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.

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A136800 Number of composites in prime gaps of size 3 or larger, in order of appearance.

Original entry on oeis.org

3, 3, 3, 5, 5, 3, 3, 5, 5, 5, 3, 5, 3, 5, 7, 3, 3, 3, 13, 3, 5, 9, 5, 5, 3, 5, 5, 9, 3, 11, 11, 3, 3, 5, 9, 5, 5, 5, 5, 3, 9, 13, 3, 3, 13, 5, 9, 3, 5, 7, 5, 5, 3, 5, 7, 3, 7, 9, 9, 5, 3, 5, 7, 3, 3, 11, 7, 3, 7, 3, 5, 11
Offset: 1

Views

Author

Enoch Haga, Jan 22 2008

Keywords

Comments

The sequence counts the terms in the runs of composites associated with A136798-A136799.
A129856 is obtained by removing the composites (9, 15 etc.) from this sequence.
This is sequence A046933, with the zero and all the 1's deleted. - R. J. Mathar, Jan 24 2008

Examples

			a(1)=3 because in the run 8, 9, 10 there are three terms.

Crossrefs

Cf. A136798, A136799, A136801.

Programs

  • Mathematica
    Select[#[[2]]-#[[1]]-1&/@Partition[Prime[Range[100]],2,1],#>2&] (* Harvey P. Dale, Apr 08 2015 *)

Formula

a(n)=A136799(n)-A136798(n)+1.

Extensions

Edited by R. J. Mathar, May 27 2009

A204100 Number of integers between successive twin primes, divided by 3.

Original entry on oeis.org

0, 1, 1, 3, 3, 5, 3, 9, 1, 9, 3, 9, 3, 1, 9, 3, 9, 3, 9, 11, 23, 3, 9, 19, 15, 9, 5, 7, 5, 49, 3, 1, 9, 7, 45, 3, 5, 3, 9, 19, 25, 15, 3, 3, 5, 35, 7, 9, 1, 39, 3, 15, 9, 7, 21, 27, 1, 17, 5, 15, 9, 17, 1, 7, 5, 3, 31, 9, 13, 9, 13, 55, 13, 21, 9, 7, 5, 19
Offset: 1

Views

Author

Michel Lagneau, Jan 10 2012

Keywords

Examples

			a(2) = 1 because there exists three numbers 8, 9 and 10 between (5,7) and (11,13) => a(2) = 3/3 = 1.

Links

Crossrefs

Cf. A028334, A063091, A046933, A204099.

Programs

  • Maple
    T:=array(1..100,1..2):k:=0:for n from 1 to 1000 do:p1:=ithprime(n):p2:=ithprime(n+1):if p2-p1 = 2 then k:=k+1:T[k,1]:=p1:T[k,2]:=p2:else fi:od: for p from 2 to k do:x:= T[p+1,1]- T[p,2]: printf(`%d, `,(x-1)/3):od:
  • Mathematica
    Join[{0},Rest[(#[[2]]-#[[1]]-1)/3&/@Partition[Rest[Flatten[Select[ Partition[ Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&]]],2]]] (* Harvey P. Dale, Jan 10 2016 *)

Formula

a(n) = (A063091(n+1)- A063091(n)-3)/3 = A204099(n)/3

A211005 Pair (i, j) where i = number of adjacent nonprimes and j = number of adjacent primes.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 5, 1, 3, 1, 1, 1, 3, 1, 5, 1, 5, 1, 1, 1, 5, 1, 3, 1, 1, 1, 5, 1, 3, 1, 5, 1, 7, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 13, 1, 3, 1, 5, 1, 1, 1, 9, 1, 1, 1, 5, 1, 5, 1, 3, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Aug 11 2012

Keywords

Comments

Also number of consecutive occurrences of n-1 in A069754. - Reinhard Zumkeller, Dec 04 2012
Run lengths of A010051. - Paolo Xausa, Jan 17 2023

