A202472 -id:A202472 - cp's OEIS frontend

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

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

Jaycob Coleman, Sep 08 2013

nonn

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

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

Jaycob Coleman, Table of n, a(n) for n = 1..265

Cf. A202472.

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

Jaycob Coleman, Sep 08 2013

nonn

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.

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

Amiram Eldar, Table of n, a(n) for n = 1..84

Cf. A002182, A117825, A202472.

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

More terms from Amiram Eldar, Nov 03 2024

A205601 Goldbach's problem extended to division: number of decompositions of 2n into the floor of unordered ratios of two primes, floor(q/p) = 2n, where p < 2n < q.

Original entry on oeis.org

0, 1, 3, 5, 4, 5, 10, 5, 10, 16, 12, 17, 18, 16, 19, 27, 23, 22, 34, 27, 34, 39, 39, 45, 51, 41, 50, 51, 44, 57, 68, 71, 63, 74, 63, 76, 87, 84, 89, 104, 94, 108, 111, 99, 117, 116, 120, 104, 126, 114, 133, 146, 149, 146, 166, 148, 190, 178, 182, 170, 179, 173

Offset: 1

James D. Klein, Jan 29 2012

nonn

			For n = 3, a(n) = 3 because 6 is the floor of 13/2, 19/3, and 31/5. - _T. D. Noe_, Jan 31 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002375, A202472.

Table[Length[Flatten[Table[Select[2*n*p + Range[p - 1], PrimeQ], {p, Prime[Range[PrimePi[2*n - 1]]]}]]], {n, 62}] (* T. D. Noe, Jan 31 2012 *)

a(n)=n*=2;my(s,t);forprime(p=2,n-1,t=n*p;while(n==(t=nextprime(t+1))\p,s++));s \\ Charles R Greathouse IV, Jan 30 2012

a(21)-a(62) from Charles R Greathouse IV, Jan 31 2012

cp's OEIS Frontend

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

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

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A205601 Goldbach's problem extended to division: number of decompositions of 2n into the floor of unordered ratios of two primes, floor(q/p) = 2n, where p < 2n < q.

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