Nicolas Bělohoubek - cp's OEIS frontend

A374233 Irregular triangle read by rows where row n lists the primes containing at least one digit not seen in any smaller prime in base n, for n >= 2.

2, 2, 3, 2, 3, 5, 17, 2, 3, 5, 19, 2, 3, 5, 7, 29, 37, 2, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 37, 53, 67, 2, 3, 5, 7, 11, 13, 17, 59, 83, 2, 3, 5, 7, 11, 19, 41, 61, 83, 101, 2, 3, 5, 7, 11, 17, 19, 31, 37, 43, 2, 3, 5, 7, 11, 13, 53, 73, 97, 109, 127, 149

Offset: 2

Nicolas Bělohoubek, Jul 01 2024

			Row n=4 is 2, 3, 5, 17, which is 2_4, 3_4, 11_4, 101_4.
First few rows:
       k=1  2  3   4   5   6   7   8
  n=2:  [2],
  n=3:  [2, 3],
  n=4:  [2, 3, 5, 17],
  n=5:  [2, 3, 5, 19],
  n=6:  [2, 3, 5,  7, 29, 37],
  n=7:  [2, 3, 5,  7, 11, 13],
  n=8:  [2, 3, 5,  7, 11, 37, 53, 67],
  ...

Cf. A033274.

isok(d, digs) = for (i=1, #d, if (!vecsearch(digs, d[i]), return(1)));
row(n) = my(digs=List(), v=List()); forprime(p=2, , my(d = digits(p, n)); if (isok(d, Vec(digs)), listput(v, p); for (i=1, #d, listput(digs, d[i])); listsort(digs, 1); if (#digs == n, return(Vec(v))););); \\ Michel Marcus, Jul 02 2024

from sympy.ntheory import digits, nextprime
def row(n):
    if n == 2: return [2]
    p, r, used = 2, [2], {2}
    while len(used) < n:
        while (ds:=set(digits(p:=nextprime(p), n)[1:])) <= used: pass
        r.append(p)
        used |= ds
    return r
print([an for b in range(2, 13) for an in row(b)]) # Michael S. Branicky, Jul 01 2024

A372855 Number of ways two dihexes can be placed on an n-th regular hexagonal board.

Original entry on oeis.org

0, 33, 702, 3630, 11409, 27603, 56748, 104352, 176895, 281829, 427578, 623538, 880077, 1208535, 1621224, 2131428, 2753403, 3502377, 4394550, 5447094, 6678153, 8106843, 9753252, 11638440, 13784439, 16214253, 18951858, 22022202, 25451205, 29265759, 33493728

Offset: 1

Nicolas Bělohoubek, May 15 2024

			Regular hexagonal boards n = 1...4:
. ___
./   \
.\___/
.     ___
. ___/   \___
./   \___/   \
.\___/   \___/
./   \___/   \
.\___/   \___/
.    \___/
.         ___
.     ___/   \___
. ___/   \___/   \___
./   \___/   \___/   \
.\___/   \___/   \___/
./   \___/   \___/   \
.\___/   \___/   \___/
./   \___/   \___/   \
.\___/   \___/   \___/
.    \___/   \___/
.        \___/
.             ___
.         ___/   \___
.     ___/   \___/   \___
. ___/   \___/   \___/   \___
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
./   \___/   \___/   \___/   \
.\___/   \___/   \___/   \___/
.    \___/   \___/   \___/
.        \___/   \___/
.            \___/
For n = 2 the a(2) = 33: (without grid)
. . . . . . . . . . . . . . . . . . .
.   x---x   .   x---x   .   x---x   .
.           .           .           .
. x---x   o . o   x---x . o   o   o .
.           .           .           .
.   o   o   .   o   o   .   x---x   .
. . . . . . . . . . . . . . . . . . .
.   x---x   .   x---x   .   x---x   .
.           .           .           .
. x   o   o . o   x   o . o   x   o .
.  \        .      \    .    /      .
.   x   o   .   o   x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x---x   .   o   o   .   o   x   .
.           .           .        \  .
. o   o   x . x---x   o . x---x   x .
.        /  .           .           .
.   o   x   .   x---x   .   o   o   .
. . . . . . . . . . . . . . . . . . .
.   o   o   .   o   o   .   o   o   .
.           .           .           .
. x---x   x . o   x---x . x   x---x .
.        /  .           .  \        .
.   o   x   .   x---x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   o   .   o   x   .
.  /        .    \      .        \  .
. x   x---x . o   x   o . o   o   x .
.           .           .           .
.   o   o   .   x---x   .   x---x   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   o   x   .   x   x   .
.  /        .      /    .    \   \  .
. x   o   o . o   x   o . o   x   x .
.           .           .           .
.   x---x   .   x---x   .   o   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   o   .   o   x   .
.    \      .    \      .        \  .
. x   x   o . o   x   x . x   o   x .
.  \        .        /  .  \        .
.   x   o   .   o   x   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   o   x   .   x   x   .   o   x   .
.        \  .  /     \  .        \  .
. o   x   x . x   o   x . o   x   x .
.      \    .           .    /      .
.   o   x   .   o   o   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   o   o   .   o   x   .   o   o   .
.           .      /    .           .
. x   x   o . x   x   o . x   o   x .
.  \   \    .  \        .  \     /  .
.   x   x   .   x   o   .   x   x   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   x   x   .   x   o   .
.  /        .  /   /    .  /        .
. x   x   o . x   x   o . x   x   o .
.      \    .           .    /      .
.   o   x   .   o   o   .   x   o   .
. . . . . . . . . . . . . . . . . . .
.   x   o   .   o   x   .   o   o   .
.  /        .      /    .           .
. x   o   x . o   x   x . o   x   x .
.        /  .        /  .    /   /  .
.   o   x   .   o   x   .   x   x   .
. . . . . . . . . . . . . . . . . . .

