A136437 -id:A136437 - cp's OEIS frontend

A212598 a(n) = n - m!, where m is the largest number such that m! <= n.

0, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

N. J. A. Sloane, May 22 2012

nonn

The m in definition is given by A084558(n).

Sequence consists of numbers 0..A001563(n)-1 followed by numbers 0..A001563(n+1)-1, and so on. - Antti Karttunen, Dec 17 2012

A. Karttunen, Table of n, a(n) for n = 1..5040

Cf. A136437, A212266, A220656.

f:=proc(n) local i; for i from 0 to n do if i! > n then break; fi; od; n-(i-1)!; end;
[seq(f(n),n=1..70)];

a(n)=my(m); while(m++!<=n,); n-(m-1)! \\ Charles R Greathouse IV, Sep 02 2015

(define (A212598 n) (- n (A000142 (A084558 n))))

a(n) = n-A000142(A084558(n)). - Antti Karttunen, Dec 17 2012

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

A137328 a(n) = prime(n) - primorial(k), where k is the greatest number for which primorial(k) <= prime(n).

Original entry on oeis.org

0, 1, 3, 1, 5, 7, 11, 13, 17, 23, 1, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 49, 53, 59, 67, 71, 73, 77, 79, 83, 97, 101, 107, 109, 119, 121, 127, 133, 137, 143, 149, 151, 161, 163, 167, 169, 1, 13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 61, 67, 71, 73, 83, 97, 101, 103, 107, 121

Offset: 1

Ctibor O. Zizka, Apr 07 2008

Conjecture: Each prime number appears in this sequence at least once.

Is there any general asymptotic formula for the appearance of prime(n) in this sequence?

			a(6) = prime(6) - primorial(2) = 13 - 6 = 7.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000040, A002110, A136437.

pn=FoldList[Times, 1, Prime[Range[5]]] (* Increase for n>343 *); a[n_]:=Module[{k=1},Until[pn[[k]]>Prime[n],k++];Prime[n]-pn[[k-1]]];Array[a,67] (* James C. McMahon, May 28 2025 *)

a(n) = {my(p=prime(n), q=1, P=1); until (P > p, q = nextprime(q+1); P *= q;); p - P/q;} \\ Michel Marcus, Mar 14 2022

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    pn, primk, pk, pkplus = 2, 2, 2, 3
    while True:
        while primk * pkplus <= pn:
            primk, pk, pkplus = primk*pkplus, pkplus, nextprime(pkplus)
        yield pn - primk
        pn = nextprime(pn)
print(list(islice(agen(), 67))) # Michael S. Branicky, Mar 14 2022

A137330 a(n) = primorial(k) - prime(n) where k is the smallest number for which prime(n) <= primorial(k).

Original entry on oeis.org

0, 3, 1, 23, 19, 17, 13, 11, 7, 1, 179, 173, 169, 167, 163, 157, 151, 149, 143, 139, 137, 131, 127, 121, 113, 109, 107, 103, 101, 97, 83, 79, 73, 71, 61, 59, 53, 47, 43, 37, 31, 29, 19, 17, 13, 11, 2099, 2087, 2083, 2081, 2077, 2071, 2069, 2059, 2053, 2047, 2041

Offset: 1

Ctibor O. Zizka, Apr 07 2008

Does each prime number appear in this sequence at least once?

Answer to previous: Neither 2 nor 5 will appear, as no prime > 5 can end in a 5 or be even. - Bill McEachen, Dec 05 2020

			a(6) = primorial(3) - prime(6) = 30-13 = 17.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A002110, A136437.

Block[{a = {}, P = 2, j = 1}, Do[AppendTo[a, If[# > P, j++; P *= Prime[j], P] - #] &@ Prime[n], {n, 57}]; a] (* Michael De Vlieger, Dec 07 2020 *)

a(n) = {my(pp=2, k=1, p=prime(n)); while (pp < p, k++; pp*=prime(k)); pp-p;} \\ Michel Marcus, Dec 06 2020

A346425 a(n) is the greatest number k such that k! <= prime(n).

