Hugo Pfoertner - cp's OEIS frontend

A387500 Median of twin prime pairs with n decimal digits.

5, 36, 420, 4644, 51060, 518238, 5206320, 52565169, 528113445, 5302625562

Offset: 1

Hugo Pfoertner, Sep 02 2025

It is assumed that a pair of twin primes is characterized by its mean. The median of an even number of values is taken as the arithmetic mean of the two central elements in their sorted list.

			a(1): 2 pairs {3,5}, {5,7}, median = mean of {4,6} = 5;
a(2): 6 pairs {11,13}, {17,19}, {29,31}, {41,43}, {59,61}, {71,73}, median = mean of {30,42} = 36.

Cf. A001097, A007508, A309329.

a[n_]:=Module[{s={},p=NextPrime[10^(n-1)]},While[p<10^n-4,If[PrimeQ[p+2],AppendTo[s,p+1]];p=NextPrime[p]];Median[s]];Array[a,7] (* James C. McMahon, Sep 02 2025 *)

A387495 Exponents k such that exp(k)/Pi is closer to an integer than for any smaller k.

Original entry on oeis.org

0, 1, 7, 22, 30, 50, 79, 103, 262, 993, 20819, 39397

Offset: 1

Hugo Pfoertner, Aug 31 2025

			a(1) = 0: exp(0)/Pi = 0.3183... = distance to nearest integer 0;
a(2) = 1: exp(1)/Pi = 0.8652559..., distance to nearest integer 1 = 0.134744...;
a(3) = 7: exp(7)/Pi = 349.0691758..., distance to nearest integer 349 = 0.0691758...;
a(4) = 22: exp(22)/Pi = 1141113200.0309559..., distance to nearest integer = 0.0309...

Cf. A061360, A079490, A080284.

A387146 Number of unlabeled biconnected cubic simple graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 5, 18, 81, 480, 3874, 39866, 497818, 7187627, 116349635

Offset: 0

Hugo Pfoertner, Aug 21 2025

Combos, Generate graphs.
Brendan McKay and Adolfo Piperno, nauty and Traces.

Cf. A002851, A005638, A058378.

A387144 a(n) is the least k>1 such that omega(k^n-1) = omega(k^n+1) with omega = A001221.

Original entry on oeis.org

3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3483, 3, 2, 2, 2343, 8, 2, 3, 420, 2, 23, 2, 65, 3, 15, 2, 724, 2, 33, 15

Offset: 1

Hugo Pfoertner, Aug 18 2025

a(32)...a(39): >1000, 2, 5, 8, >1000, 4, 8, 7.

Cf. A001221, A387145.

a387144(n) = for(k=2, oo, if(omega(k^n-1)==omega(k^n+1), return(k)))

A387145 a(n) is the least k>1 such that bigomega(k^n-1) = bigomega(k^n+1) with bigomega=A001222.

Original entry on oeis.org

4, 2, 6, 150, 3, 60, 4, 40, 19, 2, 2, 12450, 8, 2, 5, 4590, 5, 96, 5, 420, 2, 12, 2

Offset: 1

Hugo Pfoertner, Aug 18 2025

a(24)...a(39): >3300, 3, 18, 33, >1000, 2, 32, 3, >625, 6, 13, 8, >1000, 4, 32, 2.

Cf. A001222, A385315, A387144.

a387145(n) = for(k=2, oo, if(bigomega(k^n-1)==bigomega(k^n+1), return(k)))

A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jul 18 2025

Upper bound of the wobbling distance S of two rotated square lattices. See A307110 and A307731 for the special case of rotation angle Pi/4. According to Jan Fricke (1999), the angle Pi/4 is the most unfavorable case, i.e., smaller bounds can be found for all other angles.

			0.8557061686312838477748180718246837073...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Hans-Georg Carstens, Walter A. Deuber, Wolfgang Thumser, and Elke Koppenrade, Geometrical Bijections in Discrete Lattices. Combinatorics, Probability and Computing, 8(1-2), 109-129, 1999.
Jan Fricke, Symmetric Graphs have symmetric Matchings, arXiv:1607.07426 [math.GR], 25 Jul 2016, Introduction.
Klaus Nagel, A Lower Bound for Double Lattice Bijections, August 17, 2025.
Index entries for algebraic numbers, degree 4

Cf. A002163, A144981, A182168, A307110, A307731, A367150.

First[RealDigits[Sqrt[5]*Sin[Pi/8],10,100]] (* Paolo Xausa, Aug 13 2025 *)

sqrt(5)*sin(Pi/8) \\ Charles R Greathouse IV, Aug 19 2025

Equals A002163*A182168.

The minimal polynomial is 8*x^4 - 40*x^2 + 25. - Joerg Arndt, Aug 02 2025

A386242 a(n) is the least perfect power A001597 with binary weight n.

Original entry on oeis.org

1, 9, 25, 27, 121, 125, 1521, 2025, 5625, 24025, 42875, 59319, 32761, 393129, 851929, 1540081, 6275025, 15327225, 27258841, 41925625, 127893481, 243204025, 385611769, 268336125, 1979449081, 4823441401, 12870221809, 25698491351, 51354402813, 127506840561, 205822820329

Offset: 1

Hugo Pfoertner, Jul 23 2025

Cf. A000120, A001597, A231897.

upto = 10^11; L = Table[2 upto, {2 + Log2@ upto}]; Do[n = 1; While[(v = n^k) <= upto, nb = Plus @@ IntegerDigits[v, 2]; If[L[[nb]] > v, L[[nb]] = v]; n++], {k, 2, Log2[upto]}]; Take[L, Position[L, 2 upto][[1, 1]] - 1] (* Giovanni Resta, Jul 23 2025 *)

a(27)-a(31) from Giovanni Resta, Jul 23 2025

A386240 Primes containing the digit string "007" in their decimal representation.

