Sophia Lebiere - cp's OEIS frontend

A346199 a(n) is the number of permutations on [n] with at least one strong fixed point and no small descents.

1, 1, 1, 5, 19, 95, 569, 3957, 31455, 281435, 2799981, 30666153, 366646995, 4751669391, 66348304849, 992975080813, 15856445382119, 269096399032035, 4836375742967861, 91766664243841393, 1833100630242606203, 38452789552631651191, 845116020421125048153

Offset: 1

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

nonn

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

			For n = 4, the a(4) = 5 permutations on [4] with strong fixed points but no small descents: {(1*, 2*, 3*, 4*), (1*, 3, 4, 2), (1*, 4, 2, 3), (2, 3, 1, 4*), (3, 1, 2, 4*)} where * marks strong fixed points.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, On Properties of Pebble Assignment Graphs, Journal of Integer Sequences, 24(6), 2020.

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, A346204.

Python
```
See A346204.
```

a(n) = b(n-1) + Sum_{i=4..n} A346189(i-1)*b(n-i) where b(n) = A000255(n).

A346563 a(n) = n + A007978(n).

Original entry on oeis.org

3, 5, 5, 7, 7, 10, 9, 11, 11, 13, 13, 17, 15, 17, 17, 19, 19, 22, 21, 23, 23, 25, 25, 29, 27, 29, 29, 31, 31, 34, 33, 35, 35, 37, 37, 41, 39, 41, 41, 43, 43, 46, 45, 47, 47, 49, 49, 53, 51, 53, 53, 55, 55, 58, 57, 59, 59, 61, 61, 67, 63, 65, 65, 67, 67

Offset: 1

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 23 2021

Beginning at n=3, a(n) represents the maximum length of consecutive numbers that are divisible by the product of their nonzero digits in base n. In particular, if n=10, the sequence of numbers that are divisible by the product of their nonzero digits is given by A055471.

			For n=6, the least non-divisor of 6 is 4, so a(6) = 6+4 = 10. As seen in the Comments section, 55980, 55981, ..., 55989 form a sequence of length 10, where every number is divisible by the product of its nonzero digits in base n=6. Work has been done to show that 10 is the maximum length for such sequences.

Cf. A000027, A007978, A055471.

a[n_] := Module[{k = 1}, While[Divisible[n, k], k++]; n + k]; Array[a, 100] (* Amiram Eldar, Jul 23 2021 *)

a(n) = my(k=2); while(!(n % k), k++); n+k; \\ Michel Marcus, Jul 23 2021

goal = 100
these = []
n = 1
while n <= goal:
    k = 1
    while n % k == 0:
        k = k + 1
    these.append(n + k)
    n += 1
print(these)

a(2k+1) = 2k+3.

a(2k) >= 2k+3.

A346537 Squares that are divisible by the product of their nonzero digits.

Original entry on oeis.org

1, 4, 9, 36, 100, 144, 400, 900, 1024, 1296, 2304, 2500, 2916, 3600, 10000, 11664, 12100, 14400, 22500, 32400, 40000, 41616, 78400, 82944, 90000, 102400, 110224, 121104, 122500, 129600, 152100, 176400, 186624, 200704, 202500, 219024, 230400, 250000, 260100, 291600

Offset: 1

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 23 2021

			For the perfect square 1024 = 32^2 the product of its nonzero digits is 8 which divides 1024.

Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below.)
Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.

Intersection of A000290 and A055471.

Select[Range[500]^2, Divisible[#, Times @@ Select[IntegerDigits[#], #1 > 0 &]] &] (* Amiram Eldar, Jul 23 2021 *)

isok(m) = issquare(m) && !(m % vecprod(select(x->(x>0), digits(m))));
lista(nn) = for (m=1, nn, if (isok(m^2), print1(m^2, ", "))); \\ Michel Marcus, Jul 23 2021

from math import prod
def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
def ok(sqr): return sqr > 0 and sqr%nzpd(sqr) == 0
print(list(filter(ok, (i*i for i in range(541))))) # Michael S. Branicky, Jul 23 2021

A339999 Squares that are divisible by both the sum of their digits and the product of their nonzero digits.

