A007310 -id:A007310 - cp's OEIS frontend

A249745 Permutation of natural numbers: a(n) = (1 + A064989(A007310(n))) / 2.

1, 2, 3, 4, 6, 7, 9, 10, 5, 12, 15, 8, 16, 19, 21, 22, 13, 24, 11, 27, 30, 17, 31, 34, 36, 18, 37, 40, 20, 42, 28, 26, 45, 49, 51, 52, 54, 55, 29, 33, 25, 14, 57, 64, 43, 66, 69, 39, 35, 70, 75, 44, 76, 48, 79, 82, 61, 84, 23, 87, 90, 47, 46, 91, 96, 97, 99, 58, 56, 60, 100, 62, 73, 72, 106, 112, 114, 115, 65, 117, 120, 38, 94, 121

Offset: 1

Antti Karttunen, Nov 23 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249746.
Row 2 of A251721.
Cf. A007310, A048673, A064989, A249823, A249735.

a249745[n_Integer] := Module[{f, p, a064989, a007310, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a007310[x_] := Select[Range[x], MemberQ[{1, 5}, Mod[#, 6]] &];
  a[x_] := (1 + a064989 /@ a007310[x])/2;
a[n]]; a249745[252] (* Michael De Vlieger, Dec 18 2014, after Harvey P. Dale at A007310 *)

A249745(n)=A064989(A007310(n))\2+1 \\ M. F. Hasler, Jan 19 2016

(define (A249745 n) (/ (+ 1 (A064989 (A007310 n))) 2))

a(n) = (A064989(A007310(n)) + 1) / 2.
a(n) = A048673(A249823(n)), as a composition of related permutations.
A007310(n) = A249735(a(n)) for all n >= 1. (This is the permutation which sorts the terms of A249735 into an ascending order, as they occur in A007310.)

A384058 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 7, 7, 8, 5, 11, 6, 13, 7, 10, 15, 17, 8, 19, 15, 14, 11, 23, 14, 25, 13, 26, 21, 29, 10, 31, 31, 22, 17, 35, 24, 37, 19, 26, 35, 41, 14, 43, 33, 40, 23, 47, 30, 49, 25, 34, 39, 53, 26, 55, 49, 38, 29, 59, 30, 61, 31, 56, 63, 65, 22, 67, 51, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384042.
Cf. A007310, A065330, A065331, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), this sequence (5-rough).

f[p_, e_] := p^e - If[p < 5, 1, 0]; 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]^f[i,2] - if(f[i,1] < 5, 1, 0));}

Multiplicative with a(p^e) = p^e-1 if p <= 3, and p^e if p >= 5.
a(n) = n * A047994(n) / A384057(n).
a(n) = A047994(A065331(n)) * A065330(n).
Dirichlet g.f.: zeta(s-1) * ((1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s)) * ((1 - 2/3^s + 1/3^(2*s-1))/(1 - 1/3^s)).
Sum_{k=1..n} a(k) ~ (55/144) * n^2.
In general, the average order of the number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a p-rough number (i.e., not divisible by any prime smaller than the prime p) is (1/2) * Product_{q prime <= p} (1 - 1/q + 1/(q+1)) * n^2.

A135765 Distribute the odd numbers in columns based on the occurrence of "3" in each prime factorization; square array A(row, col) = 3^(row-1) * A007310(col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 5, 3, 7, 15, 9, 11, 21, 45, 27, 13, 33, 63, 135, 81, 17, 39, 99, 189, 405, 243, 19, 51, 117, 297, 567, 1215, 729, 23, 57, 153, 351, 891, 1701, 3645, 2187, 25, 69, 171, 459, 1053, 2673, 5103, 10935, 6561, 29, 75, 207, 513, 1377, 3159, 8019, 15309, 32805

Offset: 1

Alford Arnold, Nov 28 2007

The Table can be constructed by multiplying sequence A000244 by A007310.
From Antti Karttunen, Jan 26 2015: (Start)
A permutation of odd numbers. Adding one to each term and then dividing by two gives a related table A254051, which for any odd number, located in this array as x = A(row,col), gives the result at A254051(row+1,col) after one combined Collatz step (3x+1)/2 -> x (A165355) has been applied.
Each odd number n occurs here in position A(A007949(n), A126760(n)).
Compare also to A135764.
(End)

