Zhandos Mambetaliyev - cp's OEIS frontend

A342870 a(n) is the number of twin primes between A001359(n)^2 and A001359(n)*(A001359(n)+1).

1, 1, 0, 0, 1, 2, 2, 1, 2, 2, 0, 2, 4, 1, 1, 1, 5, 1, 3, 4, 2, 5, 4, 2, 4, 3, 6, 6, 3, 5, 6, 6, 4, 6, 4, 7, 9, 6, 8, 9, 6, 8, 10, 6, 11, 9, 13, 8, 12, 6, 14, 4, 7, 11, 11, 15, 9, 10, 12, 11, 10, 12, 13, 8, 15, 14, 11, 12, 9, 11, 15, 14, 18, 16, 11, 18, 10

Offset: 1

Zhandos Mambetaliyev, Mar 28 2021

nonn

Cf. A171727, A342552, A001359, A037074, A006094.

{for(k=1, 400, if(prime(k+1)-prime(k)==2, my(c=0); forprime(m=prime(k)^2, prime(k)*(prime(k)+1), c+=isprime(m+2)); print1(c, ", ")))}

A342552 The number of twin primes between prime(i)prime(i+1) and prime(i+1)prime(i+2) where prime(i) and prime(i+1) are twin primes.

Original entry on oeis.org

2, 3, 4, 3, 5, 6, 9, 11, 8, 12, 24, 19, 34, 14, 27, 14, 28, 17, 46, 26, 24, 55, 28, 14, 86, 50, 38, 66, 28, 67, 76, 41, 64, 40, 43, 93, 53, 87, 67, 48, 89, 66, 42, 72, 69, 76, 49, 76, 42, 49, 59, 73, 260, 109, 145, 169, 70, 137, 193, 292

Offset: 1

Zhandos Mambetaliyev, Mar 27 2021

nonn

Cf. A171727, A006094, A001359, A006512, A037074.

{for(n=1,200, if(prime(n+1)-prime(n)==2, my(a=prime(n)*prime(n+1), b=prime(n+1)*prime(n+2), c=0); for(m=a,b, if(isprime(m)==1&&isprime(m+2)==1, c=c+1)); print1(c, ", ")))}

A337022 a(n) is the number of positive integers <= A070826(n) with at least one odd prime divisor <= prime(n).

Original entry on oeis.org

0, 1, 7, 57, 675, 9255, 163095, 3190965, 75051075, 2212976535, 69624142665, 2606749381005, 107980344307605, 4687299592683015, 222157161929253705, 11859617311615438365, 704152383312290447535, 43210523173814533171635, 2910538720151462674819545, 207666871186142520765307695

Offset: 1

Zhandos Mambetaliyev, Aug 11 2020

nonn

The set of finite differences positive numbers up to A070826(n) with at least one odd prime divisor <= prime(n) is a palindromic set.

			a(3) = 7, p = {3, 5}, prime(n)# / 2 = 15, {3, 5, 6, 9, 10, 12, 15} - divisible by 3 or 5.

Cf. A002110, A070826, A005867.

pm=1; forprime(p=2,19,pm*=p; my(k=0); for(x=2,pm/2, forprime(q=3,p, if(x%q==0,k++;break))); print1(k,", ")) \\ Hugo Pfoertner, Aug 11 2020

a(n+1) = (prime(n+1) - 1)*a(n) + A070826(n). - Jinyuan Wang, Aug 11 2020

a(n) = A002110(n)/2 - A005867(n). - Jamie Morken, Aug 11 2021

a(8)-a(10) from Hugo Pfoertner, Aug 11 2020

More terms from Jinyuan Wang, Aug 11 2020

A331809 a(1) = 1; a(2) = 2; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2a(i) nor Sum_{j=1..n-1} ( b_ja(j) ) where 0 < i < n, b_j = 1 or 0.