Original entry on oeis.org

1, 4, 9, 36, 100, 144, 400, 900, 1296, 2304, 2916, 3600, 10000, 11664, 12100, 14400, 22500, 32400, 40000, 41616, 82944, 90000, 121104, 122500, 129600, 152100, 176400, 186624, 202500, 219024, 230400, 260100, 291600, 360000, 419904, 435600, 504100

Offset: 1

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 23 2021

			For the perfect square 144 = 12^2, the sum of its digits is 9, which divides 144, and the product of its nonzero digits is 16, which also divides 144 so 144 is a term of the sequence.

H. G. Grundman, Sequences of consecutive n-Niven numbers, Fibonacci Quarterly (1994) 32 (2): 174-175.
Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below.)
Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.

Intersection of A000290, A005349 and A055471.

Cf. A118547, A342262, A342650.

Select[Range[720]^2, And @@ Divisible[#, {Plus @@ (d = IntegerDigits[#]), Times @@ Select[d, #1 > 0 &]}] &] (* Amiram Eldar, Jul 23 2021 *)

from math import prod
def sumd(n): return sum(map(int, str(n)))
def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
def ok(sqr): return sqr > 0 and sqr%sumd(sqr) == 0 and sqr%nzpd(sqr) == 0
print(list(filter(ok, (i*i for i in range(1001)))))
# Michael S. Branicky, Jul 23 2021

A346198 a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.

Original entry on oeis.org

0, 1, 1, 8, 43, 283, 2126, 17947, 168461, 1741824, 19684171, 241506539, 3198239994, 45482655683, 691471698917, 11193266251700, 192238116358427, 3491633681792507, 66875708261486766, 1347168876070616179, 28474546456352896021, 630130731702950549248, 14570725407559756078387, 351411668456841530417027

Offset: 1

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

nonn

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

			For n = 4, the a(4) = 8 permutations on [4] with no strong fixed points but has small descents: {([2, 1], [4, 3]), (2, [4, 3], 1), ([3, 2], 4, 1), (3, 4, [2, 1]), (4, 1, [3, 2]), (4, [2, 1], 3), ([4, 3], 1, 2), (<4, 3, 2, 1>)} []small descent, <>consecutive small descents.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, On Properties of Pebble Assignment Graphs, Journal of Integer Sequences, 24(6), 2020.

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346199, A346204.

Python
```
See A346204.
```

For n > 2, a(n) = b(n)-c(n) where b(n) = A052186(n-1), c(n) = A346189(n).

A346657 Numbers that are not divisible by the product of their nonzero digits.

Original entry on oeis.org

13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 27 2021

			The product of the nonzero digits of 42 is 4*2 = 8, which does not divide 42.

Complement of A055471.

Select[Range[100], !Divisible[#, Times @@ Select[IntegerDigits[#], #1 > 0 &]] &] (* Amiram Eldar, Jul 27 2021 *)

isok(k) = k % vecprod(select(x->(x>0), digits(k))); \\ Michel Marcus, Jul 28 2021

from math import prod
def ok(n): return n > 0 and n%prod([int(d) for d in str(n) if d != '0'])
print(list(filter(ok, range(88)))) # Michael S. Branicky, Jul 27 2021

A346189 a(n) is the number of permutations on [n] with no strong fixed points or small descents.

Original entry on oeis.org

0, 0, 2, 6, 34, 214, 1550, 12730, 116874, 1187022, 13219550, 160233258, 2100360778, 29610224590, 446789311934, 7185155686666, 122690711149290, 2217055354281582, 42269657477711198, 847998698508705834, 17857221256001240458, 393839277313540073230, 9078806210245773668990, 218340709713567352161226

Offset: 1

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 09 2021

nonn

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

			For n = 4, the a(4) = 6 permutations on [4] with no strong fixed points or small descents: {(2,3,4,1),(3,4,1,2),(4,1,2,3),(3,1,4,2),(2,4,1,3),(4,2,3,1)}.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, On Properties of Pebble Assignment Graphs, Journal of Integer Sequences, 24(6), 2020.

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346198, A346199, A346204.

Python
```
See A346204.
```

For n > 3, a(n) = b(n) - b(n-1) - Sum{i=4..n}(a(i-1)*b(n-i)) where b(n) = A000255(n-1) and b(0) = 1.