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000384, A242856.

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 33, 702, 3630, 11409, 27603}, 50] (* Paolo Xausa, Aug 28 2024 *)

a(n) = (3/2)*(27*n^4 - 90*n^3 + 78*n^2 + 11*n - 24), for n > 1.
a(n) = 5*a(n - 1) - 10*a(n - 2) + 10*a(n - 3) - 5*a(n - 4) + a(n - 5) for n > 6.
G.f.: 3*x^2*(11 + 179*x + 150*x^2 - 17*x^3 + x^4)/(1 - x)^5.
E.g.f.: 36 - 3*x + 3*exp(x)*(27*x^4 + 72*x^3 - 3*x^2 + 26*x - 24)/2. - Stefano Spezia, Jun 04 2024

A370488 a(n) is the smallest integer k such that the average deviation of previous terms and k is an integer, where a(n) > a(n - 1) and a(1) = 1.

Original entry on oeis.org

1, 3, 11, 13, 57, 65, 95, 99, 124, 132, 159, 165, 246, 265, 335, 342, 353, 397, 406, 422, 426, 482, 876, 1018, 1383, 1641, 1689, 1731, 2376, 2433, 3149, 3202, 3294, 3822, 4068, 4077, 4192, 4274, 4554, 4575, 4712, 5368, 6283, 6303, 7997, 8226, 9815, 10696, 12273, 12325, 12764, 12868

Offset: 1

Nicolas Bělohoubek, Feb 19 2024

nonn

Nicolas Bělohoubek, Table of n, a(n) for n = 1..1000

avdev(v)={my(n=#v,sav=vecsum(v)/n);sum(k=1,n,abs(v[k]-sav))/n};
a370488(nterms) = {my(v=vector(nterms)); v[1]=1; for (k=2, nterms, for (j=v[k-1]+1, oo, v[k]=j; if (denominator(avdev(v[1..k])) == 1, break))); v};
a370488(52) \\ Hugo Pfoertner, Feb 20 2024

Conjecture: a(n)/n^3 tends to c, where 0.08 < c < 0.1.

A368062 Numbers k such that k = A257850(k) + A257297(k).

Original entry on oeis.org

0, 36, 655, 1258, 6208, 12508, 45715, 65455, 75385, 125008, 235297, 1250008, 2857144, 3214288, 4210528, 6545455, 6792453, 12500008, 34615386, 47058824, 87671233, 125000008, 654545455, 1250000008, 9529411765, 12500000008, 39130434783, 45714285715, 65454545455, 75384615385

Offset: 1

Nicolas Bělohoubek, Dec 10 2023

			     0 = 0*0   +   0*0;
    36 = 3*6   +   3*6;
   655 = 6*55  +  65*5;
  6208 = 6*208 + 620*8;
  ...

