A007489 -id:A007489 - cp's OEIS frontend

A325616 Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 2, 1

Offset: 0

Gus Wiseman, May 12 2019

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 1 1
  0 0 1 1 1
  0 0 0 1 1 1
  0 1 0 1 1 1 1
  0 0 1 0 1 1 1 1
  0 0 1 1 1 1 1 1 1
  0 0 0 1 1 1 1 1 1 1
  0 0 0 1 1 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 1 1 1 1
  0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 3 2 2 2 2 2 1 1 1 1 1
Row n = 12 counts the following partitions:
  (66)
  (6222)
  (62211)
  (222222) (621111)
  (2222211) (6111111)
  (22221111)
  (222111111)
  (2211111111)
  (21111111111)
  (111111111111)

Row sums are A064986.
Cf. A008284.
Factorial numbers: A000142, A007489, A076934, A108731, A115944, A227157, A284605, A322583, A325509, A325617.
Reciprocal factorial sum: A325618, A325619, A325620, A325622.

Table[SeriesCoefficient[Product[1/(1-y*x^(i!)),{i,1,n}],{x,0,n},{y,0,k}],{n,0,15},{k,0,n}]

T(n,k) is the coefficient of x^n * y^k in the expansion of Product_{i > 0} 1/(1 - y * x^(i!)).

A275732 One-based positions of 1-digits in the factorial base representation of n are converted to primes with those indices, then multiplied together.

Original entry on oeis.org

1, 2, 3, 6, 1, 2, 5, 10, 15, 30, 5, 10, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 7, 14, 21, 42, 7, 14, 35, 70, 105, 210, 35, 70, 7, 14, 21, 42, 7, 14, 7, 14, 21, 42, 7, 14, 1, 2, 3, 6, 1, 2, 5, 10, 15, 30, 5, 10, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 5, 10, 15, 30, 5, 10, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, 2, 5, 10, 15, 30

Offset: 0

Antti Karttunen, Aug 08 2016

All terms are squarefree (A005117), and each squarefree number occurs an infinitely many times.

			22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 1, as the empty product is 1.
35 has factorial base representation "1121" (= A007623(35)). 1's occur in the following positions, when counted from right, starting with 1: 1, 3 and 4. Thus a(35) = prime(1)*prime(3)*prime(4) = 2*5*7 = 70.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A000040, A001221, A001222, A002110, A005117, A007623, A007489, A048675, A257261, A257511, A275730.
Cf. A255411 (indices of ones).
Can be used to compute A275733 and A275734.
Cf. also to A275736.

nn = 105; m = 1; While[Factorial@ m < nn, m++]; m; Map[Times @@ Map[Prime, Flatten@ Position[#, 1]] &@ Reverse@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)

from operator import mul
from sympy import prime
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 19 2017

;; Recursive definition using memoizing definec-macro:
(definec (A275732 n) (cond ((zero? (A257261 n)) 1) (else (* (A000040 (A257261 n)) (A275732 (A275730bi n (- (A257261 n) 1)))))))
(define (A275732 n) (let loop ((z 1) (n n)) (let ((y (A257261 n))) (cond ((zero? y) z) (else (loop (* z (A000040 y)) (A275730bi n (- y 1))))))))
;; Code for A275730bi given in A275730.

If A257261(n) = 0, then a(n) = 1, otherwise a(n) = A000040(A257261(n)) * a(A275730(n, A257261(n)-1)). [Here A275730(n,p) is a bivariate function that "clears" the digit at zero-based position p in the factorial base representation of n].
Other identities and observations. For all n >= 0:
a(A007489(n)) = A002110(n).
a(A255411(n)) = 1.
A001221(a(n)) = A001222(a(n)) = A257511(n).
A048675(a(n)) = A275736(n).

A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42

Offset: 0

Antti Karttunen, Aug 08 2016

a(n) = product of primes whose indices are positions of nonzero-digits in factorial base representation of n (see A007623). Here positions are one-based, so that the least significant digit is the position 1, the next least significant the position 2, etc.

			For n=19, A007623(19) = 301, thus a(19) = prime(3)*prime(1) = 5*2 = 10.
For n=52, A007623(52) = 2020, thus a(52) = prime(2)*prime(4) = 3*7 = 21.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A001221, A002110, A003961, A005117, A007489, A007623, A048675, A060130, A061395, A084558, A257684, A275732.
Subsequence of A005117.
Cf. A275727.
Cf. also A275725, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(A007489(n)) = A002110(n).
A001221(a(n)) = A001222(a(n)) = A060130(n).
A048675(a(n)) = A275727(n).
A061395(a(n)) = A084558(n).

A219664 Repeating part of A220664: First differences of the numbers given as concatenation of permutations of (1,...,m) for sufficiently large m.