Original entry on oeis.org

1, 2, 5, 9, 13, 31, 35, 92, 118, 280, 350, 866, 1102, 2668, 3368, 8240, 10444, 25420, 32156, 78464, 99352, 242128, 306440, 747272, 945976, 2306128, 2919008, 7117088, 9009040, 21964144, 27802160, 67784384, 85802464, 209191168, 264795488, 645591584, 817196512, 1992379072

Offset: 1

Zhandos Mambetaliyev, Jan 27 2020

nonn

Inserting the additional term a(0) = 3 would result in a so-called complete sequence after sorting. (The sorted sequence will then meet Brown's criterion.)

Eric Weisstein's World of Mathematics, Brown's Criterion
Eric Weisstein's World of Mathematics, Complete Sequence

Cf. A126326, A331800, A331811.

PARI
```
/* a(n) for 0
				
```

a(12)-a(15) from Hugo Pfoertner, Jan 27 2020

More terms, using Rémy Sigrist's C++ at A331811 from Hugo Pfoertner, Jan 28 2020

A331800 a(1) = 1; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2a(i) nor Sum_{j=1..n-1} ( b_ja(j) ) where 0 < i < n, b_j = 0 or 1.

Original entry on oeis.org

1, 3, 5, 7, 17, 19, 50, 64, 152, 190, 470, 598, 1448, 1828, 4472, 5668, 13796, 17452, 42584, 53920, 131408, 166312, 405560, 513400, 1251584, 1584208, 3862592, 4889392, 11920400, 15088816, 36788000, 46566784, 113532416, 143710048, 350376032, 443509600, 1081305728

Offset: 1

Zhandos Mambetaliyev, Jan 26 2020

nonn

Inserting the additional term a(0) = 2 would result in a so-called complete sequence after sorting. (The sorted sequence will then meet Brown's criterion.)

Eric Weisstein's World of Mathematics, Brown's Criterion
Eric Weisstein's World of Mathematics, Complete Sequence

Cf. A331809, A331811.

/* a(n) for n>0 */
upto(lim)={my(a=[1], b=[]); for(i=1, lim, forsubset(#a, x, b=concat(b, [vecsum(vecextract(a, x))])); b=setminus(vecsort(b, , 8), a); for(j=1, #a, b=concat(b, [2*a[j]]); b=vecsort(b, , 8)); if(setsearch(b, i)==0, a=concat(a, [i]); a=vecsort(a, , 8)) ); a}
{ upto(200) }

a(13)-a(14) from Hugo Pfoertner, Jan 27 2020

More terms, using Rémy Sigrist's C++ at A331811 from Hugo Pfoertner, Jan 28 2020

A331811 a(n) is the next number after a(n-1) which cannot be represented in the form 2a(i) and Sum_{j=1..n-1} b_ja(j) where 0 < i < n, b_j = 1 or 0. The sequence starts: a(1) = 1; a(2) = 2; a(3) = 3; a(4) = 5.

Original entry on oeis.org

1, 2, 3, 5, 12, 25, 49, 73, 171, 195, 512, 658, 1560, 1950, 4826, 6142, 14868, 18768, 45920, 58204, 141660, 179196, 437264, 553672, 1349328, 1707720, 4164392, 5271736, 12851568, 16267008, 39662048, 50205520, 122401584, 154935600, 377748224, 478159264, 1165778688, 1475649888

Offset: 1

Zhandos Mambetaliyev, Jan 27 2020

nonn

This sequence is a complete sequence.

Rémy Sigrist, C++ program for A331811
Eric Weisstein's World of Mathematics, Complete Sequence

(C++) See Links section.