Kevin Ryde, Table of n, a(n) for n = 1..700
Nicolas Bělohoubek, C# program
Nicolas Bělohoubek, Subsequences of form A-(B)-C
Kevin Ryde, PARI/GP Code

Cf. A257850, A257297.

Select[Range[0,10^6], Part[digits=IntegerDigits[#],1]FromDigits[Drop[digits,1]] + FromDigits[Drop[digits,-1]]Part[digits,Length[digits]] == # &] (* Stefano Spezia, Dec 10 2023 *)

PARI
```
\\ See links.
```

def ok(n):
    if n < 10: return n == 0
    s = str(n)
    return n == int(s[0])*int(s[1:]) + (n%10)*(n//10)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Dec 10 2023

# faster for generating initial segment of sequence
from itertools import count, islice
def agen(): # generator of terms
    yield 0
    for digits in count(2):
        for first in range(1, 10):
            base = first*10**(digits-1)
            for rest in range(10**(digits-1)):
                n = base + rest
                if first*rest + (n%10)*(n//10) == n:
                    yield n
            print("...", digits, first, time()-time0, alst)
print(list(islice(agen(), 18))) # Michael S. Branicky, Dec 10 2023

a(24)-a(30) from Michael S. Branicky, Dec 10 2023

A364133 Index k of A007814(A000127(k)) at record terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 1034, 1619, 19940, 151012, 185354, 937444, 17714660, 30594058, 53467077, 401540691, 1127208901, 34761279059, 1529978475530, 12645510928325

Offset: 0

Nicolas Bělohoubek, Jul 10 2023

For polynomial sequences like A000127 we can always find a recurrence formula, and with that you can show that all polynomial sequences mod m will be periodic. Terms of this sequence were found using A000127(k) mod 2^x, with a recursion formula, treating each k separately. This method limits the size of working quantities.

This sequence is related to the question: Is there a another power of 2 among Moser's circle numbers?

Burkard Polster, Why don't they teach Newton's calculus of 'What comes next?', YouTube video, Oct 02 2021.
Kevin Ryde, PARI/GP Code
Grant Sanderson, The absurd circle division pattern (updated) | Moser's circle problem, YouTube video, Jul 02 2023.

Cf. A000127, A007814.

A355148 Numbers that are the concatenation of two palindromes and that have exactly two palindromic factors, all with the same number of decimal digits.

Original entry on oeis.org

12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 81, 3388, 7744, 101787, 101808, 111888, 151848, 212565, 212898, 232656, 313464, 313575, 353868, 383595, 383838, 414585, 434676, 454545, 505808, 515595, 525252, 555888

Offset: 1

Nicolas Bělohoubek, Jun 21 2022

All numbers of form (4/45)*(9*100^d - 29*10^d + 20) are terms (see example).
Also numbers of form (7/18)*(100^d - 13*10^d + 12) and are also terms (d>1) and of form (4/45)*(4*100^d - 19*10^d + 15) (d>1).
From Chai Wah Wu, Aug 23 2022: (Start)
Terms with 2 decompositions:
  12 = 3*4 = 2*6
  16 = 4*4 = 2*8
  18 = 2*9 = 3*6
  24 = 4*6 = 3*8
  36 = 4*9 = 6*6
  153535351846464648 = 189828981*808808808 = 172727271*888888888
  182919281817080718 = 303303303*603090306 = 201030102*909909909
  183838381816161618 = 303060303*606606606 = 202040202*909909909
  185676581814323418 = 306090603*606606606 = 204060402*909909909
  192919291807080708 = 303303303*636060636 = 212020212*909909909
  193838391806161608 = 303303303*639090936 = 213030312*909909909
  283919382716080617 = 312030213*909909909 = 303303303*936090639
  293656392403040304 = 461262164*636636636 = 363363363*808161808
  293919392706080607 = 323020323*909909909 = 303303303*969060969
  365838563634161436 = 603090306*606606606 = 402060204*909909909
  385838583614161416 = 606606606*636060636 = 424040424*909909909
  387676783612323216 = 606606606*639090936 = 426060624*909909909
  567838765432161234 = 624060426*909909909 = 606606606*936090639
  587838785412161214 = 646040646*909909909 = 606606606*969060969
Conjecture: these are an infinite number of such terms.
The following term has 3 decompositions:
  113131311886868688 = 279747972*404404404 = 254545452*444444444 = 252252252*448484844.
(End)
A subsequence of this sequence is {s(k)} where s(k) = (202/10989)*t(k)*u(k), t(k) = 10^(6*k + 3) - 1 and u(k) = 2099*10^(6*k + 1) + 988. s(k) can be decomposed in 2 different ways: the first is (202/333)*t(k) and (1/33)*u(k); the second is (101/111)*t(k) and (2/99)*u(k). And since {s(k)} is an infinite sequence, its existence proves Chai Wah Wu's conjecture to be true. - Nicolas Bělohoubek, May 20 2024

