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.

User: Jaycob Coleman

Jaycob Coleman's wiki page.

Jaycob Coleman has authored 4 sequences.

A228945 Number of ways to write highly composite numbers (A002182(n)) as the difference of two primes, both <= 2*A002182(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 6, 6, 7, 11, 19, 25, 28, 40, 61, 87, 109, 138, 184, 326, 437, 550, 721, 935, 1103, 1326, 1792, 1903, 2351, 3261, 4119, 5773, 7386, 8736, 10307, 14404, 15953, 18290, 21480, 30294, 38516, 54874, 70132, 85419, 99583, 142053, 155243, 182169, 220996
Offset: 1

Views

Author

Jaycob Coleman, Sep 08 2013

Keywords

Comments

Conjectures: (a) This sequence is strictly increasing beginning with n=7. (b) If p is the smallest prime with p > A002182(n)+1, then p-A002182(n) is prime. This is a strengthening of a conjecture regarding A117825.

Examples

			a(5) = 3, since A002182(5) = 12 = 23-11 = 19-7 = 17-5.

Links

Crossrefs

Cf. A002182, A117825, A202472.

Formula

a(n) = A202472(A002182(n)/2) for n>1.

Extensions

More terms from Amiram Eldar, Nov 03 2024

A228944 Number of ways to write highly composite numbers (A002182(n)) as the difference of two highly abundant numbers (A002093), both <= 2*A002182(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 6, 7, 7, 8, 10, 12, 13, 13, 14, 14, 11, 11, 13, 15, 16, 15, 17, 17, 18, 19, 16, 17, 19, 18, 19, 18, 24, 20, 29, 28, 23, 24, 24, 26, 26, 23, 22
Offset: 1

Views

Author

Jaycob Coleman, Sep 08 2013

Keywords

Comments

Conjecture: this sequence is always positive, analogous to sequence A202472 for strong Goldbach conjecture. - Jaycob Coleman, Sep 08 2013

Examples

			a(4)=3, since 6=12-6=10-4=8-2.

Links

Crossrefs

Cf. A202472.

A228943 Number of decompositions of highly composite numbers (A002182) into unordered sums of two primes.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 4, 5, 6, 12, 14, 18, 22, 39, 51, 68, 83, 112, 184, 251, 315, 431, 527, 652, 768, 1011, 1128, 1305, 1836, 2344, 3240, 4082, 4955, 5725, 8023, 8723, 10260, 11945, 16771, 21466, 30280, 38583, 46645, 54789, 77430, 85067, 99199, 120742, 154753
Offset: 1

Views

Author

Jaycob Coleman, Sep 08 2013

Keywords

Comments

a(n) = A045917(A002182(n)/2) for n>1.
Conjecture: (a) This sequence is strictly increasing beginning with n=5. (b) For all n>2, if p is the greatest prime with p<A002182(n)-1, then A002182(n)-p is prime. This is a strengthening of a conjecture regarding A117825. - Jaycob Coleman, Sep 08 2013

Examples

			a(6)=3, since 24=5+19=7+17=11+13.

Crossrefs

Cf. A116979, A117825.

Programs

  • PARI
    nbd(n) = my(s); forprime(p=2, n\2, s+=isprime(n-p)); s;
lista(nn) = {last = 1; print1(nbd(last), ", "); forstep(n=2, nn, 2, if(numdiv(n)> last, last=numdiv(n); print1(nbd(n), ", ")););} \\ Michel Marcus, Sep 10 2013

Extensions

More terms from Michel Marcus, Sep 10 2013

A228942 Number of decompositions of highly composite numbers (A002182) into unordered sums of two highly abundant numbers (A002093).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 7, 8, 11, 12, 12, 12, 14, 14, 12, 12, 13, 15, 17, 15, 17, 17, 18, 15, 15, 15, 17, 17, 20, 19, 25, 26, 28, 24, 25, 19, 19, 24, 20, 19, 18
Offset: 1

Views

Author

Jaycob Coleman, Sep 08 2013

Keywords

Comments

Conjecture: this sequence is always positive, analogous to sequence A045917 for the strong Goldbach conjecture. - Jaycob Coleman, Sep 08 2013

Examples

			a(6)=4, since 24=4+20=6+18=8+16=12+12.

Links

Crossrefs

Cf. A045917.

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