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 1-3 of 3 results.

A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

Original entry on oeis.org

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054
Offset: 1

Views

Author

Benoit Cloitre, Feb 28 2002

Keywords

Comments

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

Links

Crossrefs

A subset of A029707 (lesser index of twin primes).
Prime index of each (prime) term of A061351.
Positions of zeros in A061399.
For perfect power instead of squarefree we have A377436, zeros of A377432.
Positions of zeros in A377784.
The rest of the twin primes are at A378620, indices of A061368.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A061398 counts squarefree numbers between primes, zeros A068360.
A120327 gives the least nonsquarefree number >= n.
Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Programs

  • Mathematica
    Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)
  • PARI
    isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

Formula

n such that A061398(n) = prime(n+1)-prime(n)-1.
prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024

A378620 Lesser prime index of twin primes with nonsquarefree mean.

Original entry on oeis.org

2, 5, 7, 17, 20, 28, 35, 41, 43, 45, 49, 52, 57, 64, 69, 81, 83, 98, 109, 120, 140, 144, 152, 171, 173, 176, 178, 182, 190, 206, 215, 225, 230, 236, 253, 256, 262, 277, 286, 294, 296, 302, 307, 315, 318, 323, 336, 346, 373, 377, 390, 395, 405, 428, 430, 444
Offset: 1

Views

Author

Gus Wiseman, Dec 10 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.
This is a subset of A029707 (twin prime indices). The other twin primes are A068361, so A029707 is the disjoint union of A068361 and A378620.

Links

Crossrefs

The lesser of twin primes is A001359, index A029707 (complement A049579).
The greater of twin primes is A006512, index A107770 (complement appears to be A168543).
A subset of A029707 (twin prime lesser indices).
Prime indices of the primes listed by A061368.
Indices of twin primes with squarefree mean are A068361.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 finds the first position of a prime gap of 2n.
A046933 counts composite numbers between primes.
A120327 gives the least nonsquarefree number >= n.
Cf. A007674, A072284, A068360.
Cf. A057627, A061398, A061399, A067535, A070321, A071403, A112925.
Cf. A000720, A025584, A067774, A122535, A155752, A179067.

Programs

  • Mathematica
    Select[Range[100],Prime[#]+2==Prime[#+1]&&!SquareFreeQ[Prime[#]+1]&]
PrimePi/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&!SquareFreeQ[Mean[#]]&][[;;,1]] (* Harvey P. Dale, Jul 13 2025 *)

Formula

prime(a(n)) = A061368(n).

A366352 Lesser of 2 successive primes (k, k+4) sandwiching 3 consecutive nonsquarefree numbers.

Original entry on oeis.org

97, 349, 1447, 1663, 2347, 3697, 9547, 13147, 13309, 13687, 14533, 14947, 15727, 16603, 21139, 24547, 24847, 26557, 27733, 31147, 32797, 33613, 34603, 35593, 36943, 38149, 38707, 40849, 41047, 42433, 44449, 44647, 45763, 45949, 46447, 50047, 52387, 58147, 58309
Offset: 1

Views

Author

Massimo Kofler, Oct 08 2023

Keywords

Examples

			97 and 101 are prime numbers; 98 = 2 * 7^2, 99 = 3^2 * 11 and 100 = 2^2 * 5^2 are 3 consecutive nonsquarefree numbers, so 97 is a term.
349 and 353 are prime numbers; 350 = 2 * 5^2 * 7, 351 = 3^3 * 13, 352 = 2^5 * 11 are 3 consecutive nonsquarefree numbers, so 349 is a term.

Crossrefs

Cf. A000040, A013929.
Intersection of A023200 and A061400.
Cf. A061368.

Programs

  • Mathematica
    Select[Partition[Prime[Range[6000]], 2, 1], Differences[#] == {4} && AllTrue[Range[First[#] + 1, Last[#] - 1], ! SquareFreeQ[#1] &] &][[;; , 1]] (* Amiram Eldar, Oct 08 2023 *)
  • PARI
    isok(p) = isprime(p) && (nextprime(p+1) - p == 4) && (sum(k=1, 3, issquarefree(p+k)) == 0); \\ Michel Marcus, Oct 08 2023
Showing 1-3 of 3 results.

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