A212598 -id:A212598 - cp's OEIS frontend

A084558 a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.

0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5

Offset: 0

Antti Karttunen, Jun 23 2003

For n >= 1, a(n) = the number of significant digits in n's factorial base representation (A007623).
After zero, which occurs once, each n occurs A001563(n) times.
Number of iterations (...(f_4(f_3(f_2(n))))...) such that the result is < 1, where f_j(x):=x/j. - Hieronymus Fischer, Apr 30 2012
For n > 0: a(n) = length of row n in table A108731. - Reinhard Zumkeller, Jan 05 2014

			a(4) = 2 because 2! <= 4 < 3!.

F. Smarandache, "f-Inferior and f-Superior Functions - Generalization of Floor Functions", Arizona State University, Special Collections.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Yi Yuan and Zhang Wenpeng, On the Mean Value of the Analogue of Smarandache Function, Smarandache Notions J., Vol. 15.

A dual to A090529.
Cf. A084555, A084556, A084557.
Cf. A001069, A010096.
Cf. A000142, A001563, A055089, A060130, A111095, A211664, A211670, A108731, A212598, A220656, A220657, A220658, A220659, A231716, A235224, A257510.

a084558 n = a090529 (n + 1) - 1  -- Reinhard Zumkeller, Jan 05 2014

0, seq(m$(m*m!),m=1..5); # Robert Israel, Apr 27 2015

Table[m = 1; While[m! <= n, m++]; m - 1, {n, 0, 104}] (* Jayanta Basu, May 24 2013 *)
Table[Floor[Last[Reduce[x! == n && x > 0, x]]], {n, 120}] (* Eric W. Weisstein, Sep 13 2024 *)

a(n)={my(m=0);while(n\=m++,);m-1} \\ R. J. Cano, Apr 09 2018

def A084558(n):
  i=1
  while n: i+=1; n//=i
  return(i-1)
print(list(map(A084558,range(101)))) # Natalia L. Skirrow, May 28 2023

From Hieronymus Fischer, Apr 30 2012: (Start)
a(n!) = a((n-1)!)+1, for n>1.
G.f.: 1/(1-x)*Sum_{k>=1} x^(k!).
The explicit first terms of the g.f. are: (x+x^2+x^6+x^24+x^120+x^720...)/(1-x).
(End)
Other identities:
For all n >= 0, a(n) = A090529(n+1) - 1. - Reinhard Zumkeller, Jan 05 2014
For all n >= 1, a(n) = A060130(n) + A257510(n). - Antti Karttunen, Apr 27 2015
a(n) ~ log(n^2/(2*Pi)) / (2*LambertW(log(n^2/(2*Pi))/(2*exp(1)))) - 1/2. - Vaclav Kotesovec, Aug 22 2025

Name clarified by Antti Karttunen, Apr 27 2015

A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

Original entry on oeis.org

0, 1, 3, 1, 5, 7, 11, 13, 17, 5, 7, 13, 17, 19, 23, 29, 35, 37, 43, 47, 49, 55, 59, 65, 73, 77, 79, 83, 85, 89, 7, 11, 17, 19, 29, 31, 37, 43, 47, 53, 59, 61, 71, 73, 77, 79, 91, 103, 107, 109, 113, 119, 121, 131, 137, 143, 149, 151, 157, 161, 163, 173, 187, 191, 193, 197, 211, 217, 227, 229, 233, 239, 247