Original entry on oeis.org

4007, 6007, 9007, 10007, 10079, 12007, 13007, 16007, 20071, 24007, 30071, 36007, 45007, 50077, 60077, 61007, 64007, 70079, 78007, 80071, 80077, 82007, 88007, 90007, 90071, 90073, 94007, 97007, 100703, 100733, 100741, 100747, 100769, 100787, 100799, 103007, 108007

Offset: 1

Hugo Pfoertner, Jul 16 2025

Could also be called "James Bond" primes.

Alois P. Heinz, Table of n, a(n) for n = 1..36628
Karl-Heinz Hofmann, Ascii search picture. Who can find the first 5 terms?
Q-branch staff, The origin of James Bond: unveiling the history behind 007, 21 February 2024.
Reddit, We never saw Daniel Craig's Bond in his prime
Joe Roberts, How James Bond Became 007 (& How Its Meaning Has Changed), Jul 7, 2020.
Joe Roberts, How Did James Bond Get The Codename 007? A Canceled Movie Almost Gave Us The Answer, Feb 24, 2025.

Cf. A386247, A166580, A167282, A193552.

q:= n-> isprime(n) and searchtext("007", ""||n)>0:
select(q, [$1..110000])[];  # Alois P. Heinz, Jul 17 2025

Select[Prime[Range[553, 12000]], StringContainsQ[IntegerString[#], "007"] &] (* Paolo Xausa, Jul 17 2025 *)

is(n) = if(!isprime(n), return(0)); while(n > 10, if(n%1000==7, return(1)); n\=10); 0 \\ David A. Corneth, Jul 17 2025

from sympy import sieve
print([sieve[n] for n in range(1,10200) if "007" in str(sieve[n])]) # Karl-Heinz Hofmann, Jul 17 2025

A385755 Numbers k with a unique combination of bigomega(k) and sopfr(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 26, 29, 30, 31, 32, 34, 35, 36, 37, 38, 41, 43, 46, 47, 48, 53, 58, 59, 61, 62, 64, 67, 70, 71, 72, 73, 74, 79, 82, 83, 86, 89, 94, 96, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 128, 131

Offset: 1

Hugo Pfoertner, Jul 09 2025

nonn

			All primes p are in the sequence, because they are characterized by the pair [b,s] = [bigomega=1, sopfr=p], and no other numbers have this pair.
All even semiprimes 2*p are terms, because no other number can have [b,s]=[2,p+2]. p+2 is odd, and odd semiprimes p*q would have even s.
20 with [b,s]=[3,2+2+5] and 27 with [b,s]=[3,3+3+3] are not in the sequence, because both have [b,s]=[3,9].
21 and 25 are not in the sequence, because both have [b,s]=[2,10].
36 is in the sequence as it is the only number having [4, 10]. - _David A. Corneth_, Jul 11 2025
From _Michael De Vlieger_, Jul 13 2025: (Start)
Plot a(n) at (x,y) = (A001222(a(n)), A001414(a(n))):
     0    1    2    3    4    5    6    7     8     9
-----------------------------------------------------
 0:  1
 1:
 2:       2
 3:       3
 4:            4
 5:       5    6
 6:            9    8
 7:       7   10   12
 8:           15   18   16
 9:           14        24
10:                30   36   32
11:      11                  48
12:           35             72   64
13:      13   22                  96
14:                70            144  128
15:           26                      192
16:                                   288   256
17:      17                                 384
18:                                         576   512
19:      19   34                                  768
         ...  (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Mathematica code.
Michael De Vlieger, Plot a(n) at (x,y) = (A001222(a(n)), A001414(a(n))) for x <= 16 and y <= 33.

Cf. A001222, A001414, A385756, A385811.

Subsequences: A000040, A000079, A100484.

A385756 Complement of A385755.

Original entry on oeis.org

20, 21, 25, 27, 28, 33, 39, 40, 42, 44, 45, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 65, 66, 68, 69, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 90, 91, 92, 93, 95, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 117, 119, 120, 121, 123, 124, 125, 126, 129

Offset: 1

Hugo Pfoertner, Jul 09 2025

nonn

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

Cf. A001222, A001414, A385755, A385811.

cp's OEIS Frontend

User: Hugo Pfoertner

A387500 Median of twin prime pairs with n decimal digits.

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A387495 Exponents k such that exp(k)/Pi is closer to an integer than for any smaller k.

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A387146 Number of unlabeled biconnected cubic simple graphs with 2n nodes.

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A387144 a(n) is the least k>1 such that omega(k^n-1) = omega(k^n+1) with omega = A001221.

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A387145 a(n) is the least k>1 such that bigomega(k^n-1) = bigomega(k^n+1) with bigomega=A001222.

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A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

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A386242 a(n) is the least perfect power A001597 with binary weight n.

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A386240 Primes containing the digit string "007" in their decimal representation.

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A385755 Numbers k with a unique combination of bigomega(k) and sopfr(k).

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A385756 Complement of A385755.

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