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.

Previous Showing 11-20 of 34 results. Next

A377033 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16
Offset: 0

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Comments

Row n is the k-th differences of A002808 = the composite numbers.

Examples

			Array begins:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   4     6     8     9    10    12    14    15    16
  k=1:   2     2     1     1     2     2     1     1     2
  k=2:   0    -1     0     1     0    -1     0     1     0
  k=3:  -1     1     1    -1    -1     1     1    -1    -1
  k=4:   2     0    -2     0     2     0    -2     0     2
  k=5:  -2    -2     2     2    -2    -2     2     2    -2
  k=6:   0     4     0    -4     0     4     0    -4    -1
  k=7:   4    -4    -4     4     4    -4    -4     3    10
  k=8:  -8     0     8     0    -8     0     7     7   -29
  k=9:   8     8    -8    -8     8     7     0   -36    63
Triangle begins:
    4
    6    2
    8    2    0
    9    1   -1   -1
   10    1    0    1    2
   12    2    1    1    0   -2
   14    2    0   -1   -2   -2    0
   15    1   -1   -1    0    2    4    4
   16    1    0    1    2    2    0   -4   -8
   18    2    1    1    0   -2   -4   -4    0    8
   20    2    0   -1   -2   -2    0    4    8    8    0
   21    1   -1   -1    0    2    4    4    0   -8  -16  -16

Crossrefs

Initial rows: A002808, A073783, A073445.
The version for primes is A095195 or A376682.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377034, absolute version A377035.
Column n = 1 is A377036, for primes A007442 or A030016.
First position of 0 in each row is A377037.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
Cf. A065310, A065890, A084758, A173390, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Programs

  • Mathematica
    nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,PrimeQ]&,4,2*nn],k],nn],{k,0,nn}]

Formula

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2024

Keywords

Examples

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is not in the sequence.

Crossrefs

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of 0 in A080101, or 1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For at least one prime-power we have A377057.
For one instead of no prime-powers we have A377287.
For two instead of no prime-powers we have A377288.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A276781, A376597, A377051, A377282.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]
  • Python
    from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024

A376682 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the noncomposite numbers (A008578).

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 7, 2, 0, -1, -2, 11, 4, 2, 2, 3, 5, 13, 2, -2, -4, -6, -9, -14, 17, 4, 2, 4, 8, 14, 23, 37, 19, 2, -2, -4, -8, -16, -30, -53, -90, 23, 4, 2, 4, 8, 16, 32, 62, 115, 205, 29, 6, 2, 0, -4, -12, -28, -60, -122, -237, -442, 31, 2, -4, -6, -6, -2, 10, 38, 98, 220, 457, 899
Offset: 0

Views

Author

Gus Wiseman, Oct 15 2024

Keywords

Comments

Row k is the k-th differences of the noncomposite numbers.

Examples

			Array begins:
         n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  -----------------------------------------------------------
  k=0:    1     2     3     5     7    11    13    17    19
  k=1:    1     1     2     2     4     2     4     2     4
  k=2:    0     1     0     2    -2     2    -2     2     2
  k=3:    1    -1     2    -4     4    -4     4     0    -6
  k=4:   -2     3    -6     8    -8     8    -4    -6    14
  k=5:    5    -9    14   -16    16   -12    -2    20   -28
  k=6:  -14    23   -30    32   -28    10    22   -48    48
  k=7:   37   -53    62   -60    38    12   -70    96   -70
  k=8:  -90   115  -122    98   -26   -82   166  -166    86
  k=9:  205  -237   220  -124   -56   248  -332   252   -86
Triangle begins:
    1
    2    1
    3    1    0
    5    2    1    1
    7    2    0   -1   -2
   11    4    2    2    3    5
   13    2   -2   -4   -6   -9  -14
   17    4    2    4    8   14   23   37
   19    2   -2   -4   -8  -16  -30  -53  -90
   23    4    2    4    8   16   32   62  115  205
   29    6    2    0   -4  -12  -28  -60 -122 -237 -442
   31    2   -4   -6   -6   -2   10   38   98  220  457  899

Crossrefs

The version for modern primes (A000040) is A095195.
Initial rows: A008578, A075526, A036263 with 0 prepended.
Column n = 1 is A030016 (modern A007442).
A version for partitions is A175804, cf. A053445, A281425, A320590.
Antidiagonal-sums are A376683 (modern A140119), absolute A376684 (modern A376681).
First position of 0 is A376855 (modern A376678).
For composite instead of prime we have A377033.
For squarefree instead of prime we have A377038, nonsquarefree A377046.
For prime-power instead of composite we have A377051.
A000040 lists the primes, differences A001223, second A036263.
Cf. A002808, A064113, A065890, A084758, A173390, A233671, A258025, A258026, A333254, A376602 (zeros), A376651 (positives), A376652 (negatives), A376680, A377037.

