A007310 -id:A007310 - cp's OEIS frontend

A102280 Let k = n-th number == 1 or 5 mod 6 (A007310(n)); consider the generalized 3x+1 map T_k defined by T_k(n) = (3n+k)/2 if n odd, n/2 if n even; then a(n) = smallest positive number m which is in a primitive cycle of T_k.

1, 1, 5, 1, 1, 1, 5, 5, 7, 1, 13, 13, 19, 1, 1, 5, 25, 103, 1, 1, 1, 19, 17, 29, 5, 1, 1, 65, 7, 17, 1, 1, 1, 7, 5, 1, 19, 1, 13, 1, 5, 1, 1, 13, 11, 1, 11, 7, 1, 19, 41, 1, 13, 11, 11, 13, 11, 7, 29, 5, 11, 1, 5, 47, 1, 5, 13, 1, 43, 1, 37, 1, 17, 5, 35, 5, 7, 5, 11, 1, 7, 43, 1, 91, 1, 1, 29, 7, 47, 55, 17, 1, 13

Offset: 1

N. J. A. Sloane, Feb 19 2005

nonn

Geoffrey H. Morley, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arith. 56 (1990), 33-53. - See Table 3.1 on page 41.

Cf. A102284, A007310, A226633.

a(12)[k=35] and a(25)[k=73] corrected and more terms added by Geoffrey H. Morley, Mar 14 2013

A181709 Indices of primes in A007310.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 20, 21, 23, 24, 25, 27, 28, 30, 33, 34, 35, 36, 37, 38, 43, 44, 46, 47, 50, 51, 53, 55, 56, 58, 60, 61, 64, 65, 66, 67, 71, 75, 76, 77, 78, 80, 81, 84, 86, 88, 90, 91, 93, 94, 95, 98, 103, 104, 105, 106, 111, 113, 116, 117, 118

Offset: 1

Grant Garcia, Nov 07 2010

All primes but 2 and 3 are present in A007310, making this sequence an efficient method for storing large quantities of primes. To unpack this sequence into primes, use the formula (6n + (-1)^n - 3) / 2.

Indices 1 and 9 (1 and 25) are the smallest nonprimes.

			A007310(2), 5, is the first prime of the sequence.
A007310(50), 149, is also a prime, hence 50 is included.

Grant Garcia, Table of n, a(n) for n=1..10000

Cf. A007310, A000040.

Floor[Prime[Range[3,80]]/3]+1 (* Harvey P. Dale, Sep 12 2019 *)

from sympy import isprime
out = ""
for n,p in enumerate(isprime((6*n+(-1)**n-3)//2) for n in range(1,1000)):
    out+=["","%s "%str(n+1)][p]
for n,p in enumerate(out.rstrip(" ").split(" ")): print(n+1,p)

a(n) = floor(prime(n+2)/3)+1 = A144769(n+2)+1. - Gary Detlefs, Dec 11 2011

a(n) ~ n*log(n)/3. - Ilya Gutkovskiy, Jul 18 2016

A277907 a(n) = A007310(A277908(n)).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 41, 43, 59, 61, 67, 71, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 269, 271, 277, 281, 307, 311, 313, 317, 347, 349, 379, 383, 419, 421, 431, 433, 439, 443, 461, 463, 487, 491, 499, 503, 599, 601, 617, 619, 643, 647, 739, 743, 823, 827, 853, 857, 859, 863, 877, 881

Offset: 1

Antti Karttunen, Nov 05 2016

nonn

It seems that all terms are primes, and moreover, that all those primes occur in pairs with distance of 2 or 4. However, not all terms of A001359 are present.

Antti Karttunen, Table of n, a(n) for n = 1..7096

Cf. A000040, A001359, A007310, A118306, A277908, A277911.

(define (A277907 n) (A007310 (A277908 n)))

a(n) = A007310(A277908(n)).

A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 3, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 3, 6, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 6, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034444 at n = 25.

The sum of these divisors is A365211(n).

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

Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

Cf. A001620, A013661, A073002.

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

Multiplicative with a(p^e) = 2 for p = 2 and 3, and a(p^e) = e+1 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000005(A065330(n)) * A034444(A065331(n)).
Dirichlet g.f.: (1-1/4^s) * (1-1/9^s) * zeta(s)^2.
Sum_{k=1..n} a(k) ~ (2*n/3) * (log(n) + 2*gamma - 1 + 2*log(2)/3 + log(3)/4), where gamma is Euler's constant (A001620).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

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

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A158214 Smallest palindromic prime made up of 0's and k 1's, where k = A007310(n), odd numbers not divisible by 3.

Original entry on oeis.org

100111001, 110111011, 1110111110111, 10111101110111101, 100111111111111111001, 1111111111111111111, 11111111111111111111111, 10111111111101110111111111101, 1111110111111111111111110111111

Offset: 2

Lekraj Beedassy, Mar 13 2009

Smallest palindromic prime with digit sum A007310(n) and using only 0's and 1's.