Examples

			----------------------------------------------------------
.     Array from              Number of   Number of
n      A000027                nonprimes    primes    a(n)
----------------------------------------------------------
1         1;                      1          0        1
2         2, 3;                   0          2        2
3         4;                      1          0        1
4         5;                      0          1        1
5         6;                      1          0        1
6         7;                      0          1        1
7         8, 9, 10;               3          0        3
8        11;                      0          1        1
9        12;                      1          0        1
10       13;                      0          1        1
11       14, 15, 16;              3          0        3
12       17;                      0          1        1
13       18;                      1          0        1
14       19;                      0          1        1
15       20, 21, 22;              3          0        3
16       23;                      0          1        1
17       24, 25, 26, 27, 28;      5          0        5
18       29;                      0          1        1
19       30;                      1          0        1
20       31;                      0          1        1

Links

Crossrefs

1 together with A162154.
Cf. A000040, A001223, A010051, A018252, A046933, A211006, A211007.

Programs

  • Haskell
    import Data.List (group)
a211005 n = a211005_list !! (n-1)
a211005_list = map length $ group a069754_list
-- Reinhard Zumkeller, Dec 04 2012
  • Mathematica
    A211005[upto_]:=Map[Length, Most[Split[PrimeQ[Range[upto]]]]];
A211005[500] (* Paolo Xausa, Jan 17 2023 *)

Formula

a(n) = A162154(n-1), n >= 2.

A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

Original entry on oeis.org

0, 2, 1, 3, 12, 1, 12, 11, 18, 3, 3, 85, 29, 64, 1, 17, 23, 187, 47, 36, 3, 9, 97, 19, 931, 53, 106, 1, 50, 17, 291, 75, 733, 71, 54, 3, 36, 504, 35, 889, 31, 2533, 77, 148, 5, 21, 121, 1620, 65, 1011, 111, 1639, 187, 288, 1, 3, 171, 505, 11840, 59, 2197, 119, 4927, 179, 90, 5
Offset: 1

Views

Author

Antti Karttunen, May 21 2024

Keywords

Examples

			The top left corner of the array:
k=   1    2    3      4    5      6    7       8      9     10   11      12
2k=  2    4    6      8   10     12   14      16     18     20   22      24
---+-------------------------------------------------------------------------
1  | 0,   2,   3,    12,   3,    17,   9,     50,    36,    21,   3,     75,
2  | 1,  12,  11,    85,  23,    97,  17,    504,   121,   171,  29,    635,
3  | 1,  18,  29,   187,  19,   291,  35,   1620,   505,   265,  25,   2525,
4  | 3,  64,  47,   931,  75,   889,  65,  11840,   795,  1259,  93,  12503,
5  | 1,  36,  53,   733,  31,  1011,  59,  12456,  1561,   817,  89,  16853,
6  | 3, 106,  71,  2533, 111,  2197, 157,  52580,  1839,  2987, 107,  50507,
7  | 1,  54,  77,  1639, 119,  2163,  49,  41580,  3193,  3101, 127,  53357,
8  | 3, 148, 187,  4927, 113,  6197, 211, 142280,  8283,  4969, 183, 179083,
9  | 5, 288, 179, 11669, 305,  9481, 277, 414720,  6965, 13421, 239, 374459,
10 | 1,  90, 187,  4531, 131,  7685,  73, 190980, 12649,  6303, 137, 293947,
11 | 5, 376, 301, 19869, 247, 18395, 331, 919856, 17173, 17161, 425, 906981,
12 | 3, 274, 167, 16861, 255, 13189, 349, 899540, 10335, 17099, 367, 777083,

Links

Crossrefs

Cf. A000203, A003961, A246278, A286385, A355927.
Cf. A046933 (column 1).
Cf. also A355924, A372562.