Original entry on oeis.org

9, 81, 18, 81, 9, 702, 9, 171, 27, 72, 18, 693, 18, 72, 27, 171, 9, 702, 9, 81, 18, 81, 9, 5913, 9, 81, 18, 81, 9, 1602, 9, 261, 36, 63, 27, 594, 18, 162, 36, 162, 18, 603, 9, 171, 27, 72, 18, 5814, 9, 171, 27, 72, 18, 603, 9, 261, 36, 63, 27, 1584, 27, 63, 36

Offset: 1

Antti Karttunen, Dec 18 2012

First 5!-1 terms are identical to A107346, and the 9!-1 terms are identical to A209280. (Updated by M. F. Hasler, Jan 12 2013, Mar 03 2013)
Because of the self-similar nature of A220664, this sequence can be also constructed by picking appropriate terms from there with the auxiliary sequence A220655, cf. formula.
Similarly, differences between successive permutations of {1,2,...,k} in lexicographic order interpreted as decimal numbers, for any k=2..9, produce the first (k!)-1 terms of this sequence. But for k=10 the result is ill-defined, so we can consider the sequence finite, well-defined only for n=1..362879. [See however the following comment. - Editor's note]
In sequence A030299 it is clearly defined how it extends beyond index n = 1!+2!+...+9! = A007489(9), so the sequence A220664 of its first differences is well-defined up to infinity. (The "result" mentioned above is ill defined because the meaning of "interpreted" is not clear.) But the preceding comment is misleading by speaking of "self similar nature", and the sequence definition as "repeating part" is also misleading: If the sequence is defined to be of finite length 9!-1 (thus equal to A209280), then it is indeed infinitely often repeated as a subsequence (of consecutive terms) in A220664 (even when the latter was defined using concatenation for permutations of more than 9 elements, but then not as differences of the terms following 12345678910 where it was expected, but, e.g., as differences of the terms following 10123456789, etc.).
Since A030299 has been defined through a ("simpler") sum rather than concatenation, the nice mathematical properties of A220664, and even more this sequence A219664, persist beyond n=9!. - M. F. Hasler, Jan 12 2013

			The first four permutations of nine elements at A030299(A003422(9)..A003422(9)+3) (the terms A030299(46234..46237)) are: 123456789, 123456798, 123456879, 123456897. As 123456897-123456879 = 18, thus we have a(3) = 18.
We could compute the same value from any smaller set of permutations of at least three elements, for example, from the five element permutations used in A107346. In that case, the permutations A030299(A003422(5)..A003422(5)+3) (the terms A030299(34..37)) are: 12345, 12354, 12435, 12453, ... and we get the same result, a(3) = 12453-12435 = 18.

A. Karttunen, Table of n, a(n) for n = 1..5039

Cf. A107346, A220664, A033029.

A219664(n)=for(k=2,n+1, k!>n || next; k=vecsort( vector( (#k=vector(k,j,10^j)~\10)!,i,numtoperm(#k,i)*k )); return(k[n+1]-k[n]))  \\ (It is of course more efficient to calculate a whole vector of the first k!-1 terms. Also, for n>9!, this might yield incorrect terms.) - M. F. Hasler, Jan 12 2013

(define (A219664 n) (A220664 (A220655 n)))

a(n) = A220664(A220655(n)).

a(n) = 9*A217626(n).

A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = duu! + ... + d22! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.

Original entry on oeis.org

1, 3, 5, 9, 11, 21, 23, 33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119, 153, 155, 165, 167, 177, 179, 189, 191, 201, 203, 213, 215, 225, 227, 237, 239, 633, 635, 645, 647, 657, 659, 669, 671, 681, 683, 693, 695, 705, 707, 717, 719, 873, 875

Offset: 1

Antti Karttunen, Nov 12 2013

A001088(n+1) gives the number of terms x in sequence for which A084558(x)=n.

Because totatives (the reduced residue set) of each natural number k form a multiplicative group of integers modulo same k, it means that taking e.g. inverses of each digit modulo same k or multiplying them (again modulo k) by some member of that set keeps the set closed, and thus applying these operations to each digit modulo i+1 (2 for the least significant digit in factorial base, 3 for the next, and so on) yield only digits allowed in this sequence, and thus they induce various permutations of this sequence. These can be further "normalized" to be permutations of natural numbers with a suitable ranking function, which is to be submitted later.

			This can be viewed as an irregular table, where row n has A001088(n+1) elements, starts from position A231721(n) and ends at position A231722(n+1):
1;
3, 5;
9, 11, 21, 23;
33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119;
...

Antti Karttunen, Rows 1..7, flattened
Wikipedia, Totative

Positions of ones in A231715.
The first term of each row: A007489(n) = a(A231721(n)).
The last term of each row: A033312(n+1) = a(A231722(n+1)).
Subset of A227157.
Cf. also A007623, A000010, A001088, A084558.

