Onno M. Cain - cp's OEIS frontend

User: Onno M. Cain

Onno M. Cain has authored 5 sequences.

A330007 Base-24 integers whose substrings are primes (written in base 10).

2, 3, 5, 7, 11, 13, 17, 19, 23, 53, 59, 61, 67, 71, 79, 83, 89, 127, 131, 137, 139, 173, 179, 181, 191, 269, 271, 277, 281, 283, 317, 331, 419, 421, 431, 461, 463, 467, 479, 557, 563, 569, 571, 1279, 1283, 1289, 1291, 1423, 1429, 1433, 1483, 1613, 1619, 1709, 1721, 1723, 1901, 1907, 1997, 1999, 2011, 3061, 3163, 3299

Offset: 1

Onno M. Cain, Nov 26 2019

The largest such number is a(103)=266003, which is written (19|5|19|11)_24 in base 24.

We might call these numbers "substrimes" (= substring-primes) since the term (1) is concise, (2) is pronounceable, and (3) keeps these numbers distinct in communication from different but similar sequences (see Crossrefs).

			a(64) = 3299 = (5|17|11)_24 is in the sequence because 5, 17, 11, (5|17)_24=137, (17|11)=419, and 3299 itself are all prime.

Onno M. Cain, Python Code

Cf. A024773, A085823, A085822, A213300.

With[{b = 24}, Select[Range[b^4], Function[{s, n}, AllTrue[Flatten@ Array[FromDigits[#, b] & /@ Partition[s, #, 1] &, n], PrimeQ]] @@ {#, Length@ #} &@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 15 2019 *)

Python
```
# see Cain link
```

A329526 Number of 1's required to build n using +, -, *, ^ and factorial.

Original entry on oeis.org

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

Offset: 1

Onno M. Cain, Nov 15 2019

nonn

A number n may be written from the digits of its binary representation using the 5 aforementioned operations whenever a(n) <= floor(log_2(n))+1.

			a(117) = 7 since 117 = ((1+1+1)!-1)!-1-1-1 is the representation of 117 with the operations +,-,*,^ and factorial requiring the fewest 1's.

O. M. Cain, The Exceptional Selfcondensability of Powers of Five, arXiv:1910.13829 [Math.HO], 2019.

Cf. A025280, A306560, A323727, A091334.

import math
delta_bound = 10
limit = 8*2**delta_bound
log_limit = math.log(limit)
def factorial(n):
  if n <= 1: return 1
  return n * factorial(n-1)
def condensings(a, b):
  result = []
  # Addition
  if a+b < limit: result.append(a+b)
  # Subtraction
  if a > b: result.append(a-b)
  # Multiplication
  if a*b < limit: result.append(a*b)
  # Division
  if a % b == 0: result.append(a//b)
  # Exponentiation
  if b*math.log(a) < log_limit: result.append(a**b)
  for n in result:
    if 2 < n < 10:
      fac = factorial(n)
      if fac < limit: result.append(fac)
  return result
# Create delta bound.
# "delta(n) = k" means k 1's are required to build up
# n from the operations in the 'condensings' function.
delta = dict()
delta[1] = 1
# Create inverse map.
delta_inv = {i:[] for i in range(1, delta_bound)}
for a in delta: delta_inv[delta[a]].append(a)
# Calculate delta.
for D in range(2, delta_bound):
  for A in range(1, D):
    B = D - A
    for a in delta_inv[A]:
      for b in delta_inv[B]:
        for c in condensings(a, b):
          if c >= limit: continue
          if c not in delta:
            delta[c] = D
            delta_inv[D].append(c)
a = 1
while a < limit:
  if not a in delta: break
  print(a, delta[a])
  a += 1

A325480 a(n) is the largest integer m such that the product of n consecutive integers starting at m is divisible by at most n primes.

Original entry on oeis.org

16, 24, 24, 45, 48, 49, 120, 120, 125, 189, 240, 240, 350, 350, 350, 350, 374, 494, 494, 714, 714, 714, 714, 825, 832, 1078, 1078, 1078, 1078, 1425, 1440, 1440, 1856, 2175, 2175, 2175, 2175, 2175, 2175, 2175, 2870, 2870, 2870, 2871, 2880, 2880, 2880, 3219

Offset: 3

Onno M. Cain, Sep 06 2019

nonn

Each term is only conjectured and has been verified up to 10^6.

Note a(2) is undefined if there are infinitely many Mersenne primes.

			For example, a(3) = 16 because 16 * 17 * 18 = 2^5 * 3^2 * 17 admits only three prime divisors (2, 3, and 17) and appears to be the largest product of three consecutive integers with the property.

