A005235 -id:A005235 - cp's OEIS frontend

A067363 a(n)=p-n!^3, where p is the smallest prime > n!^3+1.

2, 3, 7, 5, 17, 11, 17, 23, 23, 103, 59, 17, 29, 79, 59, 23, 347, 307, 53, 227, 131, 83, 67, 223, 29, 59, 197, 83, 181, 293, 71, 71, 139, 43, 67, 103, 431, 743, 1279, 197, 419, 127, 271, 73, 229, 503, 211, 181, 1597, 151, 151, 197, 1013, 179, 587, 71, 137, 547

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 19 2002

nonn

The first 118 terms are primes. Are all terms prime? For n!^i, with 0

The first 2278 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..2278
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067362, A067364, A067365.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^3+i], Return[i]]]
spn[n_]:=Module[{c=(n!)^3},NextPrime[c+1]-c]; Array[spn,60] (* Harvey P. Dale, May 25 2023 *)

for n from 1 to 50 do f := n!^3:a := nextprime(f+2)-f:print(a) end_for

for(n=1, 100, f=n!^3; print1(nextprime(f+2)-f,", ")) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**3; say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 13 2015

Edited by Dean Hickerson, Mar 02 2002

A067364 a(n)=p-n!^4, where p is the smallest prime > n!^4+1.

Original entry on oeis.org

2, 3, 5, 5, 7, 29, 19, 29, 181, 19, 31, 173, 79, 43, 379, 61, 101, 127, 101, 83, 37, 29, 271, 233, 109, 233, 293, 1039, 137, 241, 173, 197, 613, 1933, 277, 71, 503, 449, 1667, 53, 67, 163, 179, 211, 53, 613, 1171, 1069, 359, 199, 839, 433, 1523, 463, 677

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 19 2002

nonn

The first 102 terms are primes. Are all terms prime? For n!^i, with 0

The first 1865 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..1865
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067362, A067363, A067365.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^4+i], Return[i]]]

for n from 1 to 50 do f := n!^4:a := nextprime(f+2)-f:print(a) end_for

for(n=1, 500, f=n!^4; print1(nextprime(f+2)-f, ", ")) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**4; say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 13 2015

Edited by Dean Hickerson, Mar 02 2002

A067365 a(n) = p-n!^5, where p is the smallest prime > n!^5+1.

Original entry on oeis.org

2, 5, 13, 13, 7, 7, 11, 71, 23, 19, 197, 17, 101, 53, 17, 47, 73, 97, 53, 433, 251, 251, 47, 263, 281, 353, 53, 61, 179, 41, 53, 401, 449, 79, 89, 1283, 367, 2011, 139, 227, 1597, 1657, 1123, 397, 131, 727, 137, 167, 89, 379, 421, 653, 223, 373, 2221, 1447

Offset: 1

Frank Buss (fb(AT)frank-buss.de), Jan 19 2002

nonn

The first 60 terms are primes. Are all terms prime? For n!^i, with 0

The first 1592 terms are primes. - Dana Jacobsen, May 13 2015

Dana Jacobsen, Table of n, a(n) for n = 1..1592
Cyril Banderier, Fortunate Numbers

Cf. A037153, A037153, A005235, A067362, A067363, A067364.

a[n_] := For[i=2, True, i++, If[PrimeQ[n!^5+i], Return[i]]]
spf[n_]:=Module[{c=(n!)^5},NextPrime[c+1]-c]; Array[spf,60] (* Harvey P. Dale, Feb 24 2015 *)

for n from 1 to 50 do f := n!^5:a := nextprime(f+2)-f:print(a) end_for

for(n=1, 100, f=n!^5; print1(nextprime(f+2)-f, ", ")) \\ Dana Jacobsen, May 13 2015

use ntheory ":all"; use Math::GMP qw/:constant/; for my $n (1..500) { my $f=factorial($n)**5; say "$n ", next_prime($f+1)-$f; } # Dana Jacobsen, May 13 2015

Edited by Dean Hickerson, Mar 02 2002

A087202 a(n) is the smallest m such that m > A037153(n) and n!+ m is prime.

Original entry on oeis.org

4, 5, 7, 7, 11, 13, 19, 31, 23, 19, 19, 43, 73, 41, 149, 41, 53, 61, 109, 37, 37, 71, 109, 193, 97, 173, 59, 101, 229, 163, 241, 83, 139, 103, 83, 577, 397, 47, 269, 61, 211, 107, 97, 89, 379, 149, 269, 83, 137, 167, 281, 89, 79, 443, 229, 157, 179, 563, 389, 277

Offset: 1

Farideh Firoozbakht, Sep 01 2003

nonn

a(n) is the second m (first m is A037153(n)) such that m > 1 and n!+ m is prime.
Conjecture: For n > 1, a(n) is prime (compare the conjecture about A037153).
Conjecture holds through 1200 terms.

