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 31-36 of 36 results.

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Original entry on oeis.org

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304
Offset: 1

Views

Author

Gus Wiseman, Sep 30 2024

Keywords

Comments

These are points at which the second differences are negative.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, downward concavity is positive curvature.
From Robert Israel, Oct 31 2024: (Start)
The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...

Links

Crossrefs

The version for A000002 is A025505, complement A022297. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258026.
For upward concavity we have A376560 (probably the complement).
A000961 lists the prime-powers inclusive, exclusive A246655.
A001597 lists the perfect-powers.
A007916 lists the non-perfect-powers.
A112344 counts partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

Programs

  • Maple
    N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024
  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]

A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84
Offset: 1

Views

Author

Gus Wiseman, Oct 03 2024

Keywords

Comments

These are points at which the second differences (A376562) are zero.
Non-perfect-powers (A007916) are numbers without a proper integer root.

Examples

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with zeros at (A376588):
  3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, ...

Links

Crossrefs

The version for A000002 is empty.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros in A376562, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376589 (nonzero curvature).
For second differences: A064113 (prime), A376602 (composite), {} (perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power inclusive), A376600 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A294068, A336416, A361102, A376599, A376590.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ],2],0]

A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, 106, 107, 109, 110, 111, 112, 126, 127, 150, 151, 176, 177, 195, 196, 203, 204, 220, 221, 232, 233, 264, 265, 298, 299, 316, 317, 333, 334, 371, 372, 411, 412, 453, 454, 480, 481, 496
Offset: 1

Views

Author

Gus Wiseman, Oct 03 2024

Keywords

Comments

These are points at which the second differences (A376562) are nonzero.
Non-perfect-powers (A007916) are numbers without a proper integer root.

Examples

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with nonzeros at (A376589):
  1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, ...

Links

Crossrefs

For first differences we had A375706, ones A375740, complement A375714.
These are the positions of nonzeros in A376562, complement A376588.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376588 (inflection and undulation points).
For second differences: A064113 (prime), A376602 (composite), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A182853, A294068, A333254, A336416, A376599.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Sign[Differences[Select[Range[1000],radQ],2]],1|-1]

A376656 Sorted positions of first appearances in the second differences (A036263) of consecutive primes (A000040).

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, 242, 262, 294, 296, 428, 429, 446, 708, 756, 834, 1005, 1182, 1229, 1663, 1830, 1831, 1846, 1879, 2191, 2224, 2343, 2809, 3077, 3086, 3384, 3385, 3427, 3643, 3644, 3793, 3795, 4230
Offset: 1

Views

Author

Gus Wiseman, Oct 07 2024

Keywords

Comments

The prime numbers are (A000040):
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...
with first differences (A001223):
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, ...
with second differences (A036263):
1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, ...
with sorted first appearances at (A376656):
1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, ...

Crossrefs

These are the sorted positions of first appearances in A036263.
For first differences we had A373400(n) + 1, except initial terms.
For prime-powers instead of prime numbers we have A376653/A376654.
For squarefree instead of prime numbers we have A376655, sorted firsts of A376590.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
Cf. A064113, A175632, A251092, A258025, A258026, A333214, A373402, A376305, A376311.

Programs

  • Mathematica
    q=Differences[Select[Range[1000],PrimeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Comments

Or, positive integers whose prime indices include at least one 1 or prime number.
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:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Crossrefs

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280
Offset: 0

Views

Author

Gus Wiseman, Jan 12 2025

Keywords

Comments

Also the inverse zero-based binomial transform of the odd prime numbers.

Crossrefs

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

Programs

  • Mathematica
    nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).
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