Offset: 1

Ctibor O. Zizka, Apr 02 2008

How many times does each prime appear in this sequence?
The only value (prime(n) - k!) = 0 is at n=1, where k=2.
Are n=2, k=2 and n=4, k=3 the only occurrences of (prime(n) - k!) = 1?
There exist infinitely many solutions of the form (prime(n) - k!) = prime(n-t), t < n.
Are there infinitely many solutions of the form (prime(n) - k!) = prime(r_1)*...*prime(r_i); r_i < n for all i?
From Bernard Schott, Jul 16 2021: (Start)
Answer to the second question is no: 18 other occurrences (n,k) of (prime(n) - k!) = 1 are known today; indeed, every k > 1 in A002981 that satisfies k! + 1 is prime gives an occurrence, but only a third pair (n, k) is known exactly; and this comes for n = 2428957, k = 11 because (prime(2428957) - 11!) = 1.
The next occurrence corresponds to k = 27 and n = X where prime(X) = 1+27! = 10888869450418352160768000001 but index X is not yet available (see A062701).
For the occurrences of (prime(m) - k!) = 1, integers k are in A002981 \ {0, 1}, corresponding indices m are in A062701 \ {1} (only 3 indices are known today) and prime(m) are in A088332 \ {2}. (End)

			a(1)  = prime(1)  - 2! =  2 -  2 =  0;
a(2)  = prime(2)  - 2! =  3 -  2 =  1;
a(3)  = prime(3)  - 2! =  5 -  2 =  3;
a(4)  = prime(4)  - 3! =  7 -  6 =  1;
a(5)  = prime(5)  - 3! = 11 -  6 =  5;
a(6)  = prime(6)  - 3! = 13 -  6 =  7;
a(7)  = prime(7)  - 3! = 17 -  6 = 11;
a(8)  = prime(8)  - 3! = 19 -  6 = 13;
a(9)  = prime(9)  - 3! = 23 -  6 = 17;
a(10) = prime(10) - 4! = 29 - 24 =  5.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A135996, A000040, A000142, A212598, A212266.

Cf. also A002981, A062701, A088332, A346425 (gives k).

f:=proc(n) local p,i; p:=ithprime(n); for i from 0 to p do if i! > p then break; fi; od; p-(i-1)!; end;
[seq(f(n),n=1..70)]; # N. J. A. Sloane, May 22 2012

a[n_] := Module[{p, k},p = Prime[n];k = 1;While[Factorial[k] <= p, k++];p - Factorial[k - 1]] (* James C. McMahon, May 05 2025 *)

a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p - (k-1)!; \\ Michel Marcus, Feb 19 2019

a(n) = prime(n)- k! where k is the greatest number for which k! <= prime(n).
a(n) = A212598(prime(n)). - Michel Marcus, Feb 19 2019
a(n) = A000040(n) - A346425(n). - Bernard Schott, Jul 16 2021

More terms from Jinyuan Wang, Feb 18 2019

A220655 For n with a unique factorial base representation n = duu! + ... + d22! + d11! (each di in range 0..i, cf. A007623), a(n) = (du+1)u! + ... + (d2+1)2! + (d1+1)1!; a(n) = n + A007489(A084558(n)).

Original entry on oeis.org

2, 5, 6, 7, 8, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Used for computing A107346.

Term a(n) can be obtained by adding one to each digit of factorial base representation of n (A007623(n)) and then reinterpreting it as a kind of pseudo-factorial base representation, ignoring the fact that now some of the digits might be over the maximum allowed in that position. Please see the example section. - Antti Karttunen, Nov 29 2013

			1 has a factorial base representation A007623(1) = '1', as 1 = 1*1!. Incrementing the digit 1 with 1, we get 2*1! = 2, thus a(1) = 2. (Note that although '2' is not a valid factorial base representation, it doesn't matter here.)
2 has a factorial base representation '10', as 2 = 1*2! + 0*1!. Incrementing the digits by one, we get 2*2! + 1*1! = 5, thus a(2) = 5.
3 has a factorial base representation '11', as 3 = 1*2! + 1*1!. Incrementing the digits by one, we get 2*2! + 2*1! = 6, thus a(3) = 6.

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

Complement: A220695.
One less than A220656.
Cf. A000142, A003422, A007489, A007623, A084558, A212598, A153880, A231720.

