A047845 -id:A047845 - cp's OEIS frontend

A291437 Smallest m >= 0 such that (2n)3^m + 1 is prime, or -1 if no such value exists.

0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 2, 0, 2, 1, 0, 1, 9, 0, 0, 4, 1, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 4, 0, 1, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 3, 1, 2, 4, 1, 1, 0, 2, 0, 1, 5, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 80, 0, 6, 0, 8, 2, 0, 1

Offset: 1

Martin Renner, Aug 23 2017

nonn

There exist even integers 2*n such that (2*n)*3^m + 1 is always composite.
It is conjectured that the smallest one is 125050976086 = A123159(3), therefore a(62525488043) = -1.
For the corresponding primes see A291438.
a(A005097(n)) = 0 and a(A047845(n+1)) > 0 (or = -1).

			a(4) = 2 because this is the smallest value such that 8*3^2 + 1 = 73 is prime, since 8*3^0 + 1 = 9 and 8*3^1 + 1 = 25 are not prime.

Cf. A005097, A046067, A047845, A123159, A291438.

a:=[]:
for n from 1 to 10^3 do
  t:=-1:
  for m from 0 to 10^3 do # this max value of m is sufficient up to n=10^3
    if isprime((2*n)*3^m+1) then t:=m: break: fi:
  od:
  a:=[op(a),t]:
od:
a;

Table[SelectFirst[Range[0, 10^3], PrimeQ[2 n*3^# + 1] &] /. k_ /; MissingQ@ k -> -1, {n, 104}] (* Michael De Vlieger, Aug 23 2017 *)

a(n) = {my(m = 0); while (!isprime(p=(2*n)*3^m + 1), m++); m;} \\ Michel Marcus, Aug 25 2017

A354499 Number of consecutive primes generated by adding 2n to the odd squares (A016754).

Original entry on oeis.org

2, 4, 1, 0, 2, 1, 0, 1, 1, 0, 5, 0, 0, 3, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 14, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 4, 0, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 8, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 1, 0

Offset: 1

Steven M. Altschuld, Aug 15 2022

nonn

Conjecture: a(n) <= 18 = a(326).
a(m) = 0 for m in A047845. - Michel Marcus, Aug 16 2022
I conjecture the opposite: a(n) is unbounded, and indeed for any k < 1 and any m there are >> x^k terms up to x with a(n) > m. At a very rough guess, there should be some n with 20-50 digits having a(n) > 18. - Charles R Greathouse IV, Oct 26 2022

			For n=1 we have 1^2+2*1=3 and 3^2+2*1=11 are prime but 5^2+2*1=27 is not, and thus a(1)=2.
For n=2, 1^2+2*2=5 ... 7^2+2*2=53 are prime but 9^2+2*2=85 is not, thus a(2)=4.
For n=3, 1^2+2*3=7 is prime but 3^2+2*3=15 is not thus a(3)=1.
For n=4, 1^2+2*4=9 which is not prime, thus a(4)=0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005843 (even numbers), A016754 (odd squares), A356567 (positions of records).

Cf. A047845.

f:= proc(n) local k;
  for k from 1 by 2 do
    if not isprime(k^2+2*n) then return (k-1)/2 fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Oct 26 2023

a[n_] := Module[{k = 1}, While[PrimeQ[k^2 + 2*n], k += 2]; (k - 1)/2]; Array[a, 100] (* Amiram Eldar, Aug 15 2022 *)

a(n) = my(k=1); while (isprime(k^2+2*n), k+=2); (k-1)/2; \\ Michel Marcus, Aug 16 2022

a(n) is number of consecutive primes generated by (2x-1)^2+2n for x=1,2,3,4,

A380549 List of numbers of the form i + 3j + 4i*j for i, j >= 1.

