Keith F. Lynch - cp's OEIS frontend

A385023 Number of cuboids (rectangular prisms) that can be formed from the points of Z^3 (a cubical grid of n X n X n points).

0, 1, 36, 372, 2032, 8107, 24986, 66688, 155896, 332657, 653708, 1216076, 2135220, 3604679, 5845214, 9160864, 13947880, 20778029, 30205036, 43114824, 60340252, 83145027, 112870514, 151270988, 199965096, 261491409

Offset: 1

Keith F. Lynch, Jun 15 2025

Skew cuboids are allowed. The number of orthogonal cuboids is simply binomial(n, 2)^3.

The first 15 terms were independently computed by Keith Lynch and Michael Beeler. Terms 16 through 26 are from Michael Beeler.

			The only solution for n=2 is:
  0,0,0; 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0; 1,1,1
The 36 solutions for n=3 are:
  0,0,0; 0,0,1; 0,1,0; 0,1,1; 2,0,0; 2,0,1; 2,1,0; 2,1,1
  0,0,0; 0,0,1; 0,2,0; 0,2,1; 1,0,0; 1,0,1; 1,2,0; 1,2,1
  0,0,0; 0,0,1; 0,2,0; 0,2,1; 2,0,0; 2,0,1; 2,2,0; 2,2,1
  0,0,0; 0,0,2; 0,1,0; 0,1,2; 1,0,0; 1,0,2; 1,1,0; 1,1,2
  0,0,0; 0,0,2; 0,1,0; 0,1,2; 2,0,0; 2,0,2; 2,1,0; 2,1,2
  0,0,0; 0,0,2; 0,2,0; 0,2,2; 1,0,0; 1,0,2; 1,2,0; 1,2,2
  0,0,1; 0,0,2; 0,1,1; 0,1,2; 2,0,1; 2,0,2; 2,1,1; 2,1,2
  0,0,1; 0,0,2; 0,2,1; 0,2,2; 1,0,1; 1,0,2; 1,2,1; 1,2,2
  0,0,1; 0,0,2; 0,2,1; 0,2,2; 2,0,1; 2,0,2; 2,2,1; 2,2,2
  0,0,1; 0,1,0; 0,1,2; 0,2,1; 1,0,1; 1,1,0; 1,1,2; 1,2,1
  0,0,1; 0,1,0; 0,1,2; 0,2,1; 2,0,1; 2,1,0; 2,1,2; 2,2,1
  0,0,1; 0,1,1; 1,0,0; 1,0,2; 1,1,0; 1,1,2; 2,0,1; 2,1,1
  0,0,1; 0,2,1; 1,0,0; 1,0,2; 1,2,0; 1,2,2; 2,0,1; 2,2,1
  0,1,0; 0,1,1; 0,2,0; 0,2,1; 2,1,0; 2,1,1; 2,2,0; 2,2,1
  0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,2,0; 1,2,1; 2,1,0; 2,1,1
  0,1,0; 0,1,2; 0,2,0; 0,2,2; 1,1,0; 1,1,2; 1,2,0; 1,2,2
  0,1,0; 0,1,2; 0,2,0; 0,2,2; 2,1,0; 2,1,2; 2,2,0; 2,2,2
  0,1,0; 0,1,2; 1,0,0; 1,0,2; 1,2,0; 1,2,2; 2,1,0; 2,1,2
  0,1,1; 0,1,2; 0,2,1; 0,2,2; 2,1,1; 2,1,2; 2,2,1; 2,2,2
  0,1,1; 0,1,2; 1,0,1; 1,0,2; 1,2,1; 1,2,2; 2,1,1; 2,1,2
  0,1,1; 0,2,1; 1,1,0; 1,1,2; 1,2,0; 1,2,2; 2,1,1; 2,2,1
  1,0,0; 1,0,1; 1,2,0; 1,2,1; 2,0,0; 2,0,1; 2,2,0; 2,2,1
  0,1,1; 0,2,1; 1,1,0; 1,1,2; 1,2,0; 1,2,2; 2,1,1; 2,2,1
  1,0,0; 1,0,1; 1,2,0; 1,2,1; 2,0,0; 2,0,1; 2,2,0; 2,2,1
  1,0,0; 1,0,2; 1,1,0; 1,1,2; 2,0,0; 2,0,2; 2,1,0; 2,1,2
  1,0,0; 1,0,2; 1,2,0; 1,2,2; 2,0,0; 2,0,2; 2,2,0; 2,2,2
  1,0,1; 1,0,2; 1,2,1; 1,2,2; 2,0,1; 2,0,2; 2,2,1; 2,2,2
  1,0,1; 1,1,0; 1,1,2; 1,2,1; 2,0,1; 2,1,0; 2,1,2; 2,2,1
  1,1,0; 1,1,2; 1,2,0; 1,2,2; 2,1,0; 2,1,2; 2,2,0; 2,2,2
  0,0,0; 0,0,1; 0,1,0; 0,1,1; 1,0,0; 1,0,1; 1,1,0; 1,1,1
  0,0,1; 0,0,2; 0,1,1; 0,1,2; 1,0,1; 1,0,2; 1,1,1; 1,1,2
  0,1,0; 0,1,1; 0,2,0; 0,2,1; 1,1,0; 1,1,1; 1,2,0; 1,2,1
  0,1,1; 0,1,2; 0,2,1; 0,2,2; 1,1,1; 1,1,2; 1,2,1; 1,2,2
  1,0,0; 1,0,1; 1,1,0; 1,1,1; 2,0,0; 2,0,1; 2,1,0; 2,1,1
  1,0,1; 1,0,2; 1,1,1; 1,1,2; 2,0,1; 2,0,2; 2,1,1; 2,1,2
  1,1,0; 1,1,1; 1,2,0; 1,2,1; 2,1,0; 2,1,1; 2,2,0; 2,2,1
  1,1,1; 1,1,2; 1,2,1; 1,2,2; 2,1,1; 2,1,2; 2,2,1; 2,2,2