Block[{nn = 66, m = 1}, While[Factorial@ m < nn, m++]; m = MixedRadix[Reverse@ Range[2, m]]; Array[FromDigits[1 + IntegerDigits[#, m], m] &, nn]] (* Michael De Vlieger, Jan 20 2020 *)

;; Standalone iterative implementation (Nov 29 2013):
(define (A220655 n) (let loop ((n n) (z 0) (i 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n i) (+ (* f (+ 1 (remainder n i))) z) (+ 1 i) (* f i))))))
;; Alternative implementation:
(define (A220655 n) (+ n (A007489 (A084558 n))))

a(n) = A220656(n)-1 = A003422(A084558(n)+1) + A000142(A084558(n)) + A212598(n) - 1. [The original definition]
a(n) = n + A007489(A084558(n)). [The above formula reduces to this, which proves that the original Dec 17 2012 description and the new main description produce the same sequence. Essentially, we are adding to n a factorial base repunit '1...111' with as many fact.base digits as n has.] - Antti Karttunen, Nov 29 2013
For n >= 1, A231720(n) = a(A153880(n)).

Name changed by Antti Karttunen, Nov 29 2013

A220656 The positions of those permutations in A030298 where the first element is not fixed.

Original entry on oeis.org

3, 6, 7, 8, 9, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Correspondingly, gives the positions of those terms in A030299 whose first digit is not 1, as long as the decimal encoding system employed is valid.

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

Complement: A220696. Cf. A081291.

(define (A220656 n) (+ (A003422 (+ 1 (A084558 n))) (A000142 (A084558 n)) (A212598 n)))

a(n) = A003422(1+A084558(n)) + A000142(A084558(n)) + A212598(n).

a(n) = A220655(n)+1.

A212266 Primes p such that p - m! is composite, where m is the greatest number such that m! < p.

Original entry on oeis.org

59, 73, 79, 89, 101, 109, 197, 211, 239, 241, 263, 281, 307, 337, 367, 373, 379, 409, 419, 421, 439, 443, 449, 461, 463, 491, 523, 547, 557, 571, 593, 601, 613, 617, 631, 647, 653, 659, 673, 701, 709, 769, 797, 811, 839, 853, 863, 881, 907, 929, 937, 941, 967

Offset: 1

Balarka Sen, May 12 2012

The first five terms 59, 73, 79, 89, 101 belong to A023209. The terms 409, 419, 421, 439, 443, 449 also belong to A127209.

It seems likely that a(n) ~ n log n, can this be proved? - Charles R Greathouse IV, Sep 20 2012

			29 is not a member because 29 - 4! = 5 is prime.
59 is a member because 59 - 4! = 35 is composite.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A212600, A212598, A136437, A023209, A127209.

Select[Prime[Range[200]],Module[{m=9},CompositeQ[While[m!>=#,m--];#-m!]]&] (* The initial m constant (set at 9 in the program) needs to be increased if the prime Range constant (set at 200 in the program) is increased beyond 30969. *) (* Harvey P. Dale, Dec 01 2023 *)

for(n=3,5,N=n!;forprime(p=N+3,N*(n+1),if(!isprime(p-N), print1(p", ")))) \\ Charles R Greathouse IV, May 12 2012

is_A212266(p)=isprime(p) && for(n=1,p, n!1))  \\ M. F. Hasler, May 20 2012

cp's OEIS Frontend

A084558 a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.

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A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

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A220655 For n with a unique factorial base representation n = du*u! + ... + d2*2! + d1*1! (each di in range 0..i, cf. A007623), a(n) = (du+1)*u! + ... + (d2+1)*2! + (d1+1)*1!; a(n) = n + A007489(A084558(n)).

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A220656 The positions of those permutations in A030298 where the first element is not fixed.

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Formula

A212266 Primes p such that p - m! is composite, where m is the greatest number such that m! < p.

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Mathematica

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PARI

A220655 For n with a unique factorial base representation n = duu! + ... + d22! + d11! (each di in range 0..i, cf. A007623), a(n) = (du+1)u! + ... + (d2+1)2! + (d1+1)1!; a(n) = n + A007489(A084558(n)).