Original entry on oeis.org

8, 13, 15, 18, 22, 23, 24, 28, 29, 33, 35, 36, 38, 42, 43, 46, 48, 50, 51, 53, 57, 58, 60, 61, 63, 64, 68, 69, 71, 73, 74, 78, 79, 80, 83, 85, 87, 88, 90, 92, 93, 96, 97, 98, 99, 100, 101, 103, 105, 106, 108, 112, 113, 114, 118, 120, 123, 126, 127, 128, 131, 132, 133, 134, 137, 138, 139, 141, 143, 145, 148, 150

Offset: 1

Peter Bala, Jan 26 2025

This is a companion sequence to A380509. If N != 6 is a positive integer not in this list then 4*N + 3 is either a prime or three times a prime. See A380550.

Compare with A072668, numbers of the form i + j + i*j, and A047845, numbers of the form i + j + 2*i*j.

Cf. A047845, A072668, A380509, A380550.

L := 150:  S := {}:
for i from 1 to L do
  for j from 1 to L do
    if i + 3*j + 4*i*j <= L then S := `union`(S, {i+3*j+4*i*j}) end if
  end do;
end do:
S;

A117092 Numbers n such that nextprime(2n) > 2nextprime(n) (here nextprime = A007918; if p is prime then nextprime(p) = p).

Original entry on oeis.org

2, 3, 4, 5, 7, 10, 11, 12, 13, 16, 17, 19, 22, 23, 27, 28, 29, 31, 37, 40, 41, 42, 43, 45, 46, 47, 52, 53, 57, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 79, 82, 83, 87, 88, 89, 97, 100, 101, 102, 103, 106, 107, 108, 109, 112, 113, 126, 127, 129, 130, 131, 136

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Apr 18 2006

Contains all primes (A000040), and the intersection of A006093 and A047845. - Robert Israel, Mar 30 2016

			nextprime(2*12)=29 and nextprime(2)*nextprime(12)=2*13 then 12 is member because 29>26.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000040, A006093, A007918, A047845.

select(t -> nextprime(2*t) > 2*nextprime(t-1),[$1..100]); # Robert Israel, Mar 30 2016

Select[Range@100, NextPrime[2 #] > 2 NextPrime[# - 1] &] (* Ivan Neretin, Mar 30 2016 *)

PARI
```
for(i=1,100,if(nextprime(2*i)
				
```

Corrected by T. D. Noe, Oct 25 2006

A166160 a(n) = (n-th odd prime + n-th odd nonprime)/2 - 1.

Original entry on oeis.org

1, 6, 10, 15, 18, 21, 25, 28, 33, 37, 42, 45, 48, 51, 57, 61, 64, 70, 73, 76, 81, 84, 89, 94, 97, 100, 105, 109, 113, 121, 124, 128, 130, 136, 139, 144, 148, 153, 157, 161, 163, 171, 173, 177, 179, 187, 195, 198, 201, 204, 210, 212, 218, 222, 228, 234, 236, 240, 243

Offset: 1

Juri-Stepan Gerasimov, Oct 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001477, A005097, A014076, A047845, A065091.

A065091 := proc(n) ithprime(n+1) ; end proc:
A014076 := proc(n) if n = 1 then 1; else for a from procname(n-1)+2 by 2 do if not isprime(a) then return a ; end if; end do: end if; end proc:
A166160 := proc(n) (A065091(n)+A014076(n))/2-1 ; end proc: seq(A166160(n),n=1..120) ; # R. J. Mathar, May 21 2010

A014076 := Select[Range@250, ! PrimeQ@# && OddQ@# &]; A166160 := Prime[Range[2, 150]]; Table[(A166160[[n]] + A014076[[n]])/2 - 1, {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)
Module[{nn=250,op,onp},op=Prime[Range[2,nn]];onp=Select[Range[1,nn,2],!PrimeQ[#]&];Total[#]/2-1&/@Thread[{Take[op,Length[ onp]],onp}]] (* Harvey P. Dale, Mar 02 2025 *)

a(n) = A005097(n) + A047845(n), where A005097 U A047845 = A001477.

a(n) = (A065091(n) + A014076(n))/2 - 1.

