cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A212600 Composite numbers p-m! arising in A212266.

Original entry on oeis.org

35, 49, 55, 65, 77, 85, 77, 91, 119, 121, 143, 161, 187, 217, 247, 253, 259, 289, 299, 301, 319, 323, 329, 341, 343, 371, 403, 427, 437, 451, 473, 481, 493, 497, 511, 527, 533, 539, 553, 581, 589, 49, 77, 91, 119, 133, 143, 161, 187, 209, 217, 221
Offset: 1

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Author

N. J. A. Sloane, May 22 2012

Keywords

Links

Crossrefs

Cf. A212266.

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

Views

Author

Ctibor O. Zizka, Apr 02 2008

Keywords

Comments

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)

Examples

			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.

Links

Crossrefs

Cf. A135996, A000040, A000142, A212598, A212266.
Cf. also A002981, A062701, A088332, A346425 (gives k).

Programs

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

Formula

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

Extensions

More terms from Jinyuan Wang, Feb 18 2019

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, May 22 2012

Keywords

Comments

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

Links

Crossrefs

Cf. A136437, A212266, A220656.

Programs

  • Maple
    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)];
  • PARI
    a(n)=my(m); while(m++!<=n,); n-(m-1)! \\ Charles R Greathouse IV, Sep 02 2015
  • Scheme
    (define (A212598 n) (- n (A000142 (A084558 n))))

Formula

a(n) = n-A000142(A084558(n)). - Antti Karttunen, Dec 17 2012
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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