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.

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A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

Original entry on oeis.org

8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, 112, 114, 128, 136, 144, 145, 162, 180, 188, 198, 216, 226, 235, 246, 264, 265, 275, 285, 295, 305, 316, 317, 325, 328, 338, 350, 360, 367, 373, 385, 406, 416, 417, 419, 431, 443
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2024

Keywords

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, ...
with positions of latter appearances (A376519):
  8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...

Crossrefs

These are the sorted positions of latter appearances in A053289 (union A023055).
The complement is A376268.
A053707 lists first differences of consecutive prime-powers.
A333254 lists run-lengths of differences between consecutive primes.
Other families of numbers and their first differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310.
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.

Programs

  • Mathematica
    perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],MemberQ[Take[q,#-1],q[[#]]]&]

A065308 a(n) = prime(n - pi(n)).

Original entry on oeis.org

2, 2, 2, 3, 3, 5, 5, 7, 11, 13, 13, 17, 17, 19, 23, 29, 29, 31, 31, 37, 41, 43, 43, 47, 53, 59, 61, 67, 67, 71, 71, 73, 79, 83, 89, 97, 97, 101, 103, 107, 107, 109, 109, 113, 127, 131, 131, 137, 139, 149, 151, 157, 157, 163, 167, 173, 179, 181, 181, 191, 191, 193, 197
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Examples

			For n=100, pi(100) = 25, prime(100-25) = prime(75) = 379 = a(100).

Links

Crossrefs

Cf. A000040, A000720, A065309, A065310.

Programs

A065309 a(n) = prime(n) - prime(n - pi(n)).

Original entry on oeis.org

0, 1, 3, 4, 8, 8, 12, 12, 12, 16, 18, 20, 24, 24, 24, 24, 30, 30, 36, 34, 32, 36, 40, 42, 44, 42, 42, 40, 42, 42, 56, 58, 58, 56, 60, 54, 60, 62, 64, 66, 72, 72, 82, 80, 70, 68, 80, 86, 88, 80, 82, 82, 84, 88, 90, 90, 90, 90, 96, 90, 92, 100, 110, 112, 102, 94, 108, 110, 118
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Examples

			n=100, prime(100)=541, prime(100 - Pi(100)) = p(75), p(100) - p(75) = 541 - 379 = 162 = a(100).

Links

Crossrefs

Cf. A000040, A000720, A065308-A065310.

Programs

  • Mathematica
    Table[Prime[n]-Prime[n-PrimePi[n]],{n,70}] (* Harvey P. Dale, Apr 23 2016 *)
  • PARI
    a(n) = { prime(n) - prime(n - primepi(n)) } \\ Harry J. Smith, Oct 16 2009

Formula

a(n) = A000040(n) - A000040(n - A000720(n)).

A065311 Primes which occur exactly twice in the sequence of a(n) = prime(n) - prime(n - pi(n)) = A065308(n).

Original entry on oeis.org

3, 5, 13, 17, 29, 31, 43, 67, 71, 97, 107, 109, 131, 157, 181, 191, 223, 233, 239, 269, 281, 313, 359, 379, 383, 401, 409, 431, 503, 523, 569, 571, 619, 631, 659, 691, 719, 751, 787, 797, 857, 859, 881, 883, 971, 1039, 1061, 1063, 1091, 1117, 1123, 1201
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Comments

In A065308, each odd prime seems to appear once or twice.

Links

Crossrefs

Cf. A000040, A000720, A054546, A065308, A065309, A065310, A065311.

Programs

  • Mathematica
    Most@ Select[Tally@ Array[Prime[# - PrimePi@ #] &, 300], Last@ # == 2 &][[All, 1]] (* Michael De Vlieger, Nov 03 2017 *)
  • PARI
    { n=0; p=1; f=2; m=1; for (i=1, 10^9, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); if (a==2, write("b065311.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 16 2009

A065312 Primes which occur exactly once in A065308 (prime(n - pi(n))).

Original entry on oeis.org

7, 11, 19, 23, 37, 41, 47, 53, 59, 61, 73, 79, 83, 89, 101, 103, 113, 127, 137, 139, 149, 151, 163, 167, 173, 179, 193, 197, 199, 211, 227, 229, 241, 251, 257, 263, 271, 277, 283, 293, 307, 311, 317, 331, 337, 347, 349, 353, 367, 373, 389, 397, 419, 421, 433
Offset: 1

Views

Author

Labos Elemer, Oct 29 2001

Keywords

Comments

In A065308 each odd prime seems to appear once or twice. Prime 2 arises there 3 times.

Links

Crossrefs

Cf. A000040, A000720, A054546, A065308, A065309, A065310, A065311.

Programs

  • Mathematica
    With[{s = Array[Prime[# - PrimePi[#]] &, 120]}, Most@ Select[Split[s], Length@ # == 1 &][[All, 1]] ] (* Michael De Vlieger, Jun 19 2018 *)
  • PARI
    { n=0; p=1; f=2; m=1; for (i=1, 10^9, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); if (a==1, write("b065312.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 16 2009

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[#]&]
Previous Showing 21-26 of 26 results.

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