Cf. A098928.

A376012 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = any integer.

Original entry on oeis.org

1, 1, 3, 12, 83, 1323, 63090, 14736464

Offset: 0

Keith F. Lynch, Sep 05 2024

Number of ways any integer is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form (m+1)/m.

			For n = 2, a(2) = 3, three solutions, {x_1, x_2} = {2, 3} = 2; {1, 2} = 3; {1, 1} = 4.
In other words, a(2) = 3 since 2 can be written as (3/2)(4/3), 3 can be written as (2/1)(3/2), and 4 can be written as (2/1)^2, but no other integers are the product of two superparticular ratios.
a(3) = 12 since 2 can be written in 5 ways, 3 can be written in 3 ways, and 4, 5, 6, and 8 can be written in 1 way each, as the product of three superparticular ratios.

Cf. A085098, A118086, A277608.

f(n, m, p, q)={ \\ the number of solutions in which product of the ratios is equal to p/q
    if(p<=q, return(0));
    if(n==1, return(if(q%(p-q)==0, 1, 0)));
    x=floor(1/((p/q)^(1/n)-1)); \\ x is the maximum of the least denominator of the ratios
    my(ans=0);
    for(i=m, x, ans=ans+f(n-1, i, p*i, q*(i+1)));
    \\ m indicates that the solutions require the denominators of all ratios not to be less than m
    \\ discard the ratio (i+1)/i
    return(ans);
};
a(n)={
    my(ans=0);
    for(i=2, 2^n, ans=ans+f(n, 1, i, 1));
    return(ans);
}; \\ Yifan Xie, Nov 21 2024

A376011 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 3.

Original entry on oeis.org

0, 1, 3, 19, 276, 10341, 1526969

Offset: 1

Keith F. Lynch, Sep 05 2024

Number of ways 3 is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form (m+1)/m.

			a(3) = 3 since 3 can be written as (2/1)(4/3)(9/8), (2/1)(5/4)(6/5), or (3/2)^2 (4/3) but in no other way using superparticular ratios.

Cf. A085098, A118086, A277608.

A370453 Twin prime pair sums that equal a twin prime pair product plus 1 (divided by 36).

Original entry on oeis.org

36, 144, 1764, 5184, 360000, 412164, 777924, 4536900, 5673924, 7225344, 12659364, 12830724, 20684304, 37601424, 56972304, 64160100, 81757764, 179506404, 194100624, 255104784, 309689604, 366339600, 461906064, 689062500, 689692644, 1191078144, 1495368900, 1538835984

Offset: 1

Keith F. Lynch, Feb 18 2024

nonn

A twin prime pair (other than {3,5}) is always in the form {6m-1,6m+1}, so the product of the pair is always in the form 36*m^2-1 and a twin prime sum is always in the form 12m. As such, a twin prime sum can be one more than a twin prime product, but not vice versa, nor can a sum and product ever be equal.

{71,73} and {881,883} appear both as sums and as products.

			144 is a term because 71+73 = 144 and 11*13 = 143.
5184 is a term because 2591+2593 = 5184 and 71*73 = 5183.

Keith F. Lynch, Table of n, a(n) for n = 1..197

Subset of A037072.

Cf. A152787.

With[{p = Select[Prime[Range[4200]], PrimeQ[# + 2] &]}, Select[p*(p + 2) + 1, And @@ PrimeQ[#/2 + {-1, 1}] &]] (* Amiram Eldar, Feb 19 2024 *)

A369151 Numbers with a record high excess of even over odd divisors; so indices of record lows in A048272.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 48, 96, 144, 192, 240, 480, 720, 960, 1440, 2880, 3360, 5040, 6720, 10080, 20160, 30240, 40320, 60480, 80640, 100800, 110880, 181440, 201600, 221760, 332640, 443520, 665280, 887040, 1108800, 1330560, 1995840, 2217600, 2661120, 2882880, 4324320, 5765760, 8648640, 11531520, 14414400

Offset: 1

Keith F. Lynch, Jan 14 2024

nonn

Every term is the product of primorials, i.e., this is a subsequence of A025487, i.e., no prime factor of any term has a lower exponent than the following prime has.