A346204 a(n) is the number of permutations on [n] with at least one strong fixed point and at least one small descent.

Original entry on oeis.org

0, 0, 2, 5, 24, 128, 795, 5686, 46090, 418519, 4213098, 46595650, 561773033, 7333741536, 103065052300, 1551392868821, 24902155206164, 424588270621876, 7663358926666175, 145967769353476594, 2926073829112697318, 61577929208485406331, 1357369100658321844470, 31276096500003460511422

Offset: 1

Eugene Fiorini, Jared Glassband, Garrison Lee Koch, Sophia Lebiere, Xufei Liu, Evan Sabini, Nathan B. Shank, Andrew Woldar, Jul 10 2021

A small descent in a permutation p is a position i such that p(i)-p(i+1)=1.

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i) < k for i < k and p(j) > k for j > k.

			For n=4, the a(4)=5 permutations on [4] with strong fixed points and small descents: {(1*, 2*, [4, 3]), (1*, [3, 2], 4*), (1*, <4, 3, 2>), ([2, 1], 3*, 4*), (<3, 2, 1>, 4*)}. *strong fixed point, []small descent, <>consecutive small descents.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, Vol. 1, CRC Press, 2001.

M. Lind, E. Fiorini, A. Woldar, and W. H. T. Wong, On Properties of Pebble Assignment Graphs, Journal of Integer Sequences, 24(6), 2020.

Cf. A000255, A000166, A000153, A000261, A001909, A001910, A055790, A346189, A346198, A346199.

import math
bn = [1,1,1]
wn = [0,0,0]
kn = [1,1,1]
def summation(n):
    final = bn[n] - bn[n-1]
    for k in range(4,n+1):
        final -= wn[k-1]*bn[n-k]
    return final
def smallsum(n):
    final = bn[n-1]
    for k in range(4,n+1):
        final += wn[k-1]*bn[n-k]
    return final
def derrangement(n):
    finalsum = 0
    for i in range(n+1):
        if i%2 == 0:
            finalsum += math.factorial(n)*1//math.factorial(i)
        else:
            finalsum -= math.factorial(n)*1//math.factorial(i)
    if finalsum != 0:
        return finalsum
    else:
        return 1
def fixedpoint(n):
    finalsum = math.factorial(n-1)
    for i in range(2,n):
        finalsum += math.factorial(i-i)*math.factorial(n-i-1)
        print(math.factorial(i-i)*math.factorial(n-i-1))
    return finalsum
def no_cycles(n):
    goal = n
    cycles = [0, 1]
    current = 2
    while current<= goal:
        new = 0
        k = 1
        while k<=current:
            new += (math.factorial(k-1)-cycles[k-1])*(math.factorial(current-k))
            k+=1
        cycles.append(new)
        current+=1
    return cycles
def total_func(n):
    for i in range(3,n+1):
        bn.append(derrangement(i+1)//(i))
        kn.append(smallsum(i))
        wn.append(summation(i))
    an = no_cycles(n)
    tl = [int(an[i]-kn[i]) for i in range(n+1)]
    factorial = [math.factorial(x) for x in range(0,n+1)]
    print("A346189 :" + str(wn[1:]))
    print("A346198 :" + str([factorial[i]-wn[i]-tl[i]-kn[i] for i in range(n+1)][1:]))
    print("A346199 :" + str(kn[1:]))
    print("A346204 :" + str(tl[1:]))
total_func(20)

a(n) = A006932(n) - A346199(n).

cp's OEIS Frontend

User: Sophia Lebiere

A346199 a(n) is the number of permutations on [n] with at least one strong fixed point and no small descents.

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A346563 a(n) = n + A007978(n).

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A346537 Squares that are divisible by the product of their nonzero digits.

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A339999 Squares that are divisible by both the sum of their digits and the product of their nonzero digits.

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Mathematica

Python

A346198 a(n) is the number of permutations on [n] with no strong fixed points but contains at least one small descent.

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Python

Formula

A346657 Numbers that are not divisible by the product of their nonzero digits.

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Mathematica

PARI

Python

A346189 a(n) is the number of permutations on [n] with no strong fixed points or small descents.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Python