A215848 -id:A215848 - cp's OEIS frontend

A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).

1, 5, 6, 7, 8, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 35, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 55, 56, 59, 61, 64, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 101, 102, 103, 104, 107, 109, 113, 114, 115, 119, 121, 125, 127, 131, 133, 135

Offset: 1

Griffin N. Macris, Dec 15 2020

nonn

From Peter Munn, Mar 16 2021: (Start)
The positive integers in the multiplicative subgroup of the positive rationals generated by 8, 6, and A215848 (primes greater than 3).
This subgroup, denoted H, has two cosets: 2H = (1/3)H and 3H = (1/2)H. It follows that the sequence is one part of a 3-part partition of the positive integers with the property that each part's terms are half the even terms of one of the other parts and also one third of the multiples of 3 in the remaining part.
(End)
Positions of multiples of 3 in A276085 (and in A276075). Because A276085 is completely additive, this is closed under multiplication: if m and n are in the sequence then so is m*n. - Antti Karttunen, May 27 2024
The coset sequences mentioned in Peter Munn's comment above are A373261 and A373262. - Antti Karttunen, Jun 04 2024

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

Sequences of positive integers in a multiplicative subgroup of positive rationals generated by a set S and A215848: S={}: A007310, S={6}: A064615, S={3,4}: A003159, S={2,9}: A007417, S={4,6}: A036668, S={3,8}: A191257, S={4,9}: A339690, S={6,8}: this sequence.
Positions of 0's in A373153, positions of multiples of 3 in A276085 and in A372576.
Cf. A372573 (characteristic function), A373261, A373262.
Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120, A377872.
Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8, 27, 3125: A003159, this sequence, A369002, A373140, A373138, A377872, A377878.
Cf. also A332820, A373992, A383288.

N:= 1000: # for terms <= N
R:= {}:
for k1 from 0 to floor(N/6) do
  for k0 in [1,5] do
    k:= k0 + 6*k1;
    for j from 0 while 3^j*k <= N do
      for i from (j mod 3) by 3 do
        x:= 2^i * 3^j * k;
        if x > N then break fi;
        R:= R union {x}
od od od od:
sort(convert(R,list)); # Robert Israel, Apr 08 2021

Select[Range[130], Mod[IntegerExponent[#, 2] - IntegerExponent[#, 3], 3] == 0 &]

isA339746 = A372573; \\ Antti Karttunen, Jun 04 2024

from sympy import factorint
def ok(n):
  f = factorint(n, limit=4)
  i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]
  return (i-j)%3 == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(200)) # Michael S. Branicky, Mar 26 2021

from itertools import count
def A339746(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 12 2025

Formula

a(n) ~ (91/43)*n.

A276378 Numbers k such that 6*k is squarefree.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211

Offset: 1

Juri-Stepan Gerasimov, Sep 02 2016

These are the numbers from A005117 that are not divisible by 2 and 3.
Squarefree numbers coprime to 6. - Robert Israel, Sep 02 2016
Numbers k such that A008588(k) is in A005117. - Felix Fröhlich, Sep 02 2016
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, May 22 2020
From Peter Munn, Nov 20 2020: (Start)
The products generated from each subset of A215848 (primes greater than 3).
Closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. (End)
Multiplied by 6 we have 6, 30, 42, 66, 78, 102, ..., the values that may appear in A076978 after the 1, 2. [Don Reble, Dec 02 2020] - R. J. Mathar, Dec 15 2020
By the von Staudt-Clausen theorem, denominators of Bernoulli numbers are of the form 6*a(n) for some n. - Charles R Greathouse IV, May 16 2024

			5 is in this sequence because 6*5 = 30 = 2*3*5 is squarefree.

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

Numbers m such that k*m is squarefree: A005117 (k = 1), A056911 (k = 2), A261034 (k = 3), A274546 (k = 5).
Subsequence of A007310, A300957, and A339690.
Cf. A003961, A008588, A059897, A076978, A104141, A215848.

