A207481 -id:A207481 - cp's OEIS frontend

A377518 The largest divisor of n that is a term in A207481.

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 27, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 54, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Oct 30 2024

The number of these divisors is A377519(n), and their sum is A377520(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A207481, A368329, A377515, A377519, A377520.

f[p_, e_] := p^Min[p, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(f[i,1], f[i,2]));}

Multiplicative with a(p^e) = p^min(p, e).
a(n) = n if and only if n is in A207481.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((p+1)*s) - p^(p+1) - p^(p*s) + p^p)/p^((p+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^p * (p+1))) = 0.908130438292447963703... .

A377519 The number of divisors of n that are terms in A207481.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 8, 4, 6, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 30 2024

The sum of these divisors is A377520(n), and the largest of them is A377518(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A013661, A207481, A377516, A377518, A377520.

f[p_, e_] := Min[p, e] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, min(f[i,1], f[i,2]) + 1);}

a(n) = A000005(A377518(n)).
Multiplicative with a(p^e) = min(p, e) + 1.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^((p+1)*s)).

A377520 The sum of the divisors of n that are terms in A207481.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 40, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 120, 72, 56, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68

Offset: 1

Amiram Eldar, Oct 30 2024

First differs from A284341 at n = 81 = 3^4: a(81) = 40, while A284341(81) = 121.

The number of these divisors is A377519(n), and the largest of them is A377518(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A013661, A207481, A284341, A377517, A377518, A377519.

f[p_, e_] := (p^(Min[p, e] + 1) - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,1], f[i,2]) + 1) - 1)/(f[i,1] - 1));}

a(n) = A000203(A377518(n)).
Multiplicative with a(p^e) = (p^(min(p, e)+1) - 1)/(p - 1).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((p+1)*s) - p^(p+1))/p^((p+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(p+1)) = 1.42145673335960701365... .

A380088 The largest unitary divisor of n that is a term in A207481.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Jan 12 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A054743, A077610, A185359, A207481, A377518, A380085, A380089, A380090.

f[p_, e_] := If[e <= p, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] <= f[i, 1], f[i, 1]^f[i, 2], 1));}

Multiplicative with a(p^e) = p^e if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < n if and only if n is in A185359.
a(n) = n if and only if n is in A207481.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(2*(p+1)) + p^(2*p+1) - p^(p+1) - p^p + 1)/(p^(2*p+1) * (p+1)) = 0.87453068804586281444... .

A380089 The number of unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 2, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 12 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A054743, A077610, A185359, A207481, A377519, A380086, A380088, A380090.

f[p_, e_] := If[e <= p, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= f[i,1], 2, 1));}

a(n) = A034444(A380088(n)).
Multiplicative with a(p^e) = 2 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034444(n) if and only if n is in A185359.
a(n) = A034444(n) if and only if n is in A207481.
a(n) = A377519(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((p+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(p+1))) = 0.93168306734008028353...

A380090 The sum of the unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Amiram Eldar, Jan 12 2025

First differs from A371242 at n = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A054743, A077610, A185359, A207481, A371242, A377520, A380087, A380088, A380089.

f[p_, e_] := If[e <= p, p^e, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], f[i,1]^f[i,2], 0) + 1);}

a(n) = A034448(A380088(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034448(n) if and only if n is in A185359.
a(n) = A034448(n) if and only if n is in A207481.
a(n) = A377520(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(p+2) + p^(p+1) + p^p - p - 1)/(p^(p+1) * (p+1)) = 1.2078161... .

A054743 If n = Product p_i^e_i then p_i < e_i (where e_i > 0) for all i.

Original entry on oeis.org

1, 8, 16, 32, 64, 81, 128, 243, 256, 512, 648, 729, 1024, 1296, 1944, 2048, 2187, 2592, 3888, 4096, 5184, 5832, 6561, 7776, 8192, 10368, 11664, 15552, 15625, 16384, 17496, 19683, 20736, 23328, 31104, 32768, 34992, 41472, 46656

Offset: 1

James Sellers, Apr 22 2000

Closed under multiplication. Use A104126 to construct A192135 by putting A104126(n) * prime(n)^k in a list up to some chosen bound. Create this sequence by multiplying any k elements of A192135 with distinct prime factors in a list (k>1). The last list along with A192135 is this sequence when sorted. - David A. Corneth, Jun 07 2016

			8 appears in the list because 8 = 2^3 and 2<3.
Construction of elements up to 1000: 1. Put 2^3 and 3^5 in a list; {8, 81} (The terms of A104126 up to 1000.) 2. For each element, put products the last list with their distinct prime factors up to 1000. Gives: {8, 16, 32, 64, 128, 256, 512, 81, 243, 729} (Terms from A192135 up to 1000). 3. Put products of k powers of distinct primes in a new list up to 1000: {648} (k>1). Unite {648} with {8, 16, 32, 64, 128, 256, 512, 81, 243, 729}. {8, 16, 32, 64, 128, 256, 512, 81, 243, 729, 648}. Sort the list. This gives: {8, 16, 32, 64, 81, 128, 243, 256, 512, 648, 729}, which are the elements below 1000 in this sequence. - _David A. Corneth_, Jun 07 2016

Robert Israel, Table of n, a(n) for n = 1..10001

Cf. A048103, A054744, A104126, A192135.