			The top left corner of the array:
    1,    5,    7,   11,   13,   17,   19,   23,   25,   29,   31,   35, ...
    3,   15,   21,   33,   39,   51,   57,   69,   75,   87,   93,  105, ...
    9,   45,   63,   99,  117,  153,  171,  207,  225,  261,  279,  315, ...
   27,  135,  189,  297,  351,  459,  513,  621,  675,  783,  837,  945, ...
   81,  405,  567,  891, 1053, 1377, 1539, 1863, 2025, 2349, 2511, 2835, ...
  243, 1215, 1701, 2673, 3159, 4131, 4617, 5589, 6075, 7047, 7533, 8505, ...
etc.
For n = 6, we have [A002260(6), A004736(6)] = [3, 1] (that is 6 corresponds to location 3,1 (row,col) in above table) and A(3,1) = A000244(3-1) * A007310(1) = 3^2 * 1 = 9.
For n = 9, we have [A002260(9), A004736(9)] = [3, 2] (9 corresponds to location 3,2) and A(3,2) = A000244(3-1) * A007310(2) = 3^2 * 5 = 9*5 = 45.
For n = 13, we have [A002260(13), A004736(13)] = [3, 3] (13 corresponds to location 3,3) and A(3,3) = A000244(3-1) * A007310(3) = 3^2 * 7 = 9*7 = 63.
For n = 23, we have [A002260(23), A004736(23)] = [2, 6] (23 corresponds to location 2,6) and A(2,6) = A000244(2-1) * A007310(6) = 3^1 * 17 = 51.

Row 1: A007310.
Column 1: A000244.
Cf. A007949 (row index), A126760 (column index).
Cf. also A000265, A002260, A003961, A004736, A005408, A165355, A249745.
Related arrays: A135764, A254051, A254055, A254101, A254102.

N:= 20:
B:= [seq(op([6*n+1,6*n+5]),n=0..floor((N-1)/2))]:
[seq(seq(3^j*B[i-j],j=0..i-1),i=1..N)]; # Robert Israel, Jan 26 2015

From Antti Karttunen, Jan 26 2015: (Start)
With both row and col starting from 1:
A(row, col) = A000244(row-1) * A007310(col) = 3^(row-1) * A007310(col).
a(n) = (2*A254051(n))-1.
a(n) = A003961(A254053(n)).
Above in array form:
A(row,col) = A003961(A254053(row,col)) = A003961(A135764(row,A249745(col))).
(End)

Name amended and examples edited by Antti Karttunen, Jan 26 2015

A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 26, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 50, 26, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A186099 at n = 25; a(25) = 26, while A186099(25) = 31.

The number of these divisors is A385044(n), and the largest of them is A065330(n).

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

The unitary analog of A186099.
Cf. A002117, A007310, A034448, A065330, A385044.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), this sequence (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[p <= 3, 1, 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,1] < 5, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if p <= 3, and p^e + 1 if p >= 5.
a(n) = A034448(n)/A385046(n).
a(n) <= A034448(n), with equality if and only if n is 5-rough.
a(n) <= A186099(n).
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1)) * ((1-1/2^(s-1))/(1-1/2^(2*s-1))) * ((1-1/3^(s-1))/(1-1/3^(2*s-1))).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/(91*zeta(3)) = 0.270679... .

A266635 Permutation of natural numbers: a(n) = A126760(A264965(A007310(n))).

Original entry on oeis.org

1, 3, 2, 5, 7, 9, 6, 19, 4, 8, 13, 15, 23, 11, 27, 16, 25, 17, 22, 10, 21, 20, 49, 65, 12, 73, 41, 61, 37, 24, 29, 77, 14, 55, 47, 79, 31, 26, 43, 59, 18, 67, 35, 38, 81, 33, 62, 54, 57, 46, 78, 30, 70, 53, 74, 42, 69, 34, 163, 50, 58, 45, 66, 91, 71, 28, 60, 44, 76, 39, 68, 36, 63, 52, 217, 32, 64, 51, 80, 109

Offset: 1

Antti Karttunen, Jan 04 2016

Permutation induced when A264965 is restricted to numbers neither divisible by 2 nor 3 (A007310).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A266636.
Cf. A007310, A126760, A264965.
Cf. also A266641, A266642, A266643, A266644.

(define (A266635 n) (A126760 (A264965 (A007310 n))))

a(n) = A126760(A264965(A007310(n))).

A266636 Permutation of natural numbers: a(n) = A126760(A264966(A007310(n))).

