) = 1 because 0^2 = 0 has 1 distinct digit (0).
a(5) = 2 because 5^2 = 25 has 2 distinct digits (2, 5).
a(10) = 2 because 10^2 = 100 has 2 distinct digits (0, 1).

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000290 (squares), A043537, A054039, A137214, A185679, A309431.

Array[Count[DigitCount[#^2], ?(# > 0 &)] &, 105, 0] (* _Michael De Vlieger, Jun 22 2022 *)

a(n) = if(n==0, 1, #Set(digits(n^2))); \\ Jinyuan Wang, Aug 02 2019, Jun 22 2022 [a(0)=1 corrected by Georg Fischer, Jun 22 2022]

def a(n): return len(set(str(n*n)))
print([a(n) for n in range(87)]) # Michael S. Branicky, Jun 22 2022

[len(set(Integer((n^2)).digits(padto=1))) for n in range(25) ]

A309415 Number of different numbers that are formed by permuting digits of n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 6, 12, 60, 360, 1260, 10080, 15120, 25200, 1247400, 32432400, 12612600, 6810804000, 7264857600, 185253868800, 1005663859200, 1117404288000, 4839757322400, 93504111468768000, 37401644587507200, 160787493266400000, 13023786954578400000

Offset: 0

Hauke Löffler, Jul 30 2019

a(0) = 1 because 0! = 1 has one permutation (1).
a(4) = 2 because 4! = 24 has two permutations (24, 42).
a(5) = 6 because 5! = 120 has 6 permutations (012, 021, 102, 120, 201, 210).

Alois P. Heinz, Table of n, a(n) for n = 0..460

Cf. A000142, A047726.

import Data.List (permutations, nub)
factorial n = product [1..n]
a309415 n = length $ nub $ permutations $ show $ factorial n
map a309415 [0..]

a:= n-> (l-> combinat[multinomial](add(i, i=l), l[])
      )([coeffs(add(x^i, i=convert(n!, base, 10)))]):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 30 2019

a[n_] := Block[{w = IntegerDigits[n!]}, Length[w]! / Times @@ ((Last /@ Tally[w])!)]; Array[a, 26, 0] (* Giovanni Resta, Jul 30 2019 *)

More terms from Giovanni Resta, Jul 30 2019

A309406 Absolute lucky numbers: every permutation of digits is a lucky number.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 31, 33, 37, 51, 73, 99, 111, 115, 151, 339, 393, 511, 777, 933, 9999, 33333, 55555, 111111, 777777, 7777777, 55555555

Offset: 1

Hauke Löffler, Jul 29 2019

More terms are in A031882, as A031882 is a subset of this sequence.

No more terms below 10^9. - Amiram Eldar, Nov 16 2019

			a(6) = 15 because 15 and 51 are lucky numbers.
a(14) = 115 because (115, 151, 511) are all lucky numbers.

Cf. A000959, A118561, A031882, A003459.

a(25)-a(27) from Amiram Eldar, Nov 16 2019

A309399 Number of lucky numbers l between powers of 2, 2^n < l <= 2^(n+1).

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 12, 21, 38, 71, 123, 234, 427, 791, 1477, 2774, 5222, 9849, 18659, 35412, 67410, 128644, 245959, 471166, 904186, 1738238, 3346542, 6452030, 12455921, 24076458, 46591766, 90258683, 175029533

Offset: 0

Hauke Löffler, Jul 28 2019

			a(0) = 0 because there are no lucky numbers between 1 (2^0) and 2 (2^1).
a(3) = 3 because there are 3 lucky numbers (9, 13, 15) between 8 (2^3) and 16 (2^4).

Cf. A000959, A036378.

def lucky(n):
  L=list(range(1, n+1, 2)); j=1
  while L[j] <= len(L)-1:
    L=[L[i] for i in range(len(L)) if (i+1)%L[j]!=0]
    j+=1
  return(L)
A000959=lucky(1048576)
def lucky_range(a, b):
    lucky = []
    for l in A000959:
        if l >= b:
            return lucky
        if l>=a: lucky.append(l)
[ len(lucky_range((2^n)+1,2^(n+1))) for n in range(19)]

a(19)-a(30) from Giovanni Resta, May 10 2020

a(31)-a(32) from Kevin P. Thompson, Nov 22 2021

A309396 Number of lucky numbers <= n!.

