A167747 -id:A167747 - cp's OEIS frontend

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

Original entry on oeis.org

4, 5, 0, 7, 1, 2, 6, 2, 5, 2, 2, 6, 0, 3, 9, 1, 3, 0, 8, 3, 0, 0, 0, 0, 7, 8, 9, 5, 8, 3, 5, 2, 7, 1, 5, 5, 6, 0, 4, 4, 6, 7, 8, 5, 0, 0, 5, 4, 0, 0, 8, 5, 4, 7, 4, 3, 9, 0, 4, 5, 8, 3, 4, 8, 9, 2, 4, 4, 0, 9, 6, 0, 7, 5, 4, 0, 6, 2, 9, 4, 0, 7, 8, 2, 4, 3, 5, 3, 4, 5, 3, 1, 8, 6, 0, 8, 9, 6, 2, 6, 9, 2, 7

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.45071262522603913...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A000217, A048651, A098844, A067080, A132019, A132026, A132030, A132034, A167747.

digits = 103; NProduct[1-1/(2*6^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
(1/2)*N[QPochhammer[1/12, 1/6], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=0, 1-1/(2*6^x)) \\ Altug Alkan, Dec 20 2015

Equals lim inf_{n->oo} Product_{k=0..floor(log_6(n))} floor(n/6^k)*6^k/n.
Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^(1/2*(1+floor(log_6(n)))*floor(log_6(n))).
Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^A000217(floor(log_6(n))).
Equals (1/2)*exp(-Sum_{n>0} 6^(-n)*Sum{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132030(n)/A132030(n+1).
Equals (1/2)*(1/12; 1/6){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
Equals Product_{n>=1} (1 - 1/A167747(n)). - Amiram Eldar, May 09 2023

A340302 Numbers k such that k and the least number that is larger than k and has the same prime signature as k also has the same set of distinct prime divisors as k.

Original entry on oeis.org

12, 72, 144, 420, 432, 540, 864, 1728, 1800, 2000, 2268, 2520, 2592, 5184, 5400, 6300, 7020, 10125, 10368, 10692, 10800, 11340, 12600, 15120, 15552, 16200, 17640, 20000, 20736, 21168, 21600, 24000, 24948, 25200, 26460, 31104, 37800, 40500, 41472, 42750, 43200

Offset: 1

Amiram Eldar, Jan 03 2021

nonn

Numbers k such that A007947(k) = A007947(A081761(k)).

This sequence is infinite since it includes all the numbers of the form 2*6^k for k>=1.

			12 = 2^2 * 3 is a term since the least number that is larger than 12 and has the same prime signature as 12 is 18 = 2 * 3^2 which also has the same set of distinct prime divisors as 12, {2, 3}.

Amiram Eldar, Table of n, a(n) for n = 1..500
Index entries for sequences related to prime signature

Cf. A007947, A081761, A085079, A089247, A118914, A167747.

sig[n_] := Sort@FactorInteger[n][[;; , 2]]; nextsig[n_] := Module[{sign = sig[n], k = n + 1}, While[sig[k] != sign, k++]; k]; rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; Select[Range[2, 1000], rad[#] == rad[nextsig[#]] &]

A375933 The second-largest exponent in the prime factorization of n, or 0 if it does not exist.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0

Offset: 1

Amiram Eldar, Sep 03 2024

First differs from A363127 at n = 60, and from A363131 at n = 72.

The position of the first occurrence of k = 1, 2, ..., is A167747(k+1) = 2*6^k.

			12 = 2^2 * 3^1 has 2 exponents in its prime factorization: 1 and 2. 2 is the largest and 1 is the second-largest. Therefore a(12) = 1.

Cf. A051903, A054861, A072774, A167747, A375932, A375934.

Cf. A363127, A363131.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, # < Max[e] &]]]]; Array[a, 100]

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); e = select(x -> x < vecmax(e), e); if(#e == 0, 0, vecmax(e)));

a(n) = A051903(A375932(n)).
a(n) = 0 if and only if n is a power of a squarefree number (A072774).
a(n) = 1 if and only if n is in A375934.
a(n) <= A051903(n), with equality if and only if n = 1.
a(n!) = A054861(n) for n != 3.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{i >= 1} i * d(i) = 0.42745228287872473252..., where d(i) = Sum_{j >= i+1} d_2(i, j) and d_2(i, j) = Product_{p prime} (1 - 1/p^(i+1) + 1/p^j - 1/p^(j+1)) - Product_{p prime} (1 - 1/p^(i+1)) + [i > 1] * (Product_{p prime} (1 - 1/p^i) - Product_{p prime} (1 - 1/p^i + 1/p^j - 1/p^(j+1))), and [] is the Iverson bracket.

A171501 Inverse binomial transform of A084640.

Original entry on oeis.org

0, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881

Offset: 0

Paul Curtz, Dec 10 2009

a(n) and differences are
0,     1,   3,  -1,    7,   -9,
1,     2,  -4,   8,  -16,   32,  =(-1)^(n+1) * A171449(n),
1,    -6,  12, -24,   48,  -96,
-7,   18, -36,  72, -144,  288,
25,  -54, 108, -216, 432, -864,
Vertical: 1) 0 followed with A168589(n).
          2) (-1 followed with A008776(n) ) signed. See A025192(n).
Main diagonal: see A167747(1+n). - Paul Curtz, Jun 16 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2).