Programs

  • PARI
    up_to = 66;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A286385(n) = (A003961(n)-sigma(n));
A372563sq(row,col) = A286385(A246278sq(row,col));
A372563list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372563sq(col,(a-(col-1))))); (v); };
v372563 = A372563list(up_to);
A372563(n) = v372563[n];

Formula

A(n, k) = A286385(A246278(n, k)) = A246278(1+n, k) - A355927(n, k).

A373826 Sorted positions of first appearances in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

Original entry on oeis.org

1, 4, 38, 6781, 23238, 26100
Offset: 1

Views

Author

Gus Wiseman, Jun 22 2024

Keywords

Comments

Sorted positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

Examples

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with sorted positions of first appearances a(n).

Crossrefs

Sorted positions of first appearances in A373820, cf. A027833.
For runs we have A373824 (unsorted A373825), sorted firsts of A373819.
The unsorted version is A373827.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A001359, A006512, A025584, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

Programs

  • Mathematica
    t=Length/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ],#1+2!=#2&]];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A373827 Position of first appearance of n in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

Original entry on oeis.org

4, 1, 38, 6781, 26100, 23238
Offset: 1

Views

Author

Gus Wiseman, Jun 22 2024

Keywords

Comments

Positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

Examples

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with positions of first appearances a(n).

Crossrefs

Positions of first appearances in A373820.
For runs instead of antiruns we have A373825, sorted A373824.
The sorted version is A373826.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A001359, A006512, A025584, A027833, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

Programs

  • Mathematica
    t=Length/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ],#1+2!=#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A377433 Number of non-perfect-powers x in the range prime(n) < x < prime(n+1).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 3, 3, 1, 3, 3, 1, 3, 4, 5, 1, 4, 3, 1, 5, 2, 5, 7, 2, 1, 3, 1, 3, 11, 2, 5, 1, 8, 1, 5, 5, 3, 4, 5, 1, 9, 1, 2, 1, 11, 10, 2, 1, 3, 5, 1, 8, 4, 5, 5, 1, 5, 3, 1, 8, 13, 3, 1, 3, 12, 5, 8, 1, 3, 5, 6, 5, 5, 3, 5, 7, 2, 7, 9, 1, 9, 1, 5, 2
Offset: 1

Views

Author

Gus Wiseman, Nov 02 2024

Keywords

Comments

Non-perfect-powers (A007916) are numbers without a proper integer root.
Positions of terms > 1 appear to be A049579.

Examples

			Between prime(4) = 7 and prime(5) = 11 the only non-perfect-power is 10, so a(4) = 1.

Crossrefs

Positions of 1 are latter terms of A029707.
Positions of terms > 1 appear to be A049579.
For prime-powers instead of non-perfect-powers we have A080101.
For non-prime-powers instead of non-perfect-powers we have A368748.
Perfect-powers in the same range are counted by A377432.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect-powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706.
A065514 gives the greatest prime-power < prime(n), difference A377289.
A081676 gives the greatest perfect-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053706, A053607, A304521, A377286.
A377468 gives the least perfect-power > n.
Cf. A000015, A002808, A024619, A031218, A064113, A065890, A244508, A377282, A377434, A377436, A377466, A377467.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Table[Length[Select[Range[Prime[n]+1, Prime[n+1]-1],radQ]],{n,100}]

Formula

a(n) + A377432(n) = A046933(n) = prime(n+1) - prime(n) - 1.

A126436 Number of composites between successive values of A014612.

Original entry on oeis.org

2, 3, 0, 5, 0, 0, 8, 0, 0, 3, 1, 7, 2, 0, 1, 2, 0, 1, 10, 4, 0, 1, 1, 2, 2, 1, 0, 6, 0, 3, 5, 7, 0, 2, 0, 7, 0, 3, 0, 0, 0, 0, 4, 3, 1, 1, 2, 9, 3, 9, 4, 0, 3, 1, 1, 1, 0, 0, 7, 1, 2, 3, 1, 2, 1, 2, 1, 0, 0, 0, 3, 1
Offset: 1

Views

Author

Jonathan Vos Post, Mar 12 2007

Keywords

Examples

			a(1) = 2 because there are two composites {9,10} between A014612(1)=8 and A014612(2)=12.
a(2) = 3 because there are two composites {14, 15, 16} between A014612(2)=12 and A014612(3)=18.
a(3) = 0 because there are no composites between A014612(3)=18 and A014612(4)=20, only the prime 19.
a(7) = 8 because {32,33,34,35,36,38,39,40} between A014612(7)=30 and A014612(8)=42.

