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 1-5 of 5 results.

A353285 Consider the number of divisors tau(k) of every composite k between prime p >= 3 and the next prime; if the largest tau(k) is a square, then p is in the sequence.

Original entry on oeis.org

5, 7, 31, 97, 113, 167, 193, 199, 211, 263, 269, 277, 311, 317, 373, 383, 401, 439, 461, 509, 541, 547, 569, 593, 613, 631, 677, 701, 709, 727, 743, 757, 769, 857, 907, 941, 947, 1021, 1031, 1063, 1123, 1153, 1217, 1229, 1249, 1259, 1283, 1289, 1291, 1301, 1321, 1361
Offset: 1

Views

Author

Claude H. R. Dequatre, Apr 09 2022

Keywords

Examples

			97 is a term because up to the next prime 101, tau(98) = 6, tau(99) = 6, tau(100) = 9, thus the greatest tau is 9 and 9 is a square (3^2).
127 is prime but not a term because up to the next prime 131, tau(128) = 8, tau(129) = 4, tau(130) = 8, thus the greatest tau(k) is 8 and 8 is not a square.

Crossrefs

Cf. A000005, A000040, A000290.
Cf. A353284, A353286.

Programs

  • Mathematica
    Select[Prime[Range[2, 220]], IntegerQ[Sqrt[Max[DivisorSigma[0, Range[# + 1, NextPrime[#] - 1]]]]] &] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    forprime(p=3,2000,my(v=vector(nextprime(p+1)-p-1,k,numdiv(p+k))); if(ispower(vecmax(v),2), print1(p", ")))

A353286 Consider the number of divisors tau(k) of every composite k between prime p >= 3 and the next prime; if the largest tau(k) is a cube, then p is in the sequence.

Original entry on oeis.org

23, 29, 37, 41, 53, 61, 67, 73, 101, 103, 127, 137, 163, 181, 229, 241, 281, 353, 421, 433, 601, 617, 641, 821, 887, 1093, 1433, 1489, 1697, 1759, 1877, 2081, 2083, 2237, 2297, 2381, 2657, 2801, 2953, 3461, 3529, 3557, 3917, 4153, 4349, 4637, 4721, 4789, 5441, 5689
Offset: 1

Views

Author

Claude H. R. Dequatre, Apr 09 2022

Keywords

Examples

			37 is a term because up to the next prime 41, tau(38) = 4, tau(39) = 4, tau(40) = 8, thus the greatest tau is 8 and 8 is a cube (2^3).
47 is prime but not a term because up to the next prime 53, tau(48) = 10, tau(49) = 3, tau(50) = 6, tau(51) = 4, tau(52) = 6, thus the greatest tau is 10 and 10 is not a cube.

Crossrefs

Cf. A000005, A000040, A000578.
Cf. A353284, A353285.

Programs

  • Mathematica
    Select[Prime[Range[2, 800]], IntegerQ[Surd[Max[DivisorSigma[0, Range[# + 1, NextPrime[#] - 1]]], 3]] &] (* Amiram Eldar, Jun 10 2022 *)
  • PARI
    forprime(p=3,2000,my(v=vector(nextprime(p+1)-p-1,k,numdiv(p+k))); if(ispower(vecmax(v),3), print1(p", ")))

A356833 Primes p such that the minimum number of divisors among the numbers between p and NextPrime(p) is a square.

Original entry on oeis.org

5, 13, 19, 31, 37, 43, 53, 61, 67, 73, 79, 83, 89, 103, 109, 127, 131, 139, 151, 157, 163, 173, 181, 193, 199, 211, 223, 233, 241, 251, 257, 263, 269, 271, 277, 293, 307, 311, 313, 317, 331, 337, 353, 367, 373, 379, 383, 389, 397, 401, 409, 421, 433, 443, 449, 457, 461, 463, 467, 479
Offset: 1

Views

Author

Claude H. R. Dequatre, Sep 16 2022

Keywords

Examples

			13 is a term because up to the next prime 17, tau(14) = 4, tau(15) = 4, tau(16) = 5, thus the smallest tau(k) is 4 and 4 is a square (2^2).
23 is prime but not a term because up to the next prime 29, tau(24) = 8, tau(25) = 3, tau(26) = 4, tau(27) = 4, tau(28) = 6, thus the smallest tau(k) = 3 and 3 is not a square.

Crossrefs

Cf. A000005, A000040, A000290, A061112, A036436.
Cf. A353284, A353285, A353286, A357170, A357175.

Programs

  • PARI
    isok(p)=issquare(vecmin(apply(numdiv, [p+1..nextprime(p+1)-1])));
forprime(p=3, 2000, if(isok(p), print1(p", ")))

A357170 Primes p such that the minimum number of divisors among the numbers between p and NextPrime(p) is a prime power.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 73, 79, 83, 89, 101, 103, 109, 113, 127, 131, 137, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 349, 353, 359, 367, 373
Offset: 1

Views

Author

Claude H. R. Dequatre, Sep 16 2022

Keywords

Examples

			19 is a term because up to the next prime 23, tau(20) = 6, tau(21) = 4, tau(22) = 4, thus the smallest tau(k) is 4 and 4 is a prime power (2^2).
97 is prime but not a term because up to the next prime 101, tau(98) = 6, tau(99) = 6, tau(100) = 9, thus the smallest tau(k) is 6 and 6 is not a prime power.

Crossrefs

Cf. A000005, A000040, A061112, A246655.
Cf. A353284, A353285, A353286, A356833, A357175.

Programs

  • PARI
    isok(p)=isprimepower(vecmin(apply(numdiv, [p+1..nextprime(p+1)-1])));
forprime(p=3, 2000, if(isok(p), print1(p", ")))

A357175 Primes p such that the minimum of the number of divisors among the numbers between p and NextPrime(p) is a cube.

Original entry on oeis.org

29, 41, 101, 137, 229, 281, 349, 439, 617, 641, 643, 739, 821, 823, 853, 967, 1087, 1423, 1429, 1447, 1549, 1579, 1597, 1693, 1697, 1783, 1877, 1999, 2081, 2131, 2237, 2239, 2293, 2377, 2381, 2539, 2617, 2657, 2683, 2693, 2713, 2749, 2791, 2801, 3079, 3319
Offset: 1

Views

Author

Claude H. R. Dequatre, Sep 16 2022

Keywords

Examples

			349 is a term because up to the next prime 353, tau(350) = 12, tau(351) = 8, tau(352) = 12, thus the smallest tau(k) = 8 and 8 is a cube (2^3).
379 is prime but not a term because up to the next prime 383, tau(380) = 12, tau(381) = 4, tau(382) = 4, thus the smallest tau(k) is 4 and 4 is not a cube.

Crossrefs

Cf. A000005, A000040, A000578. A061112.
Cf. A353284, A353285, A353286, A356833, A357170.

Programs

  • PARI
    isok(p)=ispower(vecmin(apply(numdiv, [p+1..nextprime(p+1)-1])), 3);
forprime(p=3, 10000, if(isok(p), print1(p", ")))
Showing 1-5 of 5 results.

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