Original entry on oeis.org

1, 3, 2, 9, 4, 7, 5, 10, 6, 20, 14, 25, 11, 33, 12, 16, 18, 41, 8, 22, 21, 19, 13, 30, 17, 38, 15, 66, 31, 52, 37, 76, 46, 58, 43, 72, 29, 44, 70, 82, 27, 56, 39, 68, 62, 50, 35, 84, 23, 60, 78, 74, 54, 48, 34, 85, 49, 61, 40, 67, 28, 47, 73, 77, 24, 63, 42, 71, 57, 53, 65, 83, 26, 55, 81, 69, 32, 51, 36, 79, 45

Offset: 1

Antti Karttunen, Jan 04 2016

Permutation induced when A264966 is restricted to numbers neither divisible by 2 nor 3 (A007310).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A266635.
Cf. A007310, A126760, A264966.
Cf. also A266641, A266642, A266643, A266644.

(define (A266636 n) (A126760 (A264966 (A007310 n))))

a(n) = A126760(A264966(A007310(n))).

A277911 Self-inverse permutation of natural numbers induced when A118306 is restricted to A007310.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 10, 17, 8, 13, 26, 11, 15, 14, 18, 9, 16, 31, 21, 20, 40, 24, 23, 27, 12, 25, 30, 45, 28, 19, 54, 34, 33, 36, 35, 38, 37, 68, 22, 57, 115, 44, 43, 29, 47, 46, 74, 73, 51, 50, 87, 55, 32, 53, 58, 41, 56, 180, 61, 60, 96, 83, 65, 64, 67, 66, 39, 101, 100, 75, 110, 49, 48, 71, 77, 76, 80, 124, 78, 84, 283, 63, 81, 126, 88, 52

Offset: 1

Antti Karttunen, Nov 05 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A007310, A118306, A126760.

Cf. A277908, A277909, A277910 (for "almost fixed points"), also A277907.

A118306(n) = { if(1==n, 1, my(f = factor(n)); my(d = (-1)^primepi(f[1, 1])); for(i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-d)); factorback(f)); };
A007310(n) = n\2*6-(-1)^n; \\ From M. F. Hasler, Oct 31 2014
A126760(n) = {n&&n\=3^valuation(n, 3)<M. F. Hasler, Jan 19 2016
A277911(n) = A126760(A118306(A007310(n)));
for(n=1, 10001, write("b277911.txt", n, " ", A277911(n)));

(define (A277911 n) (A126760 (A118306 (A007310 n))))

a(n) = A126760(A118306(A007310(n))).

A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 14, 16, 18, 16, 18, 18, 20, 24, 24, 24, 24, 24, 31, 28, 27, 32, 30, 36, 32, 32, 36, 36, 48, 36, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 48, 57, 62, 54, 56, 54, 54, 72, 64, 60, 60, 60, 72, 62, 64, 72, 64, 84, 72, 68, 72

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A007310, A013661, A064987, A072044, A072045, A186099.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p > 3, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p > 3, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385137(n).
Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.

A254050 Permutation of odd numbers: a(n) = (2*(A249745(n))) - 1 = A064989(A007310(n)).

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 9, 23, 29, 15, 31, 37, 41, 43, 25, 47, 21, 53, 59, 33, 61, 67, 71, 35, 73, 79, 39, 83, 55, 51, 89, 97, 101, 103, 107, 109, 57, 65, 49, 27, 113, 127, 85, 131, 137, 77, 69, 139, 149, 87, 151, 95, 157, 163, 121, 167, 45, 173, 179, 93, 91, 181, 191, 193, 197, 115, 111, 119, 199, 123

Offset: 1

Antti Karttunen, Jan 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A007310, A064989, A249745, A254049, A254051, A254053.

a(n) = (2*(A249745(n))) - 1.

a(n) = A064989(A007310(n)).

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; 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,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

cp's OEIS Frontend

A249745 Permutation of natural numbers: a(n) = (1 + A064989(A007310(n))) / 2.

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A384058 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a 5-rough number (A007310).

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A135765 Distribute the odd numbers in columns based on the occurrence of "3" in each prime factorization; square array A(row, col) = 3^(row-1) * A007310(col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

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A266635 Permutation of natural numbers: a(n) = A126760(A264965(A007310(n))).

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A266636 Permutation of natural numbers: a(n) = A126760(A264966(A007310(n))).

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A277911 Self-inverse permutation of natural numbers induced when A118306 is restricted to A007310.

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A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

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