Ray Chandler, Table of n, a(n) for n=1..1200

Cf. A037153, A005235, A087200.

A037153[n_] := (For[m=Prime[PrimePi[n]+1], !PrimeQ[n!+m], m++ ]; m); a[n_] := (For[m=A037153[n]+1, !PrimeQ[n!+m], m++ ]; m); Table[a[n], {n, 60}]

Edited by Ray Chandler, Mar 08 2010

A264050 a(n) = least m > 1 such that m + 2^n is prime.

Original entry on oeis.org

3, 3, 3, 3, 5, 3, 3, 7, 9, 7, 5, 3, 17, 27, 3, 3, 29, 3, 21, 7, 17, 15, 9, 43, 35, 15, 29, 3, 11, 3, 11, 15, 17, 25, 53, 31, 9, 7, 23, 15, 27, 15, 29, 7, 59, 15, 5, 21, 69, 55, 21, 21, 5, 159, 3, 81, 9, 69, 131, 33, 15, 135, 29, 13, 131, 9, 3, 33, 29, 25, 11, 15, 29, 37, 33

Offset: 1

Alexei Kourbatov, Nov 02 2015

nonn

The definition is similar to Fortunate numbers (A005235) but uses 2^n instead of primorial A002110(n).
Terms a(n) are often but not always prime; sometimes they are prime powers or semiprimes or have a more general form.
An analog of Fortune's conjecture for this sequence would be "a(n) is either a prime power or a semiprime." But even this relaxed conjecture is disproved by, e.g., a(62)=135, a(93)=a(97)=105, a(99)=255.
By definition, a(n) >= A013597(n). The integers n such that a(n) > A013597(n) are those with A013597(n)=1, i.e., 1, 2, 4, 8, 16, and then? - Michel Marcus, Nov 06 2015

			a(56)=81 because m=81 is the least m > 1 such that m + 2^56 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000

Cf. A005235, A013597, A092131, A263925.

Table[m = 2; While[! PrimeQ[m + 2^n], m++]; m, {n, 75}] (* Michael De Vlieger, Nov 06 2015 *)

a(n)=my(m=2);while(!isprime(m+2^n),m++);m \\ Anders Hellström, Nov 02 2015

a(n)=nextprime(2^n+2)-2^n \\ Charles R Greathouse IV, Nov 02 2015

a(60) corrected by Charles R Greathouse IV, Nov 02 2015

A324656 a(n) is the number of successive primorials A002110(i) larger than n that need to be tried before sum n + A002110(i) is found to be composite.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 11 2019

nonn

a(n) = 0 if n + A002110(A235224(n)), i.e., n plus {the least primorial > n} is composite.
a(n) = 1 if n + A002110(A235224(n)) is prime, but n + A002110(1+A235224(n)) is composite.
a(n) = k if n + A002110(j+A235224(n)) is prime for j=0..k-1, but n + A002110(k+A235224(n)) is composite.

			For n=1, it is not a composite number, so we add a next larger primorial (A002110) to it, which is 2, and we see that 3 is also noncomposite, thus we try to add (to the original n, which is 1) the next larger primorial, which is 6, and 7 is also prime, as are also 31, 211 and 2311. Only with A002110(6), 30030 + 1 is not a prime, thus a(1) = 5.
For n=3, the next larger primorial is 6, but 3+6 = 9 is composite, thus a(3) = 0.
For n=29, which is prime, we try adding it to four successively larger primorial numbers 30, 210, 2310, 30030, until we find 510510 which gives sum 510539 which is composite, thus a(29) = 4. In primorial base (A049345), 29 is written as 421 and the successive sums tested are: 1421, 10421, 100421, 1000421 and 10000421.
For n=121, which is not prime, but 210+121 = 331 is, while 2310+121 = 2431 is not, a(121) = 1.

Cf. A002110, A049345 (A324550), A235224, A324657.

Cf. also A005235, A324642.

A002110(n) = prod(i=1,n,prime(i));
A235224(n, p=2) = if(!n, n, 1+A235224(n\p, nextprime(p+1)));
A324656(n) = { my(k=0,b=A235224(n)); while(isprime(n+A002110(k+b)), k++); (k); };

A038708 Primes of the form (k-th primorial) + (k+1)-st prime.

Original entry on oeis.org

5, 11, 37, 30047, 510529, 9699713, 13082761331670077, 32589158477190044789, 1922760350154212639131, 40729680599249024150621323549

Offset: 1

Labos Elemer, May 02 2000

nonn

			k=6: 6th primorial 2*3*5*7*11*13 + prime(7) = 300030 + 17 = 30047 is a prime.

Jinyuan Wang, Table of n, a(n) for n = 1..11

Cf. A002110, A035346, A005235.