Original entry on oeis.org

1, 1, 1, 2, 7, 26, 115, 614, 3866, 28339, 237017, 2227657, 23233568, 266201749

Offset: 0

Hauke Löffler, Jul 28 2019

			a(1) = 1 because there is one lucky number (1) <= 1 (1!).
a(3) = 2 because there are two lucky numbers (1, 3) <= 6 (3!).

Cf. A000959, A003604, A309399, A000142.

def lucky(n):
  L=list(range(1, n+1, 2)); j=1
  while L[j] <= len(L)-1:
    L=[L[i] for i in range(len(L)) if (i+1)%L[j]!=0]
    j+=1
  return(L)
A000959=lucky(factorial(10))
def lucky_range(a,b):
    lucky = []
    for l in A000959:
        if l >= b:
            return lucky
        if l>=a: lucky.append(l)
[ len(lucky_range(0,factorial(n)+1)) for n in range(10) ]

a(10)-a(12) from Giovanni Resta, May 10 2020

a(13) from Kevin P. Thompson, Nov 24 2021

A309377 a(n) is the product of the divisors of n^n (A000312).

Original entry on oeis.org

1, 1, 8, 729, 68719476736, 30517578125, 2444746349972956194083608044935243159422957210683702349648543934214737968217920868940091707112078529114392164827136, 459986536544739960976801, 2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397376

Offset: 0

Hauke Löffler, Jul 26 2019

nonn

Subset of A007955.

			a(0) = 1 because 0^0 = 1, whose only divisor is 1, so the product of divisors is 1.
a(1) = 1 because 1^1 = 1, so the product of divisors is 1.
a(3) = 729 because 3^3 = 27, whose divisors are (1, 3, 9, 27), and their product is 729.

Cf. A007955, A062727.

[&*Divisors(n^n): n in [0..8]]; // Marius A. Burtea, Jul 26 2019

from math import isqrt, prod
from sympy import factorint
def A309377(n): return (isqrt(n**n) if (c:=prod(n*e+1 for e in factorint(n).values())) & 1 else 1)*n**(n*(c//2)) # Chai Wah Wu, Jun 25 2022

[ product((1*i^i).divisors()) for i in range(10) ]

A309375 Unlucky palindromic primes.

Original entry on oeis.org

2, 5, 11, 101, 131, 181, 191, 313, 353, 373, 383, 757, 797, 919, 929, 10301, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18481, 19391, 19891, 19991, 30203, 30403, 30703, 30803

Offset: 1

Hauke Löffler, Jul 26 2019

			a(1) = 2 because 2 is both a palindromic prime (A002385) and an unlucky number (A050505).

Amiram Eldar, Table of n, a(n) for n = 1..5527 (terms below 10^9)

Intersection of A002385 and A050505.

Cf. A031881.

SageMath
```
[ p for p in A002385 if p in A050505 ]
```

4 wrong terms removed and more terms added by Amiram Eldar, May 14 2023

cp's OEIS Frontend

User: Hauke Löffler

A325943 a(n) = floor(n / omega(n)) where omega = A001221.

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A325938 a(n) = omega(n)^tau(n), where omega=A001221 and tau=A000005.

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A309548 Numbers k such that sigma(k)! - 1 is prime, where sigma is A000203.

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A309432 Number of distinct digits in decimal representation of n^2.

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A309415 Number of different numbers that are formed by permuting digits of n!.

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A309406 Absolute lucky numbers: every permutation of digits is a lucky number.

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A309399 Number of lucky numbers l between powers of 2, 2^n < l <= 2^(n+1).

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SageMath

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A309396 Number of lucky numbers <= n!.

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SageMath