			42 is the concatenation of 4 and 2, and is also 6*7 (all 1 digit).
3388 is the concatenation of 33 and 88, and is also 44*77 (all 2 digits).
414585 is the concatenation of 414 and 585, and is also 555*747 (all 3 digits).
131080 = 232*565 is not a term since 080 begins with 0 and hence is not a three-digit palindromic number.
79974224 = 8998*8888, 7999742224 = 89998*88888, 799997422224 = 899998*888888 (see comments).

Michael S. Branicky, Table of n, a(n) for n = 1..597
Nicolas Bělohoubek, Table of n, a(n) for n = 1..56

from sympy import divisors
from itertools import count, islice, product
def ispal(s): return s == s[::-1]
def pals(d, start0=False): # generates palindromic strings with d digits
    digits = "0123456789"
    if d == 1: yield from "0"*int(start0) + "123456789"; return
    for p in product(digits, repeat=d//2):
        if not start0 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield left + mid + right
def agen(): # generator of terms
    for d in count(1):
        found = set()
        for p1 in pals(d):
            for p2 in pals(d):
                p = int(p1)*int(p2)
                s = str(p)
                if len(s) != 2*d: continue
                if ispal(s[:d]) and s[d] != "0" and ispal(s[d:]):
                    found.add(p)
        yield from sorted(found)
print(list(islice(agen(), 51))) # Michael S. Branicky, Jun 21 2022

A352844 Smallest k > 1 such that sopfr(k) - tau(k) = n, or -1 if no such k exists.

Original entry on oeis.org

2, 3, 8, 5, 15, 7, 21, 25, 35, 11, 33, 13, 39, 117, 65, 17, 51, 19, 57, 121, 95, 23, 69, 169, 115, 483, 161, 29, 87, 31, 93, 279, 155, 651, 217, 37, 111, 333, 185, 41, 123, 43, 129, 387, 215, 47, 141, 423, 235, 954, 318, 53, 159, 477, 265, 841, 354, 59, 177, 61

Offset: 0

Nicolas Bělohoubek, Apr 05 2022

nonn

Conjecture: There is no -1 in this sequence.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 516 terms from Nicolas Bělohoubek)

Cf. A000005, A001414.

f(m) = my(fp=factor(m)); sum(k=1, #fp~, fp[k,1]*fp[k,2]) - numdiv(fp);
a(n) = my(k=2); while(f(k) != n, k++); k; \\ Michel Marcus, Apr 06 2022

It appears that the sequence satisfies these rules, for large m:
Rule 1: a(prime(m+1) - 2) = prime(m+1)
Rule 2: a(prime(m+1) - 1) = 3*prime(m+1)
Rule 3: a(prime(m+1) + 1) = 5*prime(m+1)
Rule 4: a(prime(m+1) - 3) = 6*prime(m+1)
Rule 5: a(prime(m+1) + 3) = 7*prime(m+1)
Rule 6: a(prime(m+1)) = 9*prime(m+1)
Rule 7: a(prime(m+1) - 7) = 11*prime(m+1)
Rule 8: a(prime(m+1) + 7) = 12*prime(m+1)
Rule 9: a(prime(m+1) + 9) = 13*prime(m+1)
...
Choose the first rule that applies.

A352742 a(n) is the smallest number > 1 that is not divisible by 10 but is divisible by the n-th power of the sum of its digits.