Programs

  • Mathematica
    nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,!PrimeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
(* or *)
nn=12;
q=Table[If[n==0,1,Prime[n]],{n,0,2nn}];
Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,nn},{i,nn}]

Formula

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A008578(i+k).

A379315 Number of strict integer partitions of n with a unique 1 or prime part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 3, 1, 3, 2, 7, 3, 7, 4, 10, 7, 15, 7, 17, 13, 23, 16, 31, 20, 37, 31, 48, 38, 62, 48, 76, 68, 93, 80, 119, 105, 147, 137, 175, 166, 226, 208, 267, 263, 326, 322, 407, 391, 481, 492, 586, 591, 714, 714, 849, 884, 1020, 1050, 1232, 1263
Offset: 0

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Comments

The "old" primes are listed by A008578.

Examples

			The a(10) = 2 through a(15) = 10 partitions:
  (8,2)  (11)     (9,3)    (13)     (9,5)    (8,7)
  (9,1)  (6,5)    (10,2)   (7,6)    (12,2)   (10,5)
         (7,4)    (6,4,2)  (8,5)    (8,4,2)  (11,4)
         (8,3)             (10,3)   (9,4,1)  (12,3)
         (9,2)             (12,1)            (14,1)
         (10,1)            (6,4,3)           (6,5,4)
         (6,4,1)           (8,4,1)           (8,4,3)
                                             (8,6,1)
                                             (9,4,2)
                                             (10,4,1)

Links

Crossrefs

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
For a unique composite part we have A379303, non-strict A379302 (ranks A379301).
Considering 1 nonprime gives A379305, non-strict A379304 (ranks A331915).
For squarefree instead of old prime we have A379309, non-strict A379308 (ranks A379316).
Ranked by A379312 /\ A005117 = squarefree positions of 1 in A379311.
The non-strict version is A379314.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A376682 gives k-th differences of old primes.
Cf. A000070, A023895, A034891, A036497, A038348, A095195, A175804, A204389, A257994, A302540.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]
  • PARI
    seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k)) * prod(k=4, n, 1 + if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024

A379304 Number of integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014
Offset: 0

Views

Author

Gus Wiseman, Dec 27 2024

Keywords

Examples

			The a(2) = 1 through a(9) = 9 partitions:
  (2)  (3)   (31)   (5)     (42)     (7)       (62)       (54)
       (21)  (211)  (311)   (51)     (43)      (71)       (63)
                    (2111)  (3111)   (421)     (431)      (621)
                            (21111)  (511)     (4211)     (711)
                                     (31111)   (5111)     (4311)
                                     (211111)  (311111)   (42111)
                                               (2111111)  (51111)
                                                          (3111111)
                                                          (21111111)

Crossrefs

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A331915 = positions of one in A257994.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379305.
For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
Considering 1 prime gives A379314 (strict A379315), ranks A379312.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Count[#,_?PrimeQ]==1&]],{n,0,30}]

A379305 Number of strict integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866
Offset: 0

Views

Author

Gus Wiseman, Dec 27 2024

Keywords

Examples

			The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
  (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)
       (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)
                              (421)  (431)  (621)  (631)  (74)   (B1)
                                                          (83)   (642)
                                                          (92)   (651)
                                                          (821)  (741)
                                                                 (831)
                                                                 (921)

Crossrefs

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.
For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).
The non-strict version is A379304.
For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).
Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?PrimeQ]==1&]],{n,0,30}]

A376683 Antidiagonal-sums of the array A376682(n,k) = n-th term of the k-th differences of the noncomposite numbers (A008578).

Original entry on oeis.org

1, 3, 4, 9, 6, 27, -20, 109, -182, 471, -868, 1737, -2872, 4345, -4700, 1133, 14060, -55275, 150462, -346093, 717040, -1369351, 2432872, -4002905, 5964846, -7524917, 6123130, 4900199, -40900410, 134309057, -348584552, 798958881, -1678213106, 3277459119
Offset: 0

Views

Author

Gus Wiseman, Oct 15 2024

Keywords

Examples

			The fourth anti-diagonal of A376682 is: (7, 2, 0, -1, -2), so a(4) = 6.

Crossrefs

The modern version (for A000040 instead of A008578) is A140119.
The absolute version is A376681.
Antidiagonal-sums of A376682 (modern version A095195).
For composite instead of noncomposite we have A377033.
For squarefree instead of noncomposite we have A377038, nonsquarefree A377046.
A000040 lists the modern primes, differences A001223, second A036263.
A008578 lists the noncomposites, first differences A075526.
Cf. A007442, A030016, A064113, A065890, A002808, A173390, A233671, A333214, A333254, A376678, A376684, A376855.

Programs

  • Mathematica
    nn=12;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!PrimeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 8, 14, 14, 11, 24, 10, 20, 37, -10, 56, 26, -52, 260, -659, 2393, -8128, 25703, -72318, 184486, -430901, 933125, -1888651, 3597261, -6479654, 11086964, -18096083, 28307672, -42644743, 62031050, -86466235, 110902085, -110907437, 52379, 483682985
Offset: 1

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Comments

Row-sums of the triangle version of A377033.

Examples

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 8.

Crossrefs

The version for prime instead of composite is A140119, noncomposite A376683.
This is the antidiagonal-sums of the array A377033, absolute version A377035.
For squarefree instead of composite we have A377039, absolute version A377040.
For nonsquarefree instead of composite we have A377047, absolute version A377048.
For prime-power instead of composite we have A377052, absolute version A377053.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

Programs

  • Mathematica
    q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[t[[j,i-j+1]],{i,Length[q]/2},{j,i}]

A377037 Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.

Original entry on oeis.org

1, 14, 2, 65, 1, 83, 2, 7, 1, 83, 2, 424, 12, 32, 11, 733, 10, 940, 9, 1110, 8, 1110, 7, 1110, 6, 1110, 112, 1110, 111, 1110, 110, 2192, 109, 13852, 108, 13852, 107, 13852, 106, 13852, 105, 17384, 104, 17384, 103, 17384, 102, 17384, 101, 27144, 552, 28012, 551
Offset: 2

Views

Author

Gus Wiseman, Oct 17 2024

Keywords

Examples

			The third differences of the composite numbers are:
  -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -2, 1, 0, 0, 1, -1, -1, ...
so a(3) = 14.

Links

Crossrefs

The version for prime instead of composite is A376678.
For noncomposite numbers we have A376855.
This is the first position of 0 in row n of the array A377033.
For squarefree instead of composite we have A377042, nonsquarefree A377050.
For prime-power instead of composite we have A377055.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
A377036 gives first term of the n-th differences of the composite numbers, for primes A007442 or A030016.
Cf. A018252, A064113, A065310, A065890, A140119, A173390, A233671, A258025, A258026, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377034, A377035.

Programs

  • Mathematica
    nn=10000;
u=Table[Differences[Select[Range[nn],CompositeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]

Extensions

Offset 2 from Michel Marcus, Oct 18 2024
a(17)-a(54) from Alois P. Heinz, Oct 18 2024

A377288 Numbers k such that there are exactly two prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

4, 9, 30, 327, 3512
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2024

Keywords

Comments

Is this sequence finite? For this conjecture see A053706, A080101, A366833.
Any further terms are > 10^12. - Lucas A. Brown, Nov 08 2024

Examples

			Primes 9 and 10 are 23 and 29, and the interval (24, 25, 26, 27, 28) contains the prime-powers 25 and 27, so 9 is in the sequence.

Links

Crossrefs

The interval from A008864(n) to A006093(n+1) has A046933 elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The corresponding primes are A053706.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
These are the positions of 2 in A080101, or 3 in A366833.
For at least one prime-power we have A377057, primes A053607.
For no prime-powers we have A377286.
For exactly one prime-power we have A377287.
For squarefree instead of prime-power see A377430, A061398, A377431, A068360.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A224363, A276781, A376596, A376597, A377282.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==2&]

Formula

prime(a(n)) = A053706(n).
Previous Showing 11-20 of 34 results. Next

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

Content is available under The OEIS End-User License Agreement.