A048859 -id:A048859 - cp's OEIS frontend

A007921 Numbers that are not the difference of two primes.

7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, 103, 109, 113, 115, 117, 119, 121, 123, 127, 131, 133, 139, 141, 143, 145, 151, 153, 157, 159, 163, 167, 169, 173, 175, 181, 183, 185, 187, 193

Offset: 1

R. Muller

Conjecturally, odd numbers k such that k+2 is composite.
Is this the same as A068780(2n-1) - 1? - J. Stauduhar, Aug 23 2012
A092953(a(n)) = 0. - Reinhard Zumkeller, Nov 10 2012
It seems that the sequence contains the squares of all primes except for 2 and 3. - Ivan N. Ianakiev, Aug 29 2013 [It does: For every prime p > 3, note that p^2 == 1 (mod 3), so p^2 cannot be q - r where q and r are primes. (If it were, then since p^2 is odd, q and r could not both be odd primes; r would have to be the even prime, 2, which would mean that p^2 = q - 2, so q = p^2 + 2 == 0 (mod 3), i.e., 3 would divide q, so q would not be prime -- a contradiction.) - Jon E. Schoenfield, May 03 2024]
Integers d such that A123556(d) = 1, that is, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has only one element. For each such d, the unique element of all the first largest APs with 1 element is A342309(d) = 2. - Bernard Schott, Jan 08 2023
If it exists, the least even term is > 10^12 (see 1st comment in A020483). - Bernard Schott, Jan 09 2023

F. Smarandache, Properties of Numbers, 1972. (See Smarandache odd sieve.)

T. D. Noe, Table of n, a(n) for n = 1..10000
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
C. Dumitrescu and V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 94.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.
Index entries for primes, gaps between.

Cf. A048859.
Complement of A030173. Cf. A001223.
Cf. also A005408, A010051.
Cf. A123556, A342309.
Largest AP of prime numbers with k elements: this sequence (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A206042 (k=8), A206043 (k=9), A206044 (k=10), A206045 (k=11).

a007921 n = a007921_list !! (n-1)
a007921_list = filter ((== 0) . a010051' . (+ 2)) [1, 3 ..]
-- Reinhard Zumkeller, Jul 03 2015

filter :=  d -> irem(d, 2) <> 0 and not isprime(2+d) : select(filter, [`$`(1 .. 200)]); # Bernard Schott, Jan 08 2023

Lim=200;nn=10;seq:=Complement[Range[Lim],Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]];Until[AllTrue[seq,OddQ],nn++];seq (* James C. McMahon, May 04 2024 *)

is(n)=n%2 && !isprime(n+2) \\ On Polignac's conjecture; Charles R Greathouse IV, Jun 28 2013

from sympy import isprime
print([n for n in range(1, 200) if n%2 and not isprime(n + 2)]) # Indranil Ghosh, Jun 15 2017, after Charles R Greathouse IV

A007606 Take 1, skip 2, take 3, etc.

Original entry on oeis.org

1, 4, 5, 6, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 45, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 137, 138

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

List the natural numbers: 1, 2, 3, 4, 5, 6, 7, ... . Keep the first number (1), delete the next two numbers (2, 3), keep the next three numbers (4, 5, 6), delete the next four numbers (7, 8, 9, 10) and so on.
a(A000290(n)) = A000384(n). - Reinhard Zumkeller, Feb 12 2011
A057211(a(n)) = 1. - Reinhard Zumkeller, Dec 30 2011
Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
Union of nonzero terms of A000384 and A317304. - Omar E. Pol, Aug 29 2018
The values of k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class r' mod m' (with r' in {1,...,m'}) iff mA360418. - James Propp, Feb 10 2023

			From _Omar E. Pol_, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the odd numbers the sequence begins:
    1;
    4,   5,   6;
   11,  12,  13,  14,  15;
   22,  23,  24,  25,  26,  27,  28;
   37,  38,  39,  40,  41,  42,  43,  44,  45;
   56,  57,  58,  59,  60,  61,  62 , 63,  64,  65,  66;
   79,  80,  81,  82 , 83,  84,  85,  86,  87,  88,  89,  90,  91;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120;
...
Row sums give A005917.
Column 1 gives A084849.
Column 2 gives A096376, n >= 1.
Right border gives A000384, n >= 1.
(End)

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Properties of Numbers, 1972.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
Index entries for sequences generated by sieves

Complement of A007607.
Cf. A007950, A007951, A007952, A048859, A004201.
Cf. A000384, A005917, A084849, A096376.

a007606 n = a007606_list !! (n-1)
a007606_list = takeSkip 1 [1..] where
   takeSkip k xs = take k xs ++ takeSkip (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011

