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 11-17 of 17 results.

A319613 a(n) = prime(n) * prime(2n).

Original entry on oeis.org

6, 21, 65, 133, 319, 481, 731, 1007, 1403, 2059, 2449, 3293, 4141, 4601, 5311, 6943, 8201, 9211, 10921, 12283, 13213, 15247, 16517, 19847, 22213, 24139, 25853, 28141, 29539, 31753, 37211, 40741, 43429, 46843, 52001, 54209, 58561, 62429, 66299, 70757, 75359
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2019

Keywords

Links

Crossrefs

Cf. A000040, A001358, A018819, A031215, A031368, A056239, A087897, A101417, A112798, A120641, A320340, A323092, A323093, A323094.

Programs

  • Maple
    a:= n-> (p-> p(n)*p(2*n))(ithprime):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 08 2019
  • Mathematica
    Table[Prime[n]*Prime[2*n],{n,50}]
  • PARI
    a(n) = prime(n)*prime(2*n) \\ Felix Fröhlich, Jan 09 2019

A323053 Number of integer partitions of n with no 1's such that no part is a power of any other (unequal) part.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 25, 30, 38, 47, 58, 71, 87, 106, 131, 156, 190, 228, 275, 328, 394, 468, 556, 661, 784, 923, 1089, 1283, 1507, 1766, 2068, 2416, 2821, 3284, 3822, 4438, 5148, 5961, 6898, 7968, 9195, 10593, 12198, 14019, 16102, 18472
Offset: 0

Views

Author

Gus Wiseman, Jan 04 2019

Keywords

Examples

			The a(2) = 1 through a(11) = 12 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)      (B)
            (22)  (32)  (33)   (43)   (44)    (54)    (55)     (65)
                        (222)  (52)   (53)    (63)    (64)     (74)
                               (322)  (62)    (72)    (73)     (83)
                                      (332)   (333)   (433)    (92)
                                      (2222)  (522)   (532)    (443)
                                              (3222)  (622)    (533)
                                                      (3322)   (632)
                                                      (22222)  (722)
                                                               (3332)
                                                               (5222)
                                                               (32222)

Crossrefs

Cf. A001597, A002865, A007916, A052410, A101417, A102430, A108917, A305148, A305630, A305631, A321346, A323093.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],stableQ[#,IntegerQ[Log[#1,#2]]&]]&]],{n,30}]

A371449 Numbers whose prime indices are not powers of 2.

Original entry on oeis.org

1, 5, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 211, 215
Offset: 1

Views

Author

Gus Wiseman, Mar 31 2024

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
     1: {}        85: {3,7}      169: {6,6}     253: {5,9}
     5: {3}       89: {24}       173: {40}      257: {55}
    11: {5}       97: {25}       179: {41}      263: {56}
    13: {6}      101: {26}       181: {42}      269: {57}
    17: {7}      103: {27}       185: {3,12}    271: {58}
    23: {9}      107: {28}       187: {5,7}     275: {3,3,5}
    25: {3,3}    109: {29}       191: {43}      277: {59}
    29: {10}     113: {30}       193: {44}      281: {60}
    31: {11}     115: {3,9}      197: {45}      283: {61}
    37: {12}     121: {5,5}      199: {46}      289: {7,7}
    41: {13}     125: {3,3,3}    205: {3,13}    293: {62}
    43: {14}     127: {31}       211: {47}      295: {3,17}
    47: {15}     137: {33}       215: {3,14}    299: {6,9}
    55: {3,5}    139: {34}       221: {6,7}     305: {3,18}
    59: {17}     143: {5,6}      223: {48}      307: {63}
    61: {18}     145: {3,10}     227: {49}      313: {65}
    65: {3,6}    149: {35}       229: {50}      317: {66}
    67: {19}     151: {36}       233: {51}      319: {5,10}
    71: {20}     155: {3,11}     235: {3,15}    325: {3,3,6}
    73: {21}     157: {37}       239: {52}      331: {67}
    79: {22}     163: {38}       241: {53}      335: {3,19}
    83: {23}     167: {39}       251: {54}      337: {68}

Crossrefs

Partitions of this type are counted by A101417.
For binary indices instead of prime indices we have A326781.
Requiring powers of two gives A318400, for binary indices A253317.
An opposite version is A371443.
For primes instead of powers of 2 we have A320628.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A057716 lists non-powers of 2.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
Cf. A005117, A326782, A371451, A371444.

Programs

  • Mathematica
    Select[Range[100],And@@Not/@IntegerQ/@Log[2, PrimePi/@First/@FactorInteger[#]]&]

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23
Offset: 1

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Examples

			24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

Crossrefs

Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

Programs

  • Mathematica
    nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A322547 Numbers k such that every integer partition of k contains a 1, a squarefree number, or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 79
Offset: 1

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Examples

			48 does not belong to the sequence because there are integer partitions of 48 containing no 1's, squarefree numbers, or prime powers, namely: (48), (36,12), (28,20), (24,24), (24,12,12), (18,18,12), (12,12,12,12).

Crossrefs

Cf. A000607, A002095, A005117, A023893, A023894, A064573, A078135, A101417, A246655, A322546.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A308558 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 4, 2, 2, 2, 1, 6, 3, 2, 2, 2, 1, 6, 3, 2, 2, 2, 2, 1, 10, 3, 3, 2, 2, 2, 2, 1, 10, 5, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 1, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 20, 7, 4, 3, 3, 2
Offset: 1

Views

Author

Gus Wiseman, Jun 07 2019

Keywords

Examples

			Triangle begins:
  1
  1  2
  1  2  2
  1  4  2  2
  1  4  2  2  2
  1  6  3  2  2  2
  1  6  3  2  2  2  2
  1 10  3  3  2  2  2  2
  1 10  5  3  2  2  2  2  2
  1 14  5  3  3  2  2  2  2  2
  1 14  5  3  3  2  2  2  2  2  2
  1 20  7  4  3  3  2  2  2  2  2  2
  1 20  7  4  3  3  2  2  2  2  2  2  2
Row n = 6 counts the following partitions:
  (111111)  (42)      (33)      (411)     (51)      (6)
            (222)     (3111)    (111111)  (111111)  (111111)
            (411)     (111111)
            (2211)
            (21111)
            (111111)

Crossrefs

Same as A102430 except for the k = 1 column.
Row sums are A102431(n) + 1.
Column k = 2 is A018819.
Column k = 3 is A062051.
Cf. A000961, A001597, A007916, A008284, A023894, A052410, A101417, A112344, A323053, A323054.

Programs

  • Mathematica
    Table[If[k==1,1,Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]]],{n,10},{k,n}]

A323086 Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 3, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 4, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 9, 3, 2, 1, 11, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Jan 04 2019

Keywords

Examples

			The a(72) = 14 factorizations:
  (2*2*2*3*3),
  (2*2*2*9), (2*2*3*6),
  (2*2*18), (2*3*12), (2*6*6), (3*3*8), (3*4*6),
  (2*36), (3*24), (4*18), (6*12), (8*9),
  (72).

Crossrefs

Cf. A001055, A001597, A002865, A007916, A052410, A101417, A292886, A303707, A305149, A323053, A323087.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],stableQ[Union[#],IntegerQ[Log[#1,#2]]&]&]],{n,100}]
Previous Showing 11-17 of 17 results.

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