Nest[Append[#, Block[{k = #[[-1]] + 1}, While[Nand[NoneTrue[#, k == 2 # &], FreeQ[Map[Total, Rest@ Subsets[#]], k]], k++]; k]] & @@ {#, Map[Total, Subsets[#]]} &, {1, 2, 3, 5}, 10] (* Michael De Vlieger, Jan 27 2020 *)

upto(lim)={my(a=[1, 2, 3, 5], b=[]); for(i=1, lim, forsubset(#a, x, b=concat(b, [vecsum(vecextract(a, x))])); b=setminus(vecsort(b, , 8), a); for(j=1, #a, b=concat(b, [2*a[j]]); b=vecsort(b, , 8)); if(setsearch(b, i)==0, a=concat(a, [i]); a=vecsort(a, , 8)) ); a}
{ upto(200) }

a(13)-a(14) from Hugo Pfoertner, Jan 27 2020

More terms from Rémy Sigrist, Jan 28 2020

A309945 a(n) = floor(n - sqrt(2*n-1)).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60

Offset: 1

Zhandos Mambetaliyev, Aug 24 2019

nonn

The subsequence consisting of numbers that appear twice is A007590.
Sequence as triangle:
   0;
   0;
   0;  1,  2;
   2,  3,  4;
   4,  5,  6,  7,  8;
   8,  9, 10, 11, 12;
  12, 13, 14, 15, 16, 17, 18;
  18, 19, 20, 21, 22, 23, 24;
  ...
a(1) = 0; for n > 1, a(n) is the number of squares strictly between 2*n - 2 and n^2.

			For n = 3, 2*n - 2 = 4, n^2 = 9, no square numbers strictly between 4 and 9, a(3) = 0.
For n=5, 2*n - 2 = 8, n^2 = 25, two square numbers (9, 16) strictly between 8 and 25, a(5) = 2.

Table of n, a(n) for n = 1..999

Cf. A007590, A080476, A016813.

Table[Floor[n-(2*n-1)^(1/2)],{n,73}] (* Stefano Spezia, Aug 24 2019 *)

a(n) = floor(n - sqrt(2*n-1)); \\ Jinyuan Wang, Aug 26 2019

from math import isqrt
def A309945(n): return (m:=n-1)-isqrt(m<<1) # Chai Wah Wu, Aug 04 2022

a(n) = n-1-floor(sqrt(2*n-2)). - Wesley Ivan Hurt, Dec 03 2020

A306863 a(n) is the number of primes between the n-th and (n+1)-st odd composite numbers.

Original entry on oeis.org

2, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 0, 0

Offset: 1

Zhandos Mambetaliyev, Mar 14 2019

nonn

			The first few odd composite numbers are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49. Between 9 and 15, there are two primes (11 and 13); between 15 and 21 there are also two primes (17 and 19); between 21 and 25 there is only one prime (23), etc.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A071904, A000040, A001223 (differences between primes), A164510, A001097.

Differences@ PrimePi@ Complement[Range[3, #, 2], Prime@ Range[2, PrimePi@ #]] &@ 300 (* Michael De Vlieger, Apr 21 2019 *)
Differences[PrimePi/@Select[Range[3,301,2],CompositeQ]] (* Harvey P. Dale, Sep 16 2023 *)

{ b=9; for(i=6, 150, if(isprime(2*i-1)==0, print1(primepi(2*i-1)-primepi(b), ", "); b=2*i-1)) }

from itertools import count
from sympy import primepi, isprime
def A306863(n):
    if n == 1: return 2
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return next(c for c in count(0) if not isprime(m+(c<<1)+2)) # Chai Wah Wu, Aug 02 2024

A306848 Product of first n odd nonprimes, a(n) = Product_{k=1..n} A071904(k).

Original entry on oeis.org

1, 9, 135, 2835, 70875, 1913625, 63149625, 2210236875, 86199238125, 3878965715625, 190069320065625, 9693535323346875, 533144442784078125, 30389233238692453125, 1914521694037624546875, 124443910112445595546875, 8586629797758746092734375

Offset: 0

Zhandos Mambetaliyev, Mar 13 2019

nonn

Cf. A071904, A036691, A002866, A001147.

With[{nn = 70}, FoldList[Times, Complement[Range[1, nn, 2], Prime@ Range[2, PrimePi@ nn]]]] (* Michael De Vlieger, Apr 21 2019 *)

lista(nn) = {my(p=1); print1(p, ", "); forcomposite (n=1, nn, if (n%2, p *= n; print1(p, ", ")); ); } \\ Michel Marcus, Mar 13 2019

A292410 Difference between (2n+1)^2 and highest power of 2 less than or equal to (2n+1)^2.