Flatten[ Table[i, {j, 1, 17, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
Join[{1},Flatten[With[{nn=20},Range[#[[1]],Total[#]]&/@Take[Thread[ {Accumulate[ Range[nn]]+1,Range[nn]}],{2,-1,2}]]]] (* Harvey P. Dale, Jun 23 2013 *)
With[{nn=20},Take[TakeList[Range[(nn(nn+1))/2],Range[nn]],{1,nn,2}]]//Flatten (* Harvey P. Dale, Feb 10 2023 *)

for(n=1,66,m=sqrtint(n-1);print1(n+m*(m+1),","))

a(n) = n + m*(m+1) where m = floor(sqrt(n-1)). - Klaus Brockhaus, Mar 26 2004

a(n+1) = a(n) + if n=k^2 then 2*k+1 else 1; a(1) = 1. - Reinhard Zumkeller, May 13 2009

A064801 Take 1, skip 2, take 2, skip 3, take 3, etc.

Original entry on oeis.org

1, 4, 5, 9, 10, 11, 16, 17, 18, 19, 25, 26, 27, 28, 29, 36, 37, 38, 39, 40, 41, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 81, 82, 83, 84, 85, 86, 87, 88, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 1

Robert G. Wilson v, Oct 21 2001

A253607(a(n)) < 0. - Reinhard Zumkeller, Jan 05 2015
Integers m such that A000196(m) = A079643(m). - Firas Melaih, Dec 10 2020
Also possible values of floor(x*floor(x)) for real x >= 1. - Jianing Song, Feb 16 2021

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007606, A004202, A048859.
Cf. A061885 (complement), A253607.
Cf. A136272.

a064801 n = a064801_list !! (n-1)
a064801_list = f 1 [1..] where
   f k xs = us ++ f (k + 1) (drop (k + 1) vs)
            where (us, vs) = splitAt k xs
-- Reinhard Zumkeller, May 16 2014

seq(`if`(floor(sqrt(k)) * (floor(sqrt(k)) + 1) > k, k, NULL), k = 0..2034); # a(1)..a(1000), Rainer Rosenthal, Jul 19 2024

a = Table[n, {n, 0, 200} ]; b = {}; Do[a = Drop[a, {1, n} ]; b = Append[b, Take[a, {1, n} ]]; a = Drop[a, {1, n} ], {n, 1, 14} ]; Flatten[b]
Flatten[Table[Range[n^2,n^2+n-1],{n,12}]] (* Harvey P. Dale, Dec 18 2015 *)

{ n=0; for (m=1, 10^9, s=m^2; a=0; for (k=0, m - 1, a=s+k; write("b064801.txt", n++, " ", a); if (n==1000, return)) ) } \\ Harry J. Smith, Sep 26 2009

from math import isqrt  # after Rainer Rosenthal
def isA(k: int): return k < ((s:=isqrt(k)) * (s + 1))
print([k for k in range(129) if isA(k)]) # Peter Luschny, Jul 19 2024

a(n) = A004202(n) - 1.
Can be interpreted as a table read by rows: T(n,k) = n^2 + k, 0 <= k < n. T(n,k) = 0 iff k > A000196(n); T(n,0) = A000290(n); T(n,1) = A002522(n) for n > 1; T(n,2) = A010000(n) = A059100(n) for n > 2; T(n, n-3) = A014209(n-1) for n > 2; T(n, n-2) = A028552(n) for n > 1; T(n, n-1) = A028387(n-1); T(2*n+1, n) = A001107(n+1). - Reinhard Zumkeller, Nov 18 2003
Numbers k such that floor(sqrt(k)) * (floor(sqrt(k)) + 1) > k. - Rainer Rosenthal, Jul 19 2024

A347350 Sequence obtained by writing the first 4 integers and skipping 1, then writing the next 5 integers and skipping 2, then writing the next 6 and skipping 3, etc.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 33, 34, 35, 36, 37, 38, 39, 40, 46, 47, 48, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 97, 98, 99, 100, 101, 102, 103, 104, 105

Offset: 1

Michel Marcus, Aug 28 2021

nonn

Ana Rechtman, Août 2021, 4e défi, Images des Mathématiques, CNRS, 2021.
Index entries for sequences generated by sieves

Cf. A004201, A004202, A048859.

lista(nn) = {my(va=[1..nn], list=List()); my(i=4, j=1, pos=1, ok=1); while (ok, for(m=1, i, if (pos > nn, ok=0, listput(list, va[pos])); pos++; ); pos += j; i++; j++;); Vec(list);}

cp's OEIS Frontend

A007921 Numbers that are not the difference of two primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

A007606 Take 1, skip 2, take 3, etc.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A064801 Take 1, skip 2, take 2, skip 3, take 3, etc.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A347350 Sequence obtained by writing the first 4 integers and skipping 1, then writing the next 5 integers and skipping 2, then writing the next 6 and skipping 3, etc.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI