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.

Showing 1-10 of 10 results.

A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

Original entry on oeis.org

-2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0
Offset: 1

Views

Author

Gus Wiseman, Oct 02 2024

Keywords

Comments

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, shift left once.

Examples

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...

Crossrefs

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375735, ones A375713(n) - 1.
Positions of zeros are A376600, complement A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A057820 gives first differences of prime-powers inclusive, first appearances A376341, sorted A376340.
A321346/A321378 count integer partitions without prime-powers, factorizations A322452.
For non-prime-powers: A024619/A361102 (terms), A375735/A375708 (first differences), A376600 (inflections and undulations), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Programs

  • Mathematica
    Differences[Select[Range[100],!(#==1||PrimePowerQ[#])&],2]
  • Python
    from sympy import primepi, integer_nthroot
def A376599(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A101417 Number of partitions of n into parts without powers of 2.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 3, 6, 5, 6, 10, 9, 12, 17, 17, 22, 28, 30, 37, 48, 52, 62, 78, 86, 103, 127, 141, 166, 201, 227, 266, 317, 358, 417, 492, 560, 647, 757, 860, 991, 1153, 1309, 1503, 1738, 1971, 2257, 2594, 2941, 3356, 3843, 4351, 4948, 5644, 6382, 7240
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 16 2005

Keywords

Examples

			a(12) = #{3+3+3+3, 6+3+3, 6+6, 7+5, 9+3, 12} = 6.
From _Gus Wiseman_, Jan 07 2019: (Start)
The a(3) = 1 through a(14) = 5 integer partitions (A = 10, ..., E = 14):
  (3)  (5)  (6)   (7)  (53)  (9)    (A)   (B)    (C)     (D)    (E)
            (33)             (63)   (55)  (65)   (66)    (76)   (77)
                             (333)  (73)  (533)  (75)    (A3)   (95)
                                                 (93)    (553)  (B3)
                                                 (633)   (733)  (653)
                                                 (3333)         (5333)
(End)

Links

Crossrefs

Cf. A000041, A018819, A000123.
Cf. A087897, A102430, A276431, A321346, A323053, A323092, A323093.

Programs

  • Maple
    g:= product(1-x^(2^j),j=0..15)/product(1-x^i,i=1..75): gser:= series(g, x=0,62): seq(coeff(gser,x,n),n=0..59); # Emeric Deutsch, Mar 29 2006
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And@@Not/@IntegerQ/@Log[2,#]&]],{n,20}] (* Gus Wiseman, Jan 07 2019 *)

Formula

G.f.: Product_{j>=1} (1-x^(2^j)) / Product_{i>=2} (1-x^i). - Emeric Deutsch, Mar 29 2006

A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 55, 56, 57, 58, 59, 62, 64, 66, 68, 70, 73, 74, 75, 76, 77, 80, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 107, 108, 109, 112, 114, 116
Offset: 1

Views

Author

Gus Wiseman, Oct 05 2024

Keywords

Comments

These are points at which the second differences (A376599) are zero.
Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, add 1 to all terms.

Examples

			The non-prime-powers inclusive are (A024619):
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with zeros at (A376600):
  2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, ...

Links

Crossrefs

For first differences we had A375735, ones A375713(n)-1.
These are the zeros of A376599.
The complement is A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A024619/A361102 list non-prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A057820, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2],0]

A321378 Number of integer partitions of n containing no 1's or prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 3, 2, 3, 0, 6, 1, 5, 3, 6, 1, 11, 2, 9, 6, 12, 5, 19, 4, 17, 11, 23, 9, 32, 10, 31, 22, 39, 17, 55, 21, 57, 37, 67, 33, 92, 44, 97, 65, 114, 63, 154, 78, 162, 113, 191, 117, 250, 138, 269, 194, 320
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Examples

			The a(30) = 11 integer partitions:
  (30)
  (24,6)
  (15,15)
  (18,12)
  (20,10)
  (18,6,6)
  (12,12,6)
  (14,10,6)
  (10,10,10)
  (12,6,6,6)
  (6,6,6,6,6)

Crossrefs

Cf. A000607, A000688, A000961, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321347, A321665, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1/(1-x^n)],{n,2,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, 38, 40, 42, 46, 49, 53, 54, 60, 61, 63, 65, 67, 69, 71, 72, 78, 79, 81, 82, 83, 84, 85, 89, 91, 99, 101, 105, 106, 110, 111, 113, 115, 117, 118, 122, 124, 132, 134, 136, 138, 148
Offset: 1

Views

Author

Gus Wiseman, Oct 05 2024

Keywords

Comments

These are points at which the second differences (A376599) are nonzero.
Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, subtract 1 and shift left.

Examples

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with nonzero terms (A376601) at:
  1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, ...