Entries checked by R. J. Mathar, May 21 2010

A211173 (2n)!^n (modulo 2n+1).

Original entry on oeis.org

0, 2, 1, 6, 0, 10, 1, 0, 1, 18, 0, 22, 0, 0, 1, 30, 0, 0, 1, 0, 1, 42, 0, 46, 0, 0, 1, 0, 0, 58, 1, 0, 0, 66, 0, 70, 1, 0, 0, 78, 0, 82, 0, 0, 1, 0, 0, 0, 1, 0, 1, 102, 0, 106, 1, 0, 1, 0, 0, 0, 0, 0, 0, 126, 0, 130, 0, 0, 1, 138, 0

Offset: 0

Larry Riddle (LRiddle(AT)AgnesScott.edu) and Robert G. Wilson v, Jan 31 2013

a(n)= 0, 1 or 2n. In fact, a(n) = 0 iff n belongs to A047845, a(n) = 1 iff n belongs to A104636 and a(n) = 2n iff n belongs to A104635.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept. Jul, 1999.

Cf. A047845, A104635, A104636.

f[n_] := Mod[((2 n)!)^n, 2 n + 1]; Array[f, 70]
Table[PowerMod[(2n)!,n,2n+1],{n,0,70}] (* Harvey P. Dale, Nov 02 2019 *)

a(n)=if(isprime(2*n+1),if(n%2,2*n,1),0) \\ Charles R Greathouse IV, Feb 07 2013

a(n) = (2n)!^n (modulo 2n+1).

A277870 Ordered number of unit edges needed to build every 4-orthotope from hypercubes.

Original entry on oeis.org

32, 52, 72, 84, 92, 112, 116, 132, 135, 148, 152, 160, 172, 180, 186, 192, 204, 212, 216, 232, 237, 244, 248, 252, 256, 260, 272, 276, 288, 292, 297, 308, 312, 316, 326, 332, 336, 339, 340, 352, 372, 378, 380, 384, 390, 392, 396, 404, 408, 412, 415, 424, 428

Offset: 1

Eric R. Carter, Nov 02 2016

nonn

Ordered number of edges required to construct every hyperrectangle as a union of unit hypercubes. The sequence gives the n-th smallest such number, and generalizes the two-dimensional A047845 and the three-dimensional A090767 to four dimensions.

Does a(n) ~ n? - Charles R Greathouse IV, Nov 06 2016

			a(1)=32 as this is the number of edges in the unit hypercube.

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

Cf. A047845, A090767.

Edges[x_,y_,z_,w_]:=(4*x*y*z*w)+3*((w*x*z)+(w*y*z)+(w*x*y)+(x*y*z))+2*((w*x)+(w*y)+(w*z)+(x*y)+(x*z)+(y*z))+x+y+z+w;inputs=Tuples[Range[s],4];Union[Table[Edges[inputs[[k]][[1]],inputs[[k]][[2]],inputs[[k]][[3]],inputs[[k]][[4]]],{k,1,Length[inputs]}]]
Accuracy to 170 terms is achieved for s>=5764801, and for the entire list in the limit as s approaches infinity.

list(lim)=my(v=List()); for(w=1,(lim-12)\20, for(x=1, min((lim-8*w-4)\(12*w+8),w), for(y=1,min((lim-5*w*x-3*x-3*w-1)\(7*w*x+5*x+5*w+3),x), forstep(n=((7*w+5)*y+(5*w+3))*x+(5*w + 3)*y+3*w+1, lim, ((4*w+3)*y+3*w+2)*x+(3*w+2)*y+2*w+1, listput(v,n))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2016

These numbers are of the form: 4wxyz + 3(wxz+wyz+wxy+xyz) + 2(wx+wy+wz+xy+xz+yz) + w+x+y+z for any positive integers w, x, y, z.