Chai Wah Wu, Table of n, a(n) for n = 2..167

Cf. A020449.

from _future_ import division
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime
A158214_list = []
for i in range(2,101):
    if i % 6 == 1 or i % 6 == 5:
        i2 = i//2
        l = i2
        flag = True
        while flag:
            dlist = '0'*(l-i2) + '1'*i2
            for d in multiset_permutations(dlist):
                s = ''.join(d)
                n = int(s+'1'+s[::-1])
                if isprime(n):
                    A158214_list.append(n)
                    flag = False
                    break
            else:
                l += 1 # Chai Wah Wu, Dec 17 2015

A191833 Least number k such that k^k == k+1 (mod m), or 0 if no such k exists, where m = A007310(n).

Original entry on oeis.org

1, 7, 10, 14, 19, 11, 16, 3, 27, 43, 46, 178, 55, 36, 100, 64, 33, 79, 147, 43, 56, 258, 16, 86, 135, 52, 31, 27, 398, 335, 33, 187, 213, 151, 43, 680, 163, 61, 38, 243, 29, 327, 39, 213, 2068, 72, 37, 799, 198, 223, 141, 887, 92, 304, 132, 250, 808, 217, 327, 192, 271, 538, 398, 187, 79, 38, 31, 1713, 0, 413, 24, 1287, 976, 501, 48

Offset: 1

Franklin T. Adams-Watters, Jun 17 2011

nonn

k^k == k+1 (mod m) does not have any solutions for m = 2 or 3, so only numbers in A007310 need be considered.
In general, if there is a solution, the first is less than m * phi(m), where phi is the Euler totient function A000010, since the values loop from that point (at least for units).
a(n) = 0 if and only if A007310(n) is in A191834. - Robert Israel, Sep 12 2017

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

Cf. A007310, A191834.

f:= proc(n) local m,k;
  m:= (6*n + (-1)^n - 3)/2;
  for k from 1 to ilcm(m,numtheory:-phi(m)) do
    if igcd(k,m) = 1 and k &^ k - k - 1 mod m = 0 then return k fi;
  od:
  0
end proc:
map(f, [$1..100]); # Robert Israel, Sep 12 2017

A007310[n_] := 2*n + 2*Floor[n/2] - 1; a[n_] := (For[m = A007310[n]; k = 1, k <= m^2, k++, If[PowerMod[k, k, m] == Mod[k+1, m], Return[k]]]; 0); Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Sep 13 2013 *)

a(n)=local(m);m=A007310(n);for(k=1,m^2,if(Mod(k,m)^k==k+1,return(k)));0

A251394 Indices of numbers in A098550, that are neither multiples of 2 nor multiples of 3, cf. A007310.

Original entry on oeis.org

1, 9, 11, 13, 15, 22, 23, 30, 32, 34, 36, 38, 40, 43, 45, 51, 53, 61, 62, 68, 70, 72, 74, 76, 79, 87, 88, 94, 96, 98, 101, 103, 105, 114, 116, 118, 122, 124, 126, 127, 132, 134, 142, 144, 146, 148, 150, 153, 158, 160, 166, 167, 175, 177, 179, 181, 185, 187

Offset: 1

Reinhard Zumkeller, Dec 05 2014

nonn

GCD(A098550(a(n)),6) = 1.

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

Cf. A098550, A007310, A020639, A251237, A251393.

a251394 n = a251394_list !! (n-1)
a251394_list = filter ((== 1) . gcd 6 . a098550) [1..]

A365211 The sum of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 31, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 57, 93, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034448 at n = 25.

The number of these divisors is A365210(n).

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

Cf. A000203, A007310, A034448, A065330, A065331, A069184, A186099, A365210.

f[p_, e_] := If[p <= 3 , 1 + p^e, (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~, if(f[i,1] <= 3, 1 + f[i,1]^f[i,2], (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1)));}

Multiplicative with a(p^e) = 1 + p^e for p = 2 and 3, and a(p^e) = (p^(e+1)-1)/(p-1) for a prime p >= 5.
a(n) <= A000203(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034448(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000203(A065330(n)) * A034448(A065331(n)).
Dirichlet g.f.: (1 - 1/2^(2*s-1)) * (1 - 1/3^(2*s-1)) * zeta(s)*zeta(s-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 91*Pi^2/1296 = 0.69300463... .

A366441 The number of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 3, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 10 2023

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

Cf. A000005, A001620, A007310, A366442.

Similar sequences: A048691, A072048, A076400, A099774, A358040, A363194, A363195.

a[n_] := DivisorSigma[0, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = numdiv((3*n)\2 << 1 - 1)
```

from sympy import divisor_count
def A366441(n): return divisor_count((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A007310(n)).

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1 + 2*log(6)) * n / 3, where gamma is Euler's constant (A001620).

cp's OEIS Frontend

A102280 Let k = n-th number == 1 or 5 mod 6 (A007310(n)); consider the generalized 3x+1 map T_k defined by T_k(n) = (3n+k)/2 if n odd, n/2 if n even; then a(n) = smallest positive number m which is in a primitive cycle of T_k.

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A181709 Indices of primes in A007310.

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A277907 a(n) = A007310(A277908(n)).

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A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

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A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

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A158214 Smallest palindromic prime made up of 0's and k 1's, where k = A007310(n), odd numbers not divisible by 3.

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A191833 Least number k such that k^k == k+1 (mod m), or 0 if no such k exists, where m = A007310(n).

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A251394 Indices of numbers in A098550, that are neither multiples of 2 nor multiples of 3, cf. A007310.

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