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 71-76 of 76 results.

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&)]

A373823 Half the sum of the n-th maximal run of first differences of odd primes.

Original entry on oeis.org

2, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 6, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 6, 2, 6, 1, 5, 1, 2, 1, 12, 2, 1, 2, 3, 1, 5, 9, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 6, 2, 3, 4, 2, 4, 5, 1, 5, 1, 3, 2, 3, 4, 2, 1, 2, 6, 4, 2, 4, 2, 3
Offset: 1

Views

Author

Gus Wiseman, Jun 22 2024

Keywords

Comments

Halved run-sums of A001223.

Examples

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs:
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with halved sums a(n).

Crossrefs

Halved run-sums of A001223.
For run-lengths we have A333254, run-lengths of run-lengths A373821.
Multiplying by two gives A373822.
A000040 lists the primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Programs

  • Mathematica
    Total/@Split[Differences[Select[Range[3,1000],PrimeQ]]]/2

A375928 Positions of adjacent non-prime-powers (exclusive) differing by more than 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 11, 12, 13, 14, 18, 21, 22, 25, 26, 29, 34, 35, 37, 39, 42, 43, 48, 49, 50, 55, 62, 65, 66, 69, 70, 73, 80, 83, 84, 86, 91, 92, 101, 102, 107, 112, 115, 116, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 175, 182
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2024

Keywords

Examples

			The non-prime-powers (exclusive) are 1, 6, 10, 12, 14, 15, 18, 20, ... which increase by more than 1 after positions 1, 2, 3, 4, 6, 7, ...

Crossrefs

For prime-powers inclusive (A000961) we have A376163, differences A373672.
For nonprime numbers (A002808) we have A014689, differences A046933.
First differences are A110969.
The complement is A375713.
For non-perfect-powers we have A375714, complement A375740.
The complement for prime-powers (exclusive) is A375734, differences A373671.
The complement for nonprime numbers is A375926, differences A373403.
A000040 lists the prime numbers, differences A001223.
A000961 lists prime-powers (inclusive), differences A057820.
A007916 lists non-perfect-powers, differences A375706.
A024619 lists non-prime-powers (inclusive), differences A375735.
A246655 lists prime-powers (exclusive), differences A174965.
A361102 lists non-prime-powers (exclusive), differences A375708.
Cf. A006549, A053289, A073783, A093555, A120430, A176246, A251092.

Programs

  • Mathematica
    ce=Select[Range[100],!PrimePowerQ[#]&];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

Formula

The inclusive version is a(n+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[[#]]]&]

A376163 Positions of adjacent non-prime-powers (inclusive, so 1 is a prime-power) differing by 1.

Original entry on oeis.org

4, 7, 8, 14, 15, 16, 18, 19, 22, 23, 26, 27, 29, 30, 31, 32, 35, 37, 39, 40, 43, 44, 45, 46, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 66, 67, 70, 71, 73, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105
Offset: 1

Views

Author

Gus Wiseman, Sep 13 2024

Keywords

Examples

			The non-prime-powers (inclusive) are 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ... which increase by 1 after positions 4, 7, 8, ...

Crossrefs

For prime-powers inclusive (A000961) we have A375734, differences A373671.
For nonprime numbers (A002808) we have A375926, differences A373403.
For prime-powers exclusive (A246655) we have A375734(n+1) + 1.
First differences are A373672.
The exclusive version is a(n) - 1 = A375713.
Positions of 1's in A375735.
For non-perfect-powers we have A375740.
Prime-powers inclusive:
- terms: A000961
- differences: A057820
- runs: A373675, A373673, A373674, A174965
- antiruns: A373576, A120430, A006549, A373671
Non-prime-powers inclusive:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- antiruns: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
Cf. A046933, A053289, A073783, A093555, A176246, A251092, A375714.

Programs

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

A373828 Run-sums (differing by 0) of run-lengths (differing by 2) of odd primes.

Original entry on oeis.org

3, 4, 1, 2, 1, 2, 2, 2, 1, 2, 4, 4, 3, 4, 4, 6, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 10, 4, 4, 2, 7, 2, 4, 2, 3, 2, 2, 2, 1, 2, 2, 2, 18, 6, 2, 2, 2, 2, 17, 4, 1, 4, 2, 2, 6, 2, 9, 2, 3, 2, 1, 2, 1, 2, 1, 2, 8, 2, 3, 2, 2, 4, 15, 2, 1, 2, 4, 2, 1, 2, 1, 2, 7, 2
Offset: 1

Views

Author

Gus Wiseman, Jun 23 2024

Keywords

Comments

Run-sums of A251092.

Examples

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with runs:
{3,5,7}, {11,13}, {17,19}, {23}, {29,31}, {37}, {41,43}, {47}, {53}, ...
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, ...
with runs:
{3}, {2,2}, {1}, {2}, {1}, {2}, {1,1}, {2}, {1}, {2}, {1,1,1,1}, {2,2}, ...
with sums a(n).

Crossrefs

Run-sums of A251092.
The run-lengths (instead of run-sums) are A373819, firsts A373825, A373824.
A000040 lists the primes.
A001223 gives first differences of primes.
A027833 gives antirun-lengths of primes > 3 (prepended run-lengths A373820).
A046933 counts composite numbers between primes.
A071148 gives partial sums of odd primes.
A333254 gives run-lengths of first differences of primes.
A373821 gives run-lengths of run-lengths of first differences of odd primes.
Cf. A006560, A006562, A029707, A054265, A067774, A373403, A373404, A373406.

Programs

  • Mathematica
    Total/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ], #1+2==#2&]//Most]//Most
Previous Showing 71-76 of 76 results.

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