Original entry on oeis.org

0, 1, 9, 17, 17, 57, 41, 97, 33, 105, 185, 17, 113, 217, 329, 449, 65, 201, 345, 497, 657, 825, 1001, 161, 353, 553, 761, 977, 1201, 1433, 1673, 1921, 129, 393, 665, 945, 1233, 1529, 1833, 2145, 2465, 2793, 3129, 3473, 3825, 89, 457, 833, 1217, 1609, 2009, 2417, 2833, 3257

Offset: 0

Zhandos Mambetaliyev, Sep 15 2017

			a(0) = 1^2 - 2^0 =  0.
a(1) = 3^2 - 2^3 =  1.
a(2) = 5^2 - 2^4 =  9.
a(3) = 7^2 - 2^5 = 17.
a(4) = 9^2 - 2^6 = 17.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000079 (2^n), A016754 (odd squares), A053645 (distance to power of 2), A056577.

seq((2*n+1)^2 - 2^ilog2((2*n+1)^2), n=0..100); @ Robert Israel, Oct 19 2017

Table[# - 2^Floor@ Log2@ # &[(2 n + 1)^2], {n, 0, 53}] (* Michael De Vlieger, Sep 18 2017 *)

a(n) = my(k = 0); while(2^k < (2*n+1)^2, k++); if (k, k--); (2*n+1)^2 - 2^k; \\ Michel Marcus, Sep 16 2017

a(n) = A053645(A016754(n)). - Michel Marcus, Sep 16 2017

More terms from Michel Marcus, Sep 16 2017

cp's OEIS Frontend

User: Zhandos Mambetaliyev

A342870 a(n) is the number of twin primes between A001359(n)^2 and A001359(n)*(A001359(n)+1).

Views

Author

Keywords

Crossrefs

Programs

PARI

A342552 The number of twin primes between prime(i)*prime(i+1) and prime(i+1)*prime(i+2) where prime(i) and prime(i+1) are twin primes.

Views

Author

Keywords

Crossrefs

Programs

PARI

A337022 a(n) is the number of positive integers <= A070826(n) with at least one odd prime divisor <= prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A331809 a(1) = 1; a(2) = 2; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2*a(i) nor Sum_{j=1..n-1} ( b_j*a(j) ) where 0 < i < n, b_j = 1 or 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A331800 a(1) = 1; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2*a(i) nor Sum_{j=1..n-1} ( b_j*a(j) ) where 0 < i < n, b_j = 0 or 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A331811 a(n) is the next number after a(n-1) which cannot be represented in the form 2*a(i) and Sum_{j=1..n-1} b_j*a(j) where 0 < i < n, b_j = 1 or 0. The sequence starts: a(1) = 1; a(2) = 2; a(3) = 3; a(4) = 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A309945 a(n) = floor(n - sqrt(2*n-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A306863 a(n) is the number of primes between the n-th and (n+1)-st odd composite numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A342552 The number of twin primes between prime(i)prime(i+1) and prime(i+1)prime(i+2) where prime(i) and prime(i+1) are twin primes.

A331809 a(1) = 1; a(2) = 2; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2a(i) nor Sum_{j=1..n-1} ( b_ja(j) ) where 0 < i < n, b_j = 1 or 0.

A331800 a(1) = 1; thereafter a(n) is the smallest number > a(n-1) which is neither of the form 2a(i) nor Sum_{j=1..n-1} ( b_ja(j) ) where 0 < i < n, b_j = 0 or 1.

A331811 a(n) is the next number after a(n-1) which cannot be represented in the form 2a(i) and Sum_{j=1..n-1} b_ja(j) where 0 < i < n, b_j = 1 or 0. The sequence starts: a(1) = 1; a(2) = 2; a(3) = 3; a(4) = 5.