			24 is a term because 24 has 6 even divisors, {2,4,6,8,12,24}, and 2 odd divisors, {1,3}, giving a difference of 4, more than that of any number less than 24.

Keith F. Lynch, Table of n, a(n) for n = 1..48

Cf. A048272.

If n > 2, a(n) = 2*A181808(n-2) = 4*A002182(n-2).

A359698 Least k > 0 such that the first n digits of 2^k and 3^k are identical.

Original entry on oeis.org

1, 17, 193, 619, 2016, 91958, 91958, 8186278, 45392361, 977982331, 26450915298, 91600221212, 196425900073, 14810317269038, 44430951807114, 626642721222487, 626642721222487, 102882886570917135, 874191214492184404, 3830977578643912683, 86801197487071715103

Offset: 0

Keith F. Lynch, May 20 2023

			   n    k = a(n)   1st n digits
  --  -----------  -------------
   0            1
   1           17  1...
   2          193  12...
   3          619  217...
   4         2016  7524...
   5        91958  13071...
   6        91958  130719...
   7      8186278  1701547...
   8     45392361  17179395...
   9    977982331  725070476...
  10  26450915298  2919267309...
a(3) = 619 because 2^619 = 2.175...*10^186 and 3^619 = 2.177...*10^295 both begin with the same three digits (in base ten), and this is not true of any smaller positive integer than 619.
a(0) = 1 because 2^1 and 3^1 share zero leading digits.

Zhao Hui Du, Table of n, a(n) for n = 0..1000

Cf. A000079, A000244, A088995.

a(11)-a(20) from Jon E. Schoenfield, May 21 2023

A362101 Numbers k such that 9^k starts with k.

Original entry on oeis.org

5, 69, 789, 2004, 1212215, 1766831, 437882194, 5217071661

Offset: 1

N. J. A. Sloane, Apr 10 2023, based on an email from Keith F. Lynch

For k such that b^k starts with n, for b = 2,..., 9, see A100129, A362096, A320930, A362097-A362101.

A362100 Numbers k such that 8^k starts with k.

Original entry on oeis.org

4, 10, 18, 652, 1299, 8225, 12949, 56230, 156277, 3227298, 144225157

Offset: 1

N. J. A. Sloane, Apr 10 2023, based on an email from Keith F. Lynch

For k such that b^k starts with n, for b = 2,..., 9, see A100129, A362096, A320930, A362097-A362101.

filter:= proc(n) local d1, d2, t;
  t:= 8^n;
  d1:= ilog10(t);
  d2:= ilog10(n);
  floor(t/10^(d1-d2)) = n
end proc:
select(filter, [$1..10^5]); # Robert Israel, Apr 10 2023

from itertools import count, islice
def A362100_gen(startvalue=1): # generator of terms >= startvalue
    a = 1<<3*(m:=max(startvalue,1))
    for n in count(m):
        if (s:=str(n))==str(a)[:len(s)]:
            yield n
        a <<= 3
A362100_list = list(islice(A362100_gen(),5)) # Chai Wah Wu, Apr 10 2023

A362099 Numbers k such that 7^k starts with k.

Original entry on oeis.org

3, 5560, 14350, 76972, 239123, 2067170, 19320438, 748459491, 1273027965, 6925699528, 7758433284, 8408679517

Offset: 1

N. J. A. Sloane, Apr 10 2023, based on an email from Keith F. Lynch

For k such that b^k starts with n, for b = 2,..., 9, see A100129, A362096, A320930, A362097-A362101.

A362098 Numbers k such that 6^k starts with k.

Original entry on oeis.org

13, 21, 197, 1991, 1022859, 1346570, 1351632, 15919007, 22440783, 415638567, 9345895862

Offset: 1

N. J. A. Sloane, Apr 10 2023, based on an email from Keith F. Lynch

For k such that b^k starts with n, for b = 2,..., 9, see A100129, A362096, A320930, A362097-A362101.

cp's OEIS Frontend

User: Keith F. Lynch

A385023 Number of cuboids (rectangular prisms) that can be formed from the points of Z^3 (a cubical grid of n X n X n points).

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A376012 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = any integer.

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PARI

A376011 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 3.

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A370453 Twin prime pair sums that equal a twin prime pair product plus 1 (divided by 36).

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Mathematica

A369151 Numbers with a record high excess of even over odd divisors; so indices of record lows in A048272.

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Formula

A359698 Least k > 0 such that the first n digits of 2^k and 3^k are identical.

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A362101 Numbers k such that 9^k starts with k.

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A362100 Numbers k such that 8^k starts with k.

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Maple

Python

A362099 Numbers k such that 7^k starts with k.

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A362098 Numbers k such that 6^k starts with k.

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