Crossrefs

3-almost prime analog of A046933 = number of composites between successive primes.
Cf. A002808, A014612, A046933, A070552, A114403.

Programs

  • Maple
    isA014612 :=proc(n) if numtheory[bigomega](n) = 3 then true ; else false ; fi ; end: isA002808 := proc(n) RETURN(not isprime(n) and n <> 1 ); end: A126436 := proc(nmax) local a ; a := -1 ; for n from 1 to nmax do if isA014612(n) then if a >= 0 then printf("%d,",a) ; fi ; a := 0 ; elif isA002808(n) and a>= 0 then a := a+1 ; fi ; od : end: A126436(300) : # R. J. Mathar, Apr 03 2007
  • Mathematica
    nmax = 72;
S = Select[Range[300](* increase range if a(n) unevaluated *), PrimeOmega[#] == 3&];
a[n_ /; n+1 <= Length[S]] := Count[Range[S[[n]]+1, S[[n+1]]-1], _?CompositeQ];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 26 2023 *)

Formula

a(n) <= A114403(n) - 1.

Extensions

More terms from R. J. Mathar, Apr 03 2007

A209618 Primes separated from their adjacent next primes by a composite number of successive composites.

Original entry on oeis.org

139, 181, 241, 283, 337, 409, 421, 547, 577, 631, 691, 709, 787, 811, 829, 919, 1021, 1039, 1051, 1129, 1153, 1171, 1249, 1327, 1399, 1471, 1627, 1699, 1723, 1801, 1831, 1879, 1933, 1951, 2017, 2029, 2053, 2089, 2113, 2143, 2221, 2251, 2311, 2477, 2521, 2557
Offset: 1

Views

Author

Lekraj Beedassy, Mar 21 2012

Keywords

Comments

Primes prime(k) such that A046933(k) is composite. - Robert Israel, Jan 08 2025

Examples

			a(1) = 139 is the first prime separated from the next prime (149) by a composite number (9) of successive composites, namely, 140, 141, 142, 143, 144, 145, 146, 147, 148.

Links

Crossrefs

Cf. A046933, A209619.

Programs

A373817 Positions of terms > 1 in the run-lengths of the first differences of the odd primes.

Original entry on oeis.org

2, 14, 34, 36, 42, 49, 66, 94, 98, 100, 107, 117, 147, 150, 169, 171, 177, 181, 199, 219, 250, 268, 315, 333, 361, 392, 398, 435, 477, 488, 520, 565, 570, 585, 592, 595, 628, 642, 660, 666, 688, 715, 744, 765, 772, 778, 829, 842, 897, 906, 931, 932, 961, 1025
Offset: 1

Views

Author

Gus Wiseman, Jun 23 2024

Keywords

Comments

Positions of terms > 1 in A333254. In other words, the a(n)-th run of differences of odd primes has length > 1.

Examples

			Primes 54 to 57 are {251, 257, 263, 269}, with differences (6,6,6). This is the 49th run, and the first of length > 2.

Crossrefs

Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Positions of terms > 1 in A333254, run-lengths A373821, firsts A335406.
A000040 lists the primes, differences A001223.
A027833 gives antirun lengths of odd primes, run-lengths A373820.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A006560, A006562, A029707, A031217, A037201, A038664, A069010, A073051, A089180, A251092.

Programs

  • Mathematica
    Join@@Position[Length /@ Split[Differences[Select[Range[1000],PrimeQ]]] // Most,x_Integer?(#>1&)]
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