A289945 a(n) = Sum_{k=1..n} k!^4.

Original entry on oeis.org

1, 17, 1313, 333089, 207693089, 268946253089, 645510228813089, 2643553803594573089, 17342764866576345933089, 173418555892594089945933089, 2538940579958951120707545933089, 52646414799433780559063261145933089

Offset: 1

Eric W. Weisstein, Jul 16 2017

nonn

The only prime in this sequence is a(2) = 17 since a(n) is divisible by 13 for n >= 12 and there are no other primes with n < 12.

Seiichi Manyama, Table of n, a(n) for n = 1..144
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A007489 (k!), A104344 (k!^2), A289946 (k!^6).

Table[Sum[k!^4, {k, n}], {n, 12}]
Accumulate[(Range[15]!)^4] (* Harvey P. Dale, Jul 12 2019 *)

a(n) = sum(k=1, n, k!^4); \\ Michel Marcus, Jul 16 2017

A289946 a(n) = Sum_{k=1..n} k!^6.

Original entry on oeis.org

1, 65, 46721, 191149697, 2986175149697, 139317055679149697, 16390300280131775149697, 4296598745804900241599149697, 2283384320190476620685217983149697, 2283382306976051006261597069217983149697

Offset: 1

Eric W. Weisstein, Jul 16 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..104
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018. [This thesis cites this sequence entry, but it's just a typo: the intended sequence entry is A289546.]
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A007489 (k!), A104344 (k!^2), A289945 (k!^4).

Cf. A289947 (indices giving primes).

Table[Sum[k!^6, {k, n}], {n, 10}]
Accumulate[(Range[10]!)^6] (* Harvey P. Dale, May 14 2023 *)

a(n) = sum(k=1, n, k!^6); \\ Michel Marcus, Jul 16 2017

A336496 Products of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 34560, 36864, 41472, 49152, 55296

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A317804 in having 34560, which is the first term with more than two distinct prime factors.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}

A001013 is the version for factorials, with complement A093373.
A181818 is the version for superprimorials, with complement A336426.
A336497 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178.
A174605 is the maximum prime multiplicity in A000178.
A303279 counts prime factors of superfactorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[1000],facsusing[Rest[Array[supfac,30]],#]!={}&]

A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

Original entry on oeis.org

1, 1, 2, 0, 28, 0, 768, 0, 0, 0, 42155360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

Support appears to be {0, 1, 2, 4, 6, 10}.

			The a(4) = 28 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/8/2/1   24/12/4/2/1
      24/3   24/4/1   24/8/4/1
      24/4   24/4/2   24/8/4/2
      24/8   24/8/1   24/12/2/1
      24/12  24/8/2   24/12/3/1
             24/8/4   24/12/4/1
             24/12/1  24/12/4/2
             24/12/2
             24/12/3
             24/12/4

A336867 is the complement of the support.
A336868 is the characteristic function (image under A057427).
A336942 is half the version for superprimorials (n > 1).
A337071 does not require distinct prime multiplicities.
A337104 is the case of chains ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336571 counts chains of divisors 1 < d < n using A130091.
Cf. A000110, A001055, A002033, A007489, A022559, A048656, A071626, A098859, A124010, A167865, A325617, A336416, A336424.

chnsc[n_]:=If[!UnsameQ@@Last/@FactorInteger[n],{},If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]]];
Table[Length[chnsc[n!]],{n,0,6}]

a(n) = 2*A337104(n) = 2*A336423(n!) for n > 1.

A325619 Heinz numbers of integer partitions whose reciprocal factorial sum is 1.

Original entry on oeis.org

2, 9, 375, 15625

Offset: 1

Gus Wiseman, May 13 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      9: {2,2}
    375: {2,3,3,3}
  15625: {3,3,3,3,3,3}

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A051908, A316855, A325618, A325624.

Select[Range[100000],Total[Cases[FactorInteger[#],{p_,k_}:>k/PrimePi[p]!]]==1&]

Contains prime(n)^(n!) for all n > 0, including 191581231380566414401 for n = 4.

cp's OEIS Frontend

A325616 Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

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A275732 One-based positions of 1-digits in the factorial base representation of n are converted to primes with those indices, then multiplied together.

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A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

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A219664 Repeating part of A220664: First differences of the numbers given as concatenation of permutations of (1,...,m) for sufficiently large m.

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A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = du*u! + ... + d2*2! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.

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A289945 a(n) = Sum_{k=1..n} k!^4.

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A289946 a(n) = Sum_{k=1..n} k!^6.

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A336496 Products of superfactorials (A000178).

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A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = duu! + ... + d22! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.