Links

Crossrefs

For first differences we had A375735, ones A375713(n) - 1.
These are the nonzeros of A376599.
The complement is A376600.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A024619/A361102 list non-prime-powers inclusive.
A057820 gives first differences of prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376600 (inflections and undulations).
For nonzero curvature: A333214 (prime), A376603 (composite), A376588 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Programs

  • Mathematica
    Join@@Position[Sign[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2]],1|-1]

A321347 Number of strict integer partitions of n containing no prime powers (including 1).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 4, 4, 2, 3, 4, 4, 5, 6, 5, 6, 7, 7, 9, 10, 10, 13, 12, 11, 15, 17, 16, 19, 20, 20, 25, 28, 26, 30, 33, 35, 41, 43, 42, 50, 55, 57, 64, 67, 67, 79, 86, 87, 97, 105, 109, 124, 131, 135, 151, 163, 169
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

First differs from A286221 at a(30) = 6, A286221(30) = 5.

Examples

			The a(36) = 13 strict integer partitions:
  (36),
  (21,15), (22,14), (24,12), (26,10), (30,6), (35,1),
  (14,12,10), (18,12,6), (20,10,6), (20,15,1), (21,14,1),
  (15,14,6,1).

Links

Crossrefs

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321378, A321665, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A321665 Number of strict integer partitions of n containing no 1's or prime powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 2, 2, 0, 3, 1, 3, 2, 4, 1, 5, 2, 5, 4, 6, 4, 9, 3, 8, 7, 10, 6, 13, 7, 13, 12, 16, 10, 20, 13, 22, 19, 24, 18, 32, 23, 34, 30, 37, 30, 49, 37, 50, 47, 58, 51, 73, 58, 77, 74, 89, 80, 108, 91, 116
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Examples

			The a(36) = 9 strict integer partitions:
  (36)
  (30,6)
  (21,15)
  (22,14)
  (24,12)
  (26,10)
  (18,12,6)
  (20,10,6)
  (14,12,10)

Links

Crossrefs

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A321346, A321347, A321378, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,2,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

Formula

G.f.: Product_{k>=2, k not a prime power} 1 + x^k. - Joerg Arndt, Dec 22 2020

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}]

A323088 Number of strict integer partitions of n using numbers that are not perfect powers.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 7, 9, 11, 11, 15, 16, 18, 22, 24, 27, 32, 34, 41, 45, 51, 59, 64, 75, 82, 94, 105, 116, 132, 146, 163, 183, 202, 225, 251, 277, 309, 341, 378, 417, 463, 510, 564, 622, 685, 754, 830, 914, 1001, 1103, 1207, 1325
Offset: 0

Views

Author

Gus Wiseman, Jan 04 2019

Keywords

Examples

			A list of all strict integer partitions using numbers that are not perfect powers begins:
   2: (2)        11: (6,3,2)    15: (13,2)       17: (12,5)
   3: (3)        12: (12)       15: (12,3)       17: (12,3,2)
   5: (5)        12: (10,2)     15: (10,5)       17: (11,6)
   5: (3,2)      12: (7,5)      15: (10,3,2)     17: (10,7)
   6: (6)        12: (7,3,2)    15: (7,6,2)      17: (10,5,2)
   7: (7)        13: (13)       15: (7,5,3)      17: (7,5,3,2)
   7: (5,2)      13: (11,2)     16: (14,2)       18: (18)
   8: (6,2)      13: (10,3)     16: (13,3)       18: (15,3)
   8: (5,3)      13: (7,6)      16: (11,5)       18: (13,5)
   9: (7,2)      13: (6,5,2)    16: (11,3,2)     18: (13,3,2)
   9: (6,3)      14: (14)       16: (10,6)       18: (12,6)
  10: (10)       14: (12,2)     16: (7,6,3)      18: (11,7)
  10: (7,3)      14: (11,3)     16: (6,5,3,2)    18: (11,5,2)
  10: (5,3,2)    14: (7,5,2)    17: (17)         18: (10,6,2)
  11: (11)       14: (6,5,3)    17: (15,2)       18: (10,5,3)
  11: (6,5)      15: (15)       17: (14,3)       18: (7,6,5)

Crossrefs

Cf. A001597, A007916, A025147, A052410, A087897, A305631, A321346, A323054, A323089, A323090.

Programs

  • Mathematica
    perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&FreeQ[#,1]&&And@@Not/@perpowQ/@#&]],{n,20}]

Formula

O.g.f.: Product_{n in A007916} (1 + x^n).

A321936 Number of integer partitions of n containing no 1's, prime powers, or squarefree numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 2, 0, 3, 1, 1, 0, 7, 0, 2, 0, 5, 0, 5, 0, 7, 1, 3, 0, 12, 0, 4, 2, 10, 1, 8, 0, 14, 2, 6, 0, 22, 1, 10, 3, 20, 1, 15, 0, 26, 5, 12, 2
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

Number of integer partitions of n using elements of A126706.

Examples

			The a(56) = 7 partitions:
  (56)
  (28,28)
  (36,20)
  (44,12)
  (20,18,18)
  (24,20,12)
  (20,12,12,12)

Crossrefs

Cf. A000607, A000961, A002095, A005117, A023893, A023894, A078135, A114374, A126706, A246655, A321346, A321347, A321378, A321665.

Programs

  • Mathematica
    nn=100;
ser=Product[If[n==1||PrimePowerQ[n]||SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]
Showing 1-10 of 10 results.

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