A038507 -id:A038507 - cp's OEIS frontend

A277085 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 90 degrees.

1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 4, 6, 10, 14, 20, 26, 31, 36, 40, 44, 44, 44, 40, 36, 31, 26, 20, 14, 10, 6, 4, 2, 1, 1, 2, 4, 6, 34, 62, 116, 170, 547, 924, 1624, 2324, 5572, 8820, 14616, 20412, 40509, 60606, 95004, 129402, 224406, 319410

Offset: 0

Christian Bean, Sep 28 2016

A permutation, p, can be thought of as a set of points (i, p(i)). If you plot all the points and rotate the picture by 90 degrees then you get a permutation back.

T(n,k) is the number of size k subsets that remain unchanged by a rotation of 90 degrees.

			For n = 4 and k = 2, the subsets unchanged by a 90-degree rotation are {4321,1234}, {4231,1324}, {3412,2143} and {3142,2413}. Hence T(4,2) = 4.
Triangle starts:
1, 1;
1, 1;
1, 0, 1;
1, 0, 1, 0, 1, 0, 1;

Row lengths give A038507.

Cf. A037223, A037224.

T(n,k) = Sum_( C( R(n) - T(n), i ) * Sum_(C(n! - R(n), j) * C(T(n), k - 4*i - 2*j) for j in [0..floor((k-4*i)/2)] for i in [0..floor(k/4)] ) where R(n) = A037223(n) and T(n) = A037224(n).

A304935 a(n) is the largest possible integer value for sqrt(0 1 2 ... n), where one is allowed to place any mixture of +'s and *'s in the n blank spaces.

Original entry on oeis.org

1, 0, 0, 5, 11, 6, 71, 19, 123, 33, 174, 426, 174, 233, 625, 816, 5695, 3656, 15936, 246960, 24234, 24234, 35151, 140604, 177399, 250982, 1304130, 1304130, 1304130, 1304130, 5532955, 5532955, 58136459, 8525544, 8525544, 58136459, 941988492, 58136459, 941988492

Offset: 1

Andy Niedermaier, May 21 2018

nonn

Inspired by a test ARML problem from 2018, which asked students to determine a(8).

			a(2) = a(3) = 0, since no positive squares are achievable.
Some examples:
a(7) = 71 = sqrt(0+1+2*3*4*5*6*7).
a(8) = 19 = sqrt(0+1*2+3+4*5+6*7*8).
a(20) = 246960 = sqrt(0+1*2*3*4*5*6*7*8*9*10*11+12*13*14*15*16*17*18*19*20)

Rémy Sigrist, PARI program for A304935

Upper-bounded by sqrt(A038507).

sqStrTest[n_] := Module[{bVal, bStr, i, j, iB, mVal, mStr},
  bVal = -1;
  For[i = 0, i < 2^n, i++,
   iB = IntegerDigits[i, 2];
   While[Length[iB] < n, PrependTo[iB, 0]];
   mStr = "0";
   For[j = 1, j <= n, j++,
    mStr = StringJoin[mStr, If[iB[[j]] == 0, "+", "*"], ToString[j]]];
   mVal = ToExpression[mStr];
   If[Sqrt[mVal] == Floor[Sqrt[mVal]],
    If[mVal > bVal, {bVal, bStr} = {mVal, mStr}]
    ]
   ];
  Print[{Sqrt[bVal], bVal, bStr}]]

PARI
```
See Links section.
```

More terms from Rémy Sigrist, May 22 2018

A358279 a(n) = Sum_{d|n} (d-1)! * d^(n/d).

Original entry on oeis.org

1, 3, 7, 29, 121, 747, 5041, 40433, 362935, 3629433, 39916801, 479006531, 6227020801, 87178326609, 1307674371487, 20922790212353, 355687428096001, 6402373709021811, 121645100408832001, 2432902008212950169, 51090942171709691335, 1124000727778046766849

Offset: 1

Seiichi Manyama, Nov 08 2022

Cf. A038507, A062363, A078308, A217576, A321521, A358280.

a[n_] := DivisorSum[n, (# - 1)! * #^(n/#) &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d-1)!*d^(n/d));
```

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*x^k/(1-k*x^k)))

G.f.: Sum_{k>0} k! * x^k/(1 - k * x^k).

If p is prime, a(p) = 1 + p!.

A358593 a(n) = n! * Sum_{d|n} d^(n-d) / d!^(n/d).

Original entry on oeis.org

1, 3, 7, 49, 121, 2701, 5041, 219521, 1587601, 33446701, 39916801, 17731796545, 6227020801, 2879710009177, 98069239768501, 2418218838097921, 355687428096001, 2832293713653708877, 121645100408832001, 2295597943489176040001, 71029619657111138063041

Offset: 1

Seiichi Manyama, Feb 23 2023

nonn

Cf. A038507, A061095, A354891, A356668.

a[n_] := n! * DivisorSum[n, #^(n-#) / #!^(n/#) &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)

a(n) = n!*sumdiv(n, d, d^(n-d)/d!^(n/d));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!-(k*x)^k))))

E.g.f.: Sum_{k>0} x^k / (k! - (k * x)^k).

If p is prime, a(p) = 1 + p! = A038507(p).

A180325 a(n) = k is the smallest number such that n is the number of distinct primes dividing k! + 1.

Original entry on oeis.org

1, 6, 9, 31, 16, 18, 40, 89

Offset: 1

Michel Lagneau, Jan 18 2011

A greedy inverse to A066856. - R. J. Mathar, Jan 21 2011

			a(6) = 18 because the 6 distinct primes dividing 18! + 1 = 6402373705728001
  are {19, 23, 29, 61, 67, 123610951}.

Cf. A038507.

with(numtheory):for n from 1 to 7 do:ind:=0:for k from 1 to 50 while(ind=0)
  do: x:=k!+1:y:=nops(factorset(x)):if y=n then ind:=1:printf(`%d, `,k):else fi:od:
  od:

A264890 Integers k such that k! + 1 is the sum of 2 nonzero squares.

Original entry on oeis.org

0, 1, 4, 8, 11, 12, 17, 25, 26, 27, 28, 29, 37, 38, 41, 45, 48, 54, 60, 67, 71, 73, 75, 77, 88, 92, 94, 114, 115, 116, 119, 133

Offset: 1

Altug Alkan, Nov 27 2015

			a(3) = 4 because 4! + 1 = 4^2 + 3^2.

Cf. A000404, A038507, A264665, A271186.

Flatten@ Position[Map[Length, Map[Map[Length, PowersRepresentations[#, 2, 2] &@(#! + 1) /. 0 -> Nothing] &, Range[0, 48]] /. 1 -> Nothing], n_ /; n > 0] - 1 (* Michael De Vlieger, Nov 28 2015 *)

is(n) = { for(i=1, #n=factor(n!+1)~%4, n[1, i]==3 && n[2, i]%2 && return); n && (vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2)) }

a(25)-a(32) from Jinyuan Wang, May 22 2021

A277086 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n with respect to the symmetries of the square.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 5, 5, 2, 1, 1, 7, 56, 317, 1524, 5733, 17728, 44767, 94427, 166786, 249624, 316950, 343424, 316950, 249624, 166786, 94427, 44767, 17728, 5733, 1524, 317, 56, 7, 1, 1, 23, 1012, 36125, 1035496, 23878229, 456936220, 7437730463

Offset: 0

Christian Bean, Sep 28 2016

A permutation, p, can be thought of as a set of points (i, p(i)). In this viewpoint it is natural to consider the symmetries of the square.

T(n,k) is the number of symmetry classes of subsets of size k from S_n.

			Triangle starts:
1, 1;
1, 1;
1, 1, 1;
1, 2, 5, 5, 5, 2, 1;

Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.

Rows lengths give A038507.

Cf. A277080, A277081, A277083, A277085.

T(n,k) = 1/8 * (C(n,k) + 2*A277080(n,k) + 2*A277081(n,k) + 2*A277085(n,k) + A277083(n,k)).

A344488 Numbers that start a product crescendo of record length.

Original entry on oeis.org

1, 2, 3, 7, 47, 181, 1307, 2503, 40973, 46833, 109177, 2885373, 11744311, 192968969, 899988745

Offset: 1

Jon Wild, May 20 2021

A product crescendo is a sequence of successive integers that can be written as products j * k where the j's form a strictly increasing sequence and the k's form a strictly decreasing sequence.
From Jon E. Schoenfield, May 22 2021: (Start)
a(16) <= 13399626241.
Numbers that start long product crescendos, but are not necessarily of record length, are easy to find by testing numbers of the form 1 + m*lcm(1..k) for sufficiently large m and k. E.g., the ones that start at 13399626241 = 1 + 18592*lcm(1..16), 442452890881 = 1 + 36112*lcm(1..17), and 521688126961 = 1 + 2241*lcm(1..19) have lengths 37, 39, and 41 respectively. (End)
The sequence is infinite as for any n >= 0, A038507(n) starts a product crescendo of length >= n. - Rémy Sigrist, May 22 2021

			181 is in the list because it begins a product crescendo that is longer than any beginning at any smaller number. Here is the crescendo:
    1 * 181  =  181
    2 *  91  =  182
    3 *  61  =  183
    4 *  46  =  184
    5 *  37  =  185
    6 *  31  =  186
   11 *  17  =  187
   47 *   4  =  188
   63 *   3  =  189
   95 *   2  =  190