Cf. A002182, A045619, A163264, A164799, A239035, A244656, A006549.

for r in range(3, 100):
  history = []
  M = 0
  for n in range(1, 100000):
    primes = {p for p, _ in factor(n)}
    history.append(primes)
    history = history[-r:]
    total = set()
    for s in history: total |= s
    # Skip if too many primes.
    if len(total) > r: continue
    if n > M: M = n
  print(r, M-r+1)

A309230 Positions of records in A033178.

Original entry on oeis.org

2, 5, 13, 25, 37, 41, 61, 85, 113, 181, 361, 421, 433, 613, 793, 1009, 1121, 1261, 2053, 2161, 3421, 4001, 5441, 6481, 7141, 7561, 8033, 9361

Offset: 1

Onno M. Cain, Jul 16 2019

Terms appear to have relatively few prime factors compared to their neighbors. a(28)=9361=11*23*37 is the first term with 3 factors.

Onno M. Cain, Bioperational Multisets in Various Semi-rings, arXiv:1908.03235 [math.RA], 2019.

Cf. A033178, A033179, A104173.

A308838 Orders of Parker finite fields of odd characteristic.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 31, 43, 47, 67, 243

Offset: 1

Onno M. Cain, Jun 27 2019

A field or ring is called "Parker" if no 3 X 3 magic square of 9 distinct squared elements can be formed. Conjecture: the sequence is complete.
Example: the fact that p=31 is listed is taken to mean one cannot construct a 3 X 3 magic square of distinct squared elements of the finite field of order 31.
Cain shows that each of the entries on the list corresponds to a Parker field and claims to have checked computationally that no other primes p < 1000 are on the list.
Labruna shows at least there are infinitely many primes not on the list.
The next term, if it exists, must be > 7500. - G. C. Greubel, Aug 16 2019

			Example: The prime p=29 does not appear in the sequence because one can in fact construct a 3 X 3 magic square of distinct squares over the finite field of order 29.
Construction:
   9^2 | 11^2 |  1^2
   6^2 |  0^2 | 14^2
  12^2 | 16^2 |  8^2
The square is valid evaluated mod 29 (example independently discovered by Woll and Cain). That is to say the entries of each row, column, and the two main diagonals sum to a multiple of 29.
Example: The fields corresponding to p^n = 3, 5, 7, 9, 11, and 13 are all Parker because each contains at most 7 distinct squared entries and cannot therefore provide the 9 distinct squares required for a magic square.

Onno M. Cain, Gaussian Integers, Rings, Finite Fields and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Christian Woll, A Partial Residue Categorization of the Magic Square of Squares, arXiv:1809.03067 [math.NT], 2018.
Giancarlo Labruna, Magic Squares of Squares of Order Three Over Finite Fields, (2018). Theses, Dissertations and Culminating Projects. 138.
Matt Parker & Brady Haran, The Parker Square, Numberphile video (2016).

Cf. A309810, A348263, A364264.

def msos_search(F, single=False):
    squares = {x^2 for x in F}
    MSOS = []
    E = 0
    for A, I in Subsets(squares, 2):
        if A + I != 2*E: continue
        C, G = 1, -1
        B = 3*E - A - C
        D = 3*E - A - G
        F = 3*E - C - I
        H = 3*E - G - I
        if len(squares & {B,D,F,H}) < 4: continue
        if len({A,B,C,D,E,F,G,H,I}) < 9: continue
        if single: return [A,B,C,D,E,F,G,H,I]
        MSOS.append([A,B,C,D,E,F,G,H,I])
    E = 1
    sequences = []
    for A, I in Subsets(squares, 2):
        if A + I != 2*E: continue
        for C, G in sequences:
            B = 3*E - A - C
            D = 3*E - A - G
            F = 3*E - C - I
            H = 3*E - G - I
            if len(squares & {B,D,F,H}) < 4: continue
            if len({A,B,C,D,E,F,G,H,I}) < 9: continue
            if single: return [A,B,C,D,E,F,G,H,I]
            MSOS.append([A,B,C,D,E,F,G,H,I])
        sequences.append((A,I))
    return MSOS
for q in range(3, 500, 2):
    if len(factor(q)) > 1: continue
    print(q, msos_search(GF(q), single=True))

cp's OEIS Frontend

User: Onno M. Cain

A330007 Base-24 integers whose substrings are primes (written in base 10).

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A329526 Number of 1's required to build n using +, -, *, ^ and factorial.

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A325480 a(n) is the largest integer m such that the product of n consecutive integers starting at m is divisible by at most n primes.

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SageMath

A309230 Positions of records in A033178.

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A308838 Orders of Parker finite fields of odd characteristic.

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Sage