[n: n in [1..230] | IsSquarefree(6*n)];

select(numtheory:-issqrfree, [seq(seq(6*i+j,j=[1,5]),i=0..100)]); # Robert Israel, Sep 02 2016

Select[Range@ 212, SquareFreeQ[6 #] &] (* Michael De Vlieger, Sep 02 2016 *)

is(n) = issquarefree(6*n) \\ Felix Fröhlich, Sep 02 2016

{a(n) : n >= 1} = {A003961(A003961(A005117(n))) : n >= 1} = {A003961(A056911(n)) : n >= 1}. - Peter Munn, Nov 20 2020

Sum_{n>=1} 1/a(n)^s = (6^s)*zeta(s)/((1+2^s)*(1+3^s)*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

Original entry on oeis.org

341, 5461, 1398101, 22369621, 5726623061, 91625968981, 23456248059221, 96076792050570581, 1537228672809129301, 6296488643826193618261, 1611901092819505566274901, 25790417485112089060398421, 6602346876188694799461995861

Offset: 1

Bruno Berselli, Jan 21 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

This is the case a=2 of Theorem 1 in the paper of Hamahata and Kokubun (see Links section).

Named after the Italian mathematician Michele Cipolla (1880-1947). - Amiram Eldar, Jun 15 2021

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Pseudoprimi di Fermat e numeri di Carmichael (in Italian), 2013. The sequence is on page 3.
Michele Cipolla, Sui numeri composti P, che verificano la congruenza di Fermat a^(P-1) = 1 (mod P), Annali di Matematica, Vol. 9 (1904), pp. 139-160.
Y. Hamahata and Y. Kokubun, Cipolla Pseudoprimes, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.6.

Cf. A001567, A002450, A210461, A215848, A216170.

a210454 = (`div` 3) . (subtract 1) . (4 ^) . a000040 . (+ 2)
-- Reinhard Zumkeller, Jan 22 2013

Magma
```
[(4^NthPrime(n)-1)/3: n in [3..15]];
```

P:=proc(q)local n;
for n from 3 to q do print((4^ithprime(n)-1)/3);
od; end: P(100); # Paolo P. Lava, Oct 11 2013

Mathematica
```
(4^# - 1)/3 & /@ Prime[Range[3, 15]]
```

Prime(n) := block(if n = 1 then return(2), return(next_prime(Prime(n-1))))$
makelist((4^Prime(n)-1)/3, n, 3, 15);

a(n)=4^prime(n+2)\3 \\ Charles R Greathouse IV, Jul 09 2015

A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 9, 15, 20, 24, 31, 35, 42, 49, 59, 63, 72, 76, 84, 95, 106, 110, 121, 125

Offset: 1

Omar E. Pol, Apr 16 2019

To calculate a(n) consider that levels greater than n do not exist.
The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).
For more information about the sequences that we can see in the pyramid see A262626.

			For n = 1 the first level of the stepped pyramid (starting from the top) is a cube, and a cube has six faces, so a(1) = 6.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A325301 (number of edges), A325302 (number of vertices).

Cf. A196020, A215848, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A323645, A323648.

a(n) = A325301(n) - A325302(n) + 2 (Euler's formula).
a(n) = A323645(n) + 3.
a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

A323645 a(n) is the number of visible faces in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 6, 12, 17, 21, 28, 32, 39, 46, 56, 60, 69, 73, 81, 92, 103, 107, 118, 122

Offset: 1

Omar E. Pol, Mar 20 2019

The shape of the n-th level of the pyramid allows us to know if n is prime (see the Formula section).

For more sequences that we can find in the pyramid see A262626.

Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16) and A000203

Cf. A000040, A000203, A024916, A196020, A215848, A224880, A235791, A236104, A237048, A237270, A237271, A237593, A245092, A262626, A299692 (total area).

a(n) = a(n-1) + 4 iff n is a prime > 3 (A215848).

a(n) = A325300(n) - 3. - Omar E. Pol, Apr 17 2019

a(18)-a(19) from Omar E. Pol, Apr 18 2019

A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

Original entry on oeis.org

1, 2, 136, 156, 328, 444, 584, 600, 712, 732, 776, 876, 904, 1096, 1164, 1176, 1308, 1544, 1864, 1884, 1928, 2056, 2172, 2248, 2316, 2504, 2601, 2696, 2748, 2824, 2892, 2904, 3208, 3240, 3249, 3272, 3324, 3464, 3592, 3656, 3756, 4044, 4056, 4168, 4188, 4476, 4552, 4616

Offset: 1

M. F. Hasler, Apr 15 2022

nonn

As d^2 | N-d we have N = k*d^2 + d for some k >= 0 and d > 1. So gcd(k*d^2 + d - d, (N*d^2 + d)*d) = gcd(k*d^2, k*d^3 + d^2) = gcd(k*d^2, d^2) = d^2. So for any N such that d^2 | gcd(N - d, N*d) we have gcd(N - d, N*d) = d^2. - David A. Corneth, Apr 20 2022

Since gcd(N - d, N*d) is never larger than d^2 (if N = n*g, d = f*g with gcd(n,f) = 1, then gcd(N - d, N*d) = g*gcd(n-f,n*f*g) = g*gcd(n-f, f*f*g) <= g*g, since by assumption, no factor of f divides n), so one can also replace "=" by ">=" in the definition.

			N = 1 is in the sequence because d(N) = 1, gcd(1 - 1, 1*1) = 1 = d^2.
N = 2 is in the sequence because d(N) = 2, gcd(2 - 2, 2*2) = 4 = d^2.
N = 136 = 8*17 is in the sequence because d(N) = 4*2 = 8, gcd(8*17 - 8, 8*17*8) = gcd(8*16, 8*8*17) = 8*8 = d^2. Similarly for N = 8*p with any prime p = 8*k + 1.
N = 156 = 2^2*3*13 is in the sequence because d(n) = 3*2*2 = 12, gcd(12*13 - 12, 12*13*12) = gcd(12*12, 12*12*13) = 12*12 = d^2. Similarly for any N = 12*p with prime p = 12*k + 1.
More generally, when N = m*p^k with p^k == 1 (mod m) and m = (k+1)*d(m), then d(N) = d(m)*(k+1) = m and gcd(n - d, n*d) = gcd(m*p^k - m, m*p^k*m) = m*gcd(p^k - 1, p^k*m) = m^2. This holds for m = 8 and 12 with k = 1, for m = 9, 18 and 24 with k = 2, etc: see sequence A033950 for the m-values.

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

Cf. A000005 (number of divisors), A352483 (numerator of (n-d)/(n*d)), A352482 (denominator), A049820 (n - d), A146566 (n*d is divisible by n-d), A033950 (refactorable or tau numbers: d(n) | n, supersequence of this).

Select[Range[4650], GCD[#1 - #2, #1 #2] == #2^2 & @@ {#, DivisorSigma[0, #]} &] (* Michael De Vlieger, Apr 21 2022 *)

select( {is(n, d=numdiv(n))=gcd(n-d,d^2)==d^2}, [1..10^4])

For all m in A033950, the sequence contains all numbers m*p^k with k = m/d(m) - 1, and p^k == 1 (mod m), in particular 8*A007519 and 12*A068228 (k = 1, m = 8 and 12), 9*A129805^2, 18*A129805^2 and 24*A215848^2 (k = 2, m = 9, 18 and 24, A^2 = {x^2, x in A}), etc.

cp's OEIS Frontend

A339746 Positive integers of the form 2^i*3^j*k, gcd(k,6)=1, and i == j (mod 3).

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A276378 Numbers k such that 6*k is squarefree.

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A210454 Cipolla pseudoprimes to base 2: (4^p-1)/3 for any prime p greater than 3.

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A325300 a(n) is the number of faces of the stepped pyramid with n levels described in A245092.

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A323645 a(n) is the number of visible faces in the perspective view of the stepped pyramid with n levels described in A245092.

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Extensions

A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

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A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).