  191 *   1  =  191
This set of 11 products forms a longer crescendo than the previous record (which started at 47), and is the longest until the set of 13 products it is possible to write starting from 1307 (the next entry in the sequence).
Additional example: the crescendo from 2885373 (length 27) goes:
        1 * 2885373 = 2885373
        2 * 1442687 = 2885374
        5 * 577075  = 2885375
        6 * 480896  = 2885376
       11 * 262307  = 2885377
       19 * 151862  = 2885378
       21 * 137399  = 2885379
       89 * 32420   = 2885380
      859 * 3359    = 2885381
     1458 * 1979    = 2885382
     4817 * 599     = 2885383
    12437 * 232     = 2885384
    19365 * 149     = 2885385
    33551 * 86      = 2885386
    93077 * 31      = 2885387
   131154 * 22      = 2885388
   221953 * 13      = 2885389
   288539 * 10      = 2885390
   320599 * 9       = 2885391
   360674 * 8       = 2885392
   412199 * 7       = 2885393
   480899 * 6       = 2885394
   577079 * 5       = 2885395
   721349 * 4       = 2885396
   961799 * 3       = 2885397
  1442699 * 2       = 2885398
  2885399 * 1       = 2885399

Cf. A038507.

b(n)={if(n==1, 1, my(m=1); for(k=1, oo, fordiv(n+k, d, if(d>m, m=d; break)); if(m==n+k, return(k+1))))}
lista(lim)={my(m=0); for(n=1, lim, my(t=b(n)); if(t > m, print1(n, ", "); m=t))} \\ Andrew Howroyd, May 21 2021

a(13)-a(14) from Rémy Sigrist, May 21 2021

a(15) from Jon E. Schoenfield, May 21 2021

A357296 Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).

Original entry on oeis.org

1, 3, 7, 31, 121, 851, 5041, 43261, 369601, 3748249, 39916801, 490801081, 6227020801, 87861842641, 1310800947457, 21018206008801, 355687428096001, 6419518510204801, 121645100408832001, 2435836129700029057, 51102829650622464001, 1124549558817839481601

Offset: 1

Seiichi Manyama, Feb 23 2023

nonn

Cf. A038507, A057625, A327578, A354891.

a[n_] := n! * DivisorSum[n, 1/(#^(n/#-1) * #!) &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k/k)))))

a(n) = n!*sumdiv(n, d, 1/(d^(n/d-1)*d!));

a(n) = n! * Sum_{d|n} 1 / (d^(n/d-1) * d!).

If p is prime, a(p) = 1 + p! = A038507(p).

A358178 a(n) is the cardinality of the set of distinct pairwise gcd's of {1! + 1, ..., n! + 1}.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 11, 12, 12, 12, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset: 1

Gleb Ivanov, Nov 02 2022

nonn

			For n = 6 initial set is {1+1, 2+1, 6+1, 24+1, 120+1, 720+1} and after applying gcd for each distinct pair of elements we get {1, 7} set with cardinality of a(6) = 2.

Cf. A038507, A358127, A356371, A214799.

from math import gcd, factorial
from itertools import combinations
f, terms = [2,], []
for i in range(2,100):
    f.append(factorial(i)+1)
    terms.append(len(set([gcd(*t) for t in combinations(f, 2)])))
print(terms)

from math import gcd
from itertools import count, islice
def A358178_gen(): # generator of terms
    m, f, g = 1, [], set()
    for n in count(1):
        m *= n
        g |= set(gcd(d,m+1) for d in f)
        f.append(m+1)
        yield len(g)
A358178_list = list(islice(A358178_gen(),20)) # Chai Wah Wu, Dec 15 2022

cp's OEIS Frontend

A277085 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 90 degrees.

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A304935 a(n) is the largest possible integer value for sqrt(0 1 2 ... n), where one is allowed to place any mixture of +'s and *'s in the n blank spaces.

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A358279 a(n) = Sum_{d|n} (d-1)! * d^(n/d).

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Mathematica

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PARI

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A358593 a(n) = n! * Sum_{d|n} d^(n-d) / d!^(n/d).

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Mathematica

PARI

PARI

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A180325 a(n) = k is the smallest number such that n is the number of distinct primes dividing k! + 1.

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Maple

A264890 Integers k such that k! + 1 is the sum of 2 nonzero squares.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A277086 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n with respect to the symmetries of the square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A344488 Numbers that start a product crescendo of record length.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A357296 Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).