Original entry on oeis.org

2, 2, 2, 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, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Bernard Schott, Jul 16 2021

nonn

Terms 2, 3, 4, 5, ... appear respectively 3, 6, 21, 98, ... times consecutively; indeed, 2 appears A061232(1) + A061232(2) times, then every m >= 3 appears A061232(m) times.

			prime(1) = 2 and the greatest k with k! <= 2 is 2, so a(1) = 2.
prime(4) = 7 and the greatest k with k! <= 7 is 3, so a(4) = 3.
prime(10) = 29 and the greatest k with k! <= 29 is 4 so a(10) = 4.
Rows with n, prime(n), greatest k! <=n, a(n) for n = 1..14
      n        1    2    3    4    5    6    7    8    9   10   11   12   13   14
   prime(n)    2    3    5    7   11   13   17   19   23   29   31   37   41   43
  greatest k!  2    2    2    6    6    6    6    6    6   24   24   24   24   24
    a(n)       2    2    2    3    3    3    3    3    3    4    4    4    4    4

Cf. A000040, A061232, A136437.

a(n) = my(k=1); until (k! > prime(n), k++); k-1; \\ Michel Marcus, Jul 19 2021

a(n)! = A000040(n) - A136437(n).

A137321 a(n) = prime(n)^prime(n) - k!, where prime(n) is the n-th prime number, and k is the greatest number for which k! <= prime(n)^prime(n).

Original entry on oeis.org

2, 3, 2405, 460663, 198133379411, 281952316704253, 776149319714627324177, 1357971253927074149763979, 12038706006108210079811416910567, 2195692826371309917093828766580180926483469, 16546151513256634846850635804959240464844734431, 5052512795965694464228024657195578053165744330410199442517

Offset: 1

Ctibor O. Zizka, Apr 06 2008

			a(3) = prime(3)^prime(3) - 6! = 5^5 - 720 = 3125 - 720 = 2405.

Cf. A136437, A000040, A000142.

With[{f=Range[60]!},Table[p^p-Max[Select[f,#Harvey P. Dale, Aug 11 2023 *)

a(n) = my(k=1, P=prime(n)^prime(n)); until (k! > P, k++); P - (k-1)!; \\ Michel Marcus, Mar 15 2022

Name corrected and more terms from Michel Marcus, Mar 15 2022

A137322 a(n) = k! - A051674(n), where k is the smallest number for which A051674(n) <= k! where A051674(n) = prime(n)^prime(n).

Original entry on oeis.org

2, 93, 1915, 2805257, 1022362697389, 52812321503747, 296760465891270915823, 13532790387670672394876021, 244372391812343146601953447089433, 11196066938065133911754151366849886273516531, 3328707950474207400029638710843582600755265569

Offset: 1

Ctibor O. Zizka, Apr 06 2008

			a(4) = 10! - prime(4)^prime(4) = 3628800 - 823543 = 2805257.

Cf. A000040, A000142, A051674, A136437.

a[n_] := Module[{p = Prime[n]^Prime[n], k = 1}, While[k! < p, k++]; k! - p]; Array[a, 11] (* Amiram Eldar, Mar 12 2022 *)

f(n) = my(p = prime(n)); p^p;
a(n) = my(k=1, P=f(n)); until(k! >= P, k++); k!-P; \\ Michel Marcus, Mar 12 2022

Corrected and extended by Michel Marcus, Mar 12 2022

cp's OEIS Frontend

A212598 a(n) = n - m!, where m is the largest number such that m! <= n.

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

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A137328 a(n) = prime(n) - primorial(k), where k is the greatest number for which primorial(k) <= prime(n).

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A137330 a(n) = primorial(k) - prime(n) where k is the smallest number for which prime(n) <= primorial(k).

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A346425 a(n) is the greatest number k such that k! <= prime(n).

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A137321 a(n) = prime(n)^prime(n) - k!, where prime(n) is the n-th prime number, and k is the greatest number for which k! <= prime(n)^prime(n).

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A137322 a(n) = k! - A051674(n), where k is the smallest number for which A051674(n) <= k! where A051674(n) = prime(n)^prime(n).

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