I:=[0, 1, 3]; [n le 3 select I[n] else -Self(n-1) + 2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 18 2012

CoefficientList[Series[x*(1 + 4*x)/((1 + 2*x)*(1 - x)), {x, 0, 30}], x]
LinearRecurrence[{-1,2},{0,1,3},40] (* Harvey P. Dale, Jan 14 2020 *)

a(n) = A140966(n), n>0.
G.f.: x*(1+4*x) / ( (1+2*x)*(1-x) ). - R. J. Mathar, Jun 14 2011
a(1+n)= (-1)^(1+n) * A001045(1+n) + 2. - Paul Curtz, Jun 16 2011

Mathematica program by Olivier Gérard, Jul 06 2011

A199317 a(n) = 2*6^n + 1.

Original entry on oeis.org

3, 13, 73, 433, 2593, 15553, 93313, 559873, 3359233, 20155393, 120932353, 725594113, 4353564673, 26121388033, 156728328193, 940369969153, 5642219814913, 33853318889473, 203119913336833, 1218719480020993, 7312316880125953, 43873901280755713

Offset: 0

Vincenzo Librandi, Nov 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000400, A062394, A167747.

Magma
```
[2*6^n+1: n in [0..30]];
```

2 6^Range[0,30]+1 (* or *) LinearRecurrence[{7,-6},{3,13},30] (* Harvey P. Dale, Jul 02 2023 *)

a(n) = 6*a(n-1)-5.
a(n) = 7*a(n-1)-6*a(n-2).
G.f.: (3-8*x)/((1-x)*(1-6*x)).
a(n) = 1 + A167747(n+1) = 1 + 2*A000400(n) = A000400(n) + A062394(n). - Alois P. Heinz, Jul 02 2023

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Original entry on oeis.org

72, 108, 200, 392, 432, 500, 648, 675, 968, 1125, 1323, 1352, 1372, 1800, 2000, 2312, 2592, 2700, 2888, 3087, 3267, 3528, 3888, 4232, 4500, 4563, 5000, 5292, 5324, 5400, 5488, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10125, 10584, 10952, 11979

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

			72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

Cf. A051903, A051904, A072777.
Subsequence of A001694.
Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661

Offset: 1

Clark Kimberling, Jul 02 2025

nonn

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).
The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:
    U             V         volume   squared diagonal length, ||g(n)||^2
(1, 0, 0)     (0, 1, 0)     A000079           A052548
(1, 0, 0)     (0, 1, 1)     A008776         3*A052919
(1, 0, 0)     (1, 0, 1)     A000351           A178676
(1, 0, 0)     (1, 1, 1)     A167747         5*A204061
(1, 0, 0)     (0, 2, 0)     A005054         4*A199215
(1, 0, 0)     (1, 2, 0)     A013731         8*A199552
(1, 0, 0)     (2, 1, 0)     A011557        10*A000533
(1, 0, 0)     (1, 1, 2)     A067403        18*A135423
(1, 0, 0)     (2, 1, 1)     A334603        11*A199750
(1, 0, 1)     (0, 1, 0)     A008776           this sequence
(1, 1, 0)     (0, 1, 1)     A081341         6*A199318
(1, 1, 0)     (1, 1, 1)     A270369         9*A199559
(1, 2, 3)     (3, 2, 1)   2*A009992   48 + 96*A009992

			Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000079, A052548, A008776.

f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
f[n_] := f[n] = h[n - 1];
g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
h[n_] := h[n] = Cross[f[n], g[n]];
v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
Table[v[n], {n, 1, 16}]  (* A008776 *)
u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
Join[{1},Table[1+2*(3^(n-1)+1),{n,40}]] (* or *) LinearRecurrence[{4,-3},{1,5,9},50] (* Harvey P. Dale, Jul 20 2025 *)

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.

A247936 Riordan array ((1-2x)/(1-3x), 2x).

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 9, 6, 4, 8, 27, 18, 12, 8, 16, 81, 54, 36, 24, 16, 32, 243, 162, 108, 72, 48, 32, 64, 729, 486, 324, 216, 144, 96, 64, 128, 2187, 1458, 972, 648, 432, 288, 192, 128, 256, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 512, 19683, 13122

Offset: 0

Philippe Deléham, Sep 26 2014

Mirror of A153279.

			Triangle begins:
1
1, 2
3, 2, 4
9, 6, 4, 8
27, 18, 12, 8, 16
81, 54, 36, 24, 16, 32
243, 162, 108, 72, 48, 32, 64
Production matrix begins:
1, 2
1, 0, 2
1, 0, 0, 2
1, 0, 0, 0, 2
1, 0, 0, 0, 0, 2
1, 0, 0, 0, 0, 0, 2
1, 0, 0, 0, 0, 0, 0, 2

Cf. A000079, A000244, A133494, A153279, A167747.

Sum_{k, 0<=k<=n} T(n,k) = 3^n = A000244(n).
T(n,k) = A133494(n-k)*2^k.
T(2n,n) = A167747(n).

cp's OEIS Frontend

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

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A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

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A340302 Numbers k such that k and the least number that is larger than k and has the same prime signature as k also has the same set of distinct prime divisors as k.

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A375933 The second-largest exponent in the prime factorization of n, or 0 if it does not exist.

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A171501 Inverse binomial transform of A084640.

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A199317 a(n) = 2*6^n + 1.

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A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

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A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

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