With[{nn=30},Select[Total/@Thread[{FoldList[Times,Prime[ Range[ nn]]],Prime[ Range[ 2,nn+1]]}],PrimeQ]] (* Harvey P. Dale, Oct 12 2018 *)

Next term has 149 decimal digits.

A263925 a(n) = least m > 1 such that m + (prime(n)#)^n is prime.

Original entry on oeis.org

3, 5, 11, 19, 89, 323, 29, 61, 79, 199, 563, 181, 353, 1307, 257, 709, 1237, 1277, 1609, 1237, 4157, 2017, 577, 157, 191, 1063, 239, 823, 1607, 4159, 139, 11527, 2339, 18457, 4079, 463, 1861, 1123, 8699, 16561, 719, 4327, 9311, 1693, 3067, 4243, 22397, 4079, 3989, 24071

Offset: 1

Alexei Kourbatov, Oct 30 2015

nonn

Here prime(n)# denotes the primorial A002110(n), i.e., the product of the first n primes. Terms a(n) are often (but not always) prime; out of the first fifty terms, only one (a(6)=323) is composite.

The definition is similar to Fortunate numbers (A005235); however, in A005235 the primorial is not raised to the n-th power. Unlike this sequence, all known Fortunate numbers are prime.

			(prime(2)#)^2=36. a(2)=5 because 5 is the minimal m>1 such that m+36 is prime.

Cf. A002110, A005235, A181555, A264050.

Table[m = 2; While[! PrimeQ[m + Product[Prime@ i, {i, n}]^n], m++]; m, {n, 30}] (* Michael De Vlieger, Nov 11 2015 *)

a(n)=my(s=prod(i=1,n,prime(i))^n); nextprime(s+2)-s

A375007 Numbers t which satisfy the equation: t mod k = floor((t - k)/k) mod k (1 <= k <= t) only for k = 1 and t.

Original entry on oeis.org

1, 2, 3, 4, 8, 24, 28, 40, 60, 112, 316, 508, 568, 760, 796, 1212, 1228, 2616, 5296, 6220, 8016, 12456, 14620, 16888, 21772, 23116, 23356, 25656, 30312, 30712, 30808, 32716, 33720, 38328, 46072, 52816, 59112, 61728, 67960, 69808, 72972

Offset: 1

Lechoslaw Ratajczak, Jul 27 2024

nonn

Every term greater than 3 is divisible by 4.
Let b(z) be the number of elements of this sequence <= z:
--------------
   z  |   b(z)
--------------
10^2  |    9
10^3  |   15
10^4  |   21
10^5  |   45
10^6  |  106
10^7  |  296
10^8  |  869
--------------
Conjecture: a(n) + 1 is prime for n > 6. Verified for all terms < 10^8.
Conjecture: nextprime(u(n)) - u(n), where u(n) = Product_{m=1..n} (a(m+1) - a(m)), is a noncomposite number. Verified for all terms < 10^8.

			Let T(i,j) be the triangle read by rows: T(i,j) = 1 if i mod j = floor((i - j)/j) mod j, T(i,j) = 0 otherwise, for 1 <= j <= i. The triangle begins:
i\j | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
-----------------------------------------
  1 | 1
  2 | 1 1
  3 | 1 0 1
  4 | 1 0 0 1
  5 | 1 1 0 0 1
  6 | 1 1 0 0 0 1
  7 | 1 0 1 0 0 0 1
  8 | 1 0 0 0 0 0 0 1
  9 | 1 1 0 1 0 0 0 0 1
 10 | 1 1 0 0 0 0 0 0 0  1
 11 | 1 0 1 0 1 0 0 0 0  0  1
 12 | 1 0 1 0 0 0 0 0 0  0  0  1
 13 | 1 1 0 0 0 1 0 0 0  0  0  0  1
 14 | 1 1 0 1 0 0 0 0 0  0  0  0  0  1
 15 | 1 0 0 0 0 0 1 0 0  0  0  0  0  0  1
...
The j-th column has period j^2.

Cf. A000040, A005235, A008578, A051731.

(f(i,j):=mod((i-floor((i-j)/j)),j),
(n:4, for t:4 thru 100000 step 4 do
(for k:2 while f(t,k)#0 and k

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cp's OEIS Frontend

A275272 a(n) = p - n!, where p is the second smallest prime > n!.

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A067363 a(n)=p-n!^3, where p is the smallest prime > n!^3+1.

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A067364 a(n)=p-n!^4, where p is the smallest prime > n!^4+1.

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A067365 a(n) = p-n!^5, where p is the smallest prime > n!^5+1.

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A087202 a(n) is the smallest m such that m > A037153(n) and n!+ m is prime.

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A264050 a(n) = least m > 1 such that m + 2^n is prime.

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A324656 a(n) is the number of successive primorials A002110(i) larger than n that need to be tried before sum n + A002110(i) is found to be composite.

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