Cf. A207481 (same construction, but with p_i>=e_i),

N:= 10^10: # to get all terms <= N
p:= 1:
S:= {1}:
do
  p:= nextprime(p);
  if p^(p+1) > N then break fi;
  pp:= [seq(p^j, j=p+1 .. ilog[p](N))];
  S:= S union select(`<=`,{seq(seq(s*q,s=S),q=pp)},N);
od:
sort(convert(S,list)); # Robert Israel, Jun 07 2016

okQ[n_] := AllTrue[FactorInteger[n], #[[1]] < #[[2]]&];
Join[{1}, Select[Range[50000], okQ]] (* Jean-François Alcover, Jun 08 2016 *)

lista(nn) = {for (n=1, nn, f = factor(n); ok = 1; for (i=1, #f~, if (f[i, 1] >= f[i, 2], ok = 0; break;);); if (ok, print1(n, ", ")););} \\ Michel Marcus, Jun 15 2013

Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/((p-1)*p^p) = 1.27325025767774256043... - Amiram Eldar, Nov 24 2020

1 prepended by Alec Jones, Jun 07 2016

A185359 Numbers k such that {m^m mod k: m >= 1} is not purely periodic.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 162, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 324, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400

Offset: 1

José María Grau Ribas, Jan 21 2012

nonn

k is a term if and only if k = Product_{i=1..t} p_i^e_i with e_i > p_i for some i.
A182938(a(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - 1/p^(p+1) = 0.13585792767780221591... - Amiram Eldar, Nov 24 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. Hampel, The length of the shortest period of rests of numbers n^n, Ann. Polon. Math. 1 (1955), 360-366.

Cf. A027748, A124010, A008590 (subsequence), A185358, A207481 (complement).

a185359 n = a185359_list !! (n-1)
a185359_list = [x | x <- [1..], or $ zipWith (<)
                    (a027748_row x) (map toInteger $ a124010_row x)]
-- Reinhard Zumkeller, Feb 18 2012

j[p_,e_]:=e>p;j[n_]:={False}==Union@Module[{fa=FactorInteger[n]},Table[j[fa[[i,1]],fa[[i,2]]],{i,1,Length[fa]}]];Select[Range[1000],!j[#]&]

A255805 Numbers with no zeros in base-8 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Reinhard Zumkeller, Mar 08 2015

Different from A047592, A207481.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Octal
Index entries for 2-automatic sequences.

Cf. A007094, A100970 (subsequence).

Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A248910 (base 6), A255808 (base 9), A052382 (base 10).

a255805 n = a255805_list !! (n-1)
a255805_list = iterate f 1 where
   f x = 1 + if r < 7 then x else 8 * f x'  where (x', r) = divMod x 8

Select[Range[100],DigitCount[#,8,0]==0&] (* Harvey P. Dale, Jun 08 2015 *)

isok(m) = vecmin(digits(m,8)) > 0; \\ Michel Marcus, Jan 23 2022

def ok(n): return '0' not in oct(n)[2:]
print([k for k in range(85) if ok(k)]) # Michael S. Branicky, Jan 23 2022

from sympy import integer_log
def A255805(n):
    m = integer_log(k:=6*n+1,7)[0]
    return sum(1+(k-7**m)//(6*7**j)%7<<3*j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A047592 Numbers that are congruent to {1, 2, 3, 4, 5, 6, 7} mod 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81

Offset: 1

N. J. A. Sloane

Or, numbers that are not multiples of 8. - Benoit Cloitre, Jul 11 2009
More generally the sequence of numbers not divisible by some fixed integer m >= 2 is given by a(n, m) = n - 1 + floor((n+m-2)/(m-1)). - Benoit Cloitre, Jul 11 2009
Complement of A008590. - Reinhard Zumkeller, Nov 30 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A008590, A168181, A207481.

[ n: n in [0..100] | n mod 8 in {1, 2, 3, 4, 5, 6, 7} ]; // Vincenzo Librandi, Dec 25 2010

A047592:=n->8*floor(n/7)+[1, 2, 3, 4, 5, 6, 7][(n mod 7)+1]: seq(A047592(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Complement[Range[88], 8Range[11]] (* Harvey P. Dale, Jan 22 2011 *)
CoefficientList[Series[(1 + x)*(1 + x^2)*(1 + x^4)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2013 *)

a(n)=n-1+floor((n+6)/7) \\ Benoit Cloitre, Jul 11 2009

a(n) = n - 1 + floor((n+6)/7). - Benoit Cloitre, Jul 11 2009
A168181(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
From R. J. Mathar, Mar 08 2011: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.
G.f.: x*(1+x)*(1+x^2)*(1+x^4) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). (End)
a(n) = A207481(n) for n <= 70. - Reinhard Zumkeller, Feb 18 2012
From Wesley Ivan Hurt, Jul 20 2016: (Start)
a(n) = (56*n - 28 + (n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) - 6*((n+6) mod 7))/49.
a(7k) = 8k-1, a(7k-1) = 8k-2, a(7k-2) = 8k-3, a(7k-3) = 8k-4, a(7k-4) = 8k-5, a(7k-5) = 8k-6, a(7k-6) = 8k-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (4*sqrt(sqrt(2)+2) - 2*sqrt(2) - 1)*Pi/16. - Amiram Eldar, Dec 28 2021

cp's OEIS Frontend

A377518 The largest divisor of n that is a term in A207481.

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A377519 The number of divisors of n that are terms in A207481.

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A377520 The sum of the divisors of n that are terms in A207481.

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A380088 The largest unitary divisor of n that is a term in A207481.

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A380089 The number of unitary divisors of n that are terms in A207481.

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A380090 The sum of the unitary divisors of n that are terms in A207481.

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A054743 If n = Product p_i^e_i then p_i < e_i (where e_i > 0) for all i.

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A185359 Numbers k such that {m^m mod k: m >= 1} is not purely periodic.