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 1-6 of 6 results.

A354525 Numbers k such that A354512(k) = A001221(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 62, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 155, 157
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Numbers k such that for every prime factor p of k we have gpf(k+p) = p, gpf = A006530.
Numbers k such that for every prime factor p of k, k+p is p-smooth.
If k is an even term, then k+2 is a power of 2, so k is of the form 2*(2^m-1). Those m for which 2*(2^m-1) is a term are listed in A354531.

Examples

			15 is a term since the prime factors of 15 are 3,5, and we have gpf(15+3) = 3 and gpf(15+5) = 5.

Links

Crossrefs

Cf. A001221, A354512, A006530, A354531, A354532, A354533, A354534.
Indices of 0 in A354527. Complement of A354526.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
isA354525(n) = my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])!=f[i], return(0))); 1

A354526 Numbers k such that A354512(k) < omega(k); complement of A354525.

Original entry on oeis.org

4, 8, 10, 12, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 56, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Numbers k such that there is a prime factor p of k such that gpf(k+p) != p.
Numbers k such that there is a prime factor p of k such that k+p is not p-smooth.

Examples

			57 is a term since the prime factors of 57 are 3,19, and we have gpf(57+3) != 3.

Links

Crossrefs

Cf. A001221, A354512, A006530. Indices of positive terms in A354527.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
isA354526(n) = my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])!=f[i], return(1))); 0

A354527 a(n) = A001221(n) - A354512(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 1, 0, 3, 0
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Number of distinct prime factors p of n such that gpf(n+p) != p, gpf = A006530.
Number of distinct prime factors p of n such that n+p is not p-smooth.

Examples

			a(30) = 2 since the prime factors of 30 are 2,3,5, and we have gpf(30+3) != 3 and gpf(30+5) != 5.

Links

Crossrefs

Cf. A001221, A354512, A006530.
Cf. A354525 (indices of 0), A354526 (indices of positive terms).

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
a(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])!=f[i])

A354514 Numbers k such that m - gpf(m) = k has solutions m >= 2, gpf = A006530.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Numbers k such that there is a prime p such that gpf(k+p) = p (such p must be a prime factor of n).
Numbers k such that there is a prime factor p of k such that k+p is p-smooth.
A076563 sorted and duplicates removed.

Examples

			0 is a term because 0 = p - gpf(p) for every prime p.
if k/gpf(k) <= nextprime(gpf(k)) - 2, where nextprime = A151800, then k is a term since k+gpf(k) <= gpf(k)*(nextprime(gpf(k)) - 1) implies gpf(k+gpf(k)) = gpf(k).

Links

Crossrefs

Cf. A006530, A076563, A151800.
0 together with indices of positive terms in A354512. Complement of A354515.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
isA354514(n) = if(n, my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])==f[i], return(1))); 0, 1)

A354515 Numbers k such that m - gpf(m) = k has no solution m >= 2, gpf = A006530.

Original entry on oeis.org

1, 4, 8, 12, 16, 18, 27, 32, 36, 48, 50, 54, 60, 64, 72, 80, 81, 84, 90, 96, 100, 108, 112, 125, 128, 132, 135, 144, 147, 150, 160, 162, 176, 180, 192, 196, 198, 200, 208, 210, 216, 224, 225, 234, 242, 243, 250, 252, 256, 270, 275, 280, 288, 294, 300, 306, 320, 324
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Numbers k such that there is no prime p such that gpf(k+p) = p.
Numbers k such that there is no prime factor p of k such that k+p is p-smooth.

Examples

			12 is a term since the prime factors of 12 are 2,3, and we have gpf(12+2) != 2 and gpf(12+3) != 3.

Links

Crossrefs

Cf. A006530, A076563.
Indices of 0 in A354512. Complement of A354514.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
isA354515(n) = if(n, my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])==f[i], return(0))); 1, 0)

A354516 Smallest k such that m - gpf(m) = k has exactly n solutions m >= 2, gpf = A006530; or -1 if no such k exists.

Original entry on oeis.org

1, 2, 6, 483, 1660577
Offset: 0

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Smallest k such that there are exactly n primes p such that gpf(k+p) = p (such p must be prime factors of k).
Smallest k having exactly n distinct prime factors p such that k+p is p-smooth.
Conjectures (if no term equals -1): (Start)
(1) Sequence is strictly increasing.
(2) All terms are squarefree.
(3) All terms are in A354525. (End)

Examples

			a(4) = 1660577: 1660577 = 17*23*31*127, and we have 1660577+17 = 2*13^2*17^3 is 17-smooth, 1660577+23 = 2^3*5^2*19^2*23 is 23-smooth, 1660577+31 = 2^6*3^3*31^2 is 31-smooth, 1660577+137 = 2*11*19*29*137, so m - gpf(m) = 1660577 has 4 solutions m = 1660577+17 = 1660594, 1660577+23 = 1660600, 1660577+31 = 1660608, and 1660577+137 = 1660714.

Crossrefs

Cf. A006530, A076563, A354512, A354525.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
A354512(n) = my(f=factor(n)[, 1]); sum(i=1, #f, gpf(n+f[i])==f[i]);
a(n) = my(k=1); while(omega(k)A354512(k) != n, k++); return(k)
Showing 1-6 of 6 results.

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