A286982 Smallest nonnegative k such that (1 + k)^(2^n) + k is not prime and all (1 + k)^(2^j) + k, for 0 <= j < n, are primes.

Original entry on oeis.org

6, 3, 5, 2, 1, 54131988

Offset: 1

Juri-Stepan Gerasimov, May 12 2017

			a(1) = 6 because (1 + 6)^(2^1) + 6 = 55 is semiprime and (1 + 6)^(2^0) + 6 = 13 is prime;
a(2) = 3 because (1 + 3)^(2^2) + 3 = 259 is semiprime and both (1 + 3)^(2^0) + 3 = 7 and (1 + 3)^(2^1) + 3 = 19 are primes;
a(3) = 5 because (1 + 5)^(2^3) + 5 = 167921 is semiprime and (1 + 5)^(2^0) + 5 = 11, (1 + 5)^(2^1) + 5 = 41 and (1 + 5)^(2^2) + 5 = 1301 are all primes;
a(4) = 2 because (1 + 2)^(2^4) + 2 = 43046723 is semiprime and (1 + 2)^(2^0) + 2 = 5, (1 + 2)^(2^1) + 2 = 11, (1 + 2)^(2^2) + 2 = 83 and (1 + 2)^(2^3) + 2 = 6563 are all primes;
a(5) = 1 because (1 + 1)^(2^5) + 1 = 4294967297 is semiprime and (1 + 1)^(2^0) + 1 = 3, (1 + 1)^(2^1) + 1 = 5, (1 + 1)^(2^2) + 1 = 17, (1 + 1)^(2^3) + 1 = 257 and (1 + 1)^(2^4) + 1 = 65537 are fix known Fermat primes (A019434);
a(6) = 54131988 because (1 + 54131988)^(2^6) + 54131988 is composite and (1 + 54131988)^(2^0) + 54131988 = 108263977, (1 + 54131988)^(2^1) + 54131988 = 2930272287228109, (1 + 54131988)^(2^2) + 54131988 =  8586495360054127683625679378629, (1 + 54131988)^(2^3) + 54131988 = 73727902568231063808600888120898279950965368674840612135914869, (1 + 54131988)^(2^4) + 54131988 and (1 + 54131988)^(2^5) + 54131988 are all primes.

Cf. A019434, A047845, A057726, A160027, A286680.

a[n_] := Block[{k = 1}, While[PrimeQ[(1 + k)^(2^n) + k] || ! AllTrue[(1 + k)^(2^Range[0, n-1]) + k, PrimeQ], k++]; k]; Array[a, 5] (* Giovanni Resta, May 30 2017 *)

a(6) from Robert G. Wilson v, May 14 2017

A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

Original entry on oeis.org

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

Offset: 1

Jamie Morken and Vincenzo Librandi, Feb 09 2019

Composites with least prime factor p are on that row of A083140 which begins with p
Sequence with similar values: A122005.
Sequence written as a jagged array A with new row when a(n) > a(n+1):
  1,  2,  3,
  1,  4,  5,  6,
  2,  7,  8,  9,
  3, 10, 11,
  1, 12,
  4, 13, 14, 15,
  5, 16,
  2, 17, 18,
  6, 19, 20, 21,
  7, 22, 23,
  1, 24,
  8, 25, 26,
  3, 27,
  9, 28, 29, 30.
A153196 is the list B of the first values in successive rows with length 4.
  B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)
For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).
A243811 is the list of the second values in successive rows with length 4.
A047845 is the list of values in the second column and A104279 is the list of values in the third column of the jagged array starting on the second row.
Sequence written as an irregular triangle C with new row when a(n)=1:
  1,2,3,
  1,4,5,6,2,7,8,9,3,10,11,
  1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,
  1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7,57,19,58,4,59.
A243887 is the last value in each row of C.
The second value D on the row n > 1 of the irregular triangle C is a(A053683(n)) or equivalently A084921(n). For example for row 3 of the irregular triangle:
  D = a(A053683(3)) = a(16) = 12 or D = A084921(3) = 12. This is the number of composites < A066872(3) with the same least prime factor p as the A053683(3) = 16th composite, A066872(3) = 26.
The number of values in each row of the irregular triangle C begins: 3,11,18,57,39,98,61,141,265,104,351,268,...
The second row of the irregular triangle C is A117385(b) for 3 < b < 15.
The third row of the irregular triangle C has similar values as A117385 in different order.