Original entry on oeis.org

2, 81, 512, 2401, 11101212, 34012224, 612220032, 20047612231936, 3904305912313344, 7800803212802061312, 1025300207121086650406, 213780015477322248820322, 14076019706120526112710656, 2670419511272061205254504361, 2759031540715333904109053133443, 10530400808911150200350000010411

Offset: 1

Nicolas Bělohoubek, Mar 31 2022

a(n+1) >= a(n).
When A072408(n) is not multiple of 10 then a(n) <= A072408(n).
a(n) = m * k^n where m is a positive integer and k is the sum of digits of a(n).
Conjecture: No term is a multiple of 5.
a(28) = 265^28, disproving the above conjecture. - Charles R Greathouse IV, Apr 02 2022

			For n=5, 11101212 is not divisible by 10 but is divisible by the 5th power of the sum of its digits, that being (1+1+1+0+1+2+1+2)^5 = 9^5. There is no smaller such number.

Cf. A072408.

a(7)-a(8) confirmed by Jon E. Schoenfield, Mar 31 2022

a(9)-a(16) from Charles R Greathouse IV, Apr 02 2022

A348166 a(n) = abs(A020338(n)-A338754(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 0, 90, 180, 270, 360, 450, 540, 630, 720, 180, 90, 0, 90, 180, 270, 360, 450, 540, 630, 270, 180, 90, 0, 90, 180, 270, 360, 450, 540, 360, 270, 180, 90, 0, 90, 180, 270, 360, 450, 450, 360, 270, 180, 90, 0, 90, 180, 270, 360, 540, 450, 360, 270, 180, 90, 0, 90, 180, 270, 630, 540, 450, 360, 270, 180, 90

Offset: 1

Nicolas Bělohoubek, Oct 04 2021

All terms are multiples of 90.

			a(1) = abs(11-11) = 0
a(15) = abs(1515-1155) = 360
a(1965) = abs(19651965-11996655) = 7655310

Cf. A020338, A338754, A010785.

a(n)=abs(eval(Str(n, n))-fromdigits(digits(n), 100)*11) \\ Charles R Greathouse IV, Oct 04 2021

a(A010785(n)) = 0

Corrected by Charles R Greathouse IV, Oct 04 2021

A344679 Number of 2-matchings of the n-th centered square grid graph.

Original entry on oeis.org

0, 0, 86, 544, 1854, 4688, 9910, 18576, 31934, 51424, 78678, 115520, 163966, 226224, 304694, 401968, 520830, 664256, 835414, 1037664, 1274558, 1549840, 1867446, 2231504, 2646334, 3116448, 3646550, 4241536, 4906494, 5646704, 6467638, 7374960, 8374526, 9472384, 10674774

Offset: 1

Nicolas Bělohoubek, Aug 17 2021

Number of ways two dominoes can be placed on an "other" Aztec Diamonds chessboard.

			For n=1 there is no way to place 2 dominoes in the centered square grid graphs, because they don't have enough space to be placed, so a(1)=0.
For n=2 there is no way to place 2 dominoes in the centered square grid graphs, because the first domino will cover the center square every time, so a(2)=0.

Nicolas Bělohoubek, Visualization of 3rd term.
Ron Knott, 1.2.5 The "other" Aztec Diamonds
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844, A242856.

a(n) = 2*(n-2)*(4n^3-8n^2+n+4) for n > 1.
From Stefano Spezia, Aug 17 2021: (Start)
G.f.: 2*x^3*(43 + 57*x - 3*x^2 - x^3)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 6. (End)

cp's OEIS Frontend

User: Nicolas Bělohoubek

A374233 Irregular triangle read by rows where row n lists the primes containing at least one digit not seen in any smaller prime in base n, for n >= 2.

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A372855 Number of ways two dihexes can be placed on an n-th regular hexagonal board.

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A370488 a(n) is the smallest integer k such that the average deviation of previous terms and k is an integer, where a(n) > a(n - 1) and a(1) = 1.

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PARI

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A368062 Numbers k such that k = A257850(k) + A257297(k).

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Python

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A364133 Index k of A007814(A000127(k)) at record terms.

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A355148 Numbers that are the concatenation of two palindromes and that have exactly two palindromic factors, all with the same number of decimal digits.

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Python

A352844 Smallest k > 1 such that sopfr(k) - tau(k) = n, or -1 if no such k exists.

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PARI

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A352742 a(n) is the smallest number > 1 that is not divisible by 10 but is divisible by the n-th power of the sum of its digits.

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A348166 a(n) = abs(A020338(n)-A338754(n)).

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