			First composite 4, least prime factor is 2, first case for 2 so a(1)=1.
Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.
Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.
Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.
Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.

Jamie Morken, Table of n, a(n) for n = 1..10000

Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.

counts = {}
values = {}
For[i = 2, i < 130, i = i + 1,
If[PrimeQ[i], ,
x = PrimePi[FactorInteger[i][[1, 1]]];
  If[Length[counts] >= x,
   counts[[x]] = counts[[x]] + 1;
   AppendTo[values, counts[[x]]], AppendTo[counts, 1];
   AppendTo[values, 1]]]]
   (* Print[counts] *)
   Print[values]

c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = my(c=c(n), lpf = vecmin(factor(c)[,1]), nb=0); for(k=2, c, if (!isprime(k) && vecmin(factor(k)[,1])==lpf, nb++)); nb; \\ Michel Marcus, Feb 10 2019

a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).
For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).
This gives an estimate of 67.499... and the actual value of a(n)=67.

A121206 a(n) = (2n)! mod n(2n+1).

Original entry on oeis.org

2, 4, 6, 0, 10, 12, 0, 16, 18, 0, 22, 0, 0, 28, 30, 0, 0, 36, 0, 40, 42, 0, 46, 0, 0, 52, 0, 0, 58, 60, 0, 0, 66, 0, 70, 72, 0, 0, 78, 0, 82, 0, 0, 88, 0, 0, 0, 96, 0, 100, 102, 0, 106, 108, 0, 112, 0, 0, 0, 0, 0, 0, 126, 0, 130, 0, 0, 136, 138, 0, 0, 0, 0, 148, 150, 0, 0, 156, 0, 0, 162

Offset: 1

Ben Paul Thurston, Aug 20 2006

nonn

If the zeros are removed and a 3 is inserted at the front, the first 3000 terms (or more) of the condensed sequence coincide with A039915. - R. J. Mathar, Mar 02 2007

			a(4) = 0 because 8*7*6*5*4*3*2*1 / 8+7+6+5+4+3+2+1 divides evenly (0 remainder).

Cf. A005097 gives indices of nonzero terms; A047845 gives indices of zero terms.

Cf. A039915, A006093.

Table[Mod[(2n)!, n*(2n + 1)], {n, 85}] (* Ray Chandler, Aug 23 2006 *)

a(n) = A000142(2n) mod A000217(2n).

Edited, corrected and extended by R. J. Mathar and Ray Chandler, Aug 23 2006

cp's OEIS Frontend

A291437 Smallest m >= 0 such that (2*n)*3^m + 1 is prime, or -1 if no such value exists.

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A354499 Number of consecutive primes generated by adding 2n to the odd squares (A016754).

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A380549 List of numbers of the form i + 3*j + 4*i*j for i, j >= 1.

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A117092 Numbers n such that nextprime(2*n) > 2*nextprime(n) (here nextprime = A007918; if p is prime then nextprime(p) = p).

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A166160 a(n) = (n-th odd prime + n-th odd nonprime)/2 - 1.

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A211173 (2n)!^n (modulo 2n+1).

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A277870 Ordered number of unit edges needed to build every 4-orthotope from hypercubes.

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A291437 Smallest m >= 0 such that (2n)3^m + 1 is prime, or -1 if no such value exists.

A380549 List of numbers of the form i + 3j + 4i*j for i, j >= 1.

A117092 Numbers n such that nextprime(2n) > 2nextprime(n) (here nextprime = A007918; if p is prime then nextprime(p) = p).