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.

A256387 Numbers n such that no prime can be the arithmetic mean of 2 semiprimes whose difference is 2*n.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 63, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143
Offset: 1

Views

Author

Michel Marcus, Mar 27 2015

Keywords

Comments

That is, there is no prime p, such that p+n and p-n are both semiprime.
Complement of A256389.
From Robert Israel, Apr 13 2020: (Start)
Includes odd number n if and only if n+4 is not prime or 2*n+4 is not a semiprime.
There any no even members up to 10^5. Conjecture: all members are odd. (End)

Examples

			A256383 is the list of numbers n such that n-5 and n+5 are semiprimes, and it contains no prime, hence 5 is in the sequence.

Links

Crossrefs

Cf. A256383.
Cf. A256388 (a single prime), A256389 (one or more primes).

Programs

  • Maple
    select(t -> not isprime(t+4) or numtheory:-bigomega(2*t+4) <> 2, [seq(i,i=1..1000,2)]); # Robert Israel, Apr 13 2020

A256389 Numbers n such that one or more primes can be the arithmetic mean of 2 semiprimes whose difference is 2*n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 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, 62, 64, 66, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110
Offset: 1

Views

Author

Michel Marcus, Mar 27 2015

Keywords

Comments

That is, there are several primes p, such that p+n and p-n are both semiprime.
Complement of A256387.
The terms of this sequence that do not belong to A256388 are even.

Examples

			A256381 is the list of numbers n such that n-3 and n+3 are semiprimes, and it contains a single prime, hence 3 is in the sequence.
A256382 is the list of numbers n such that n-4 and n+4 are semiprimes, and it contains several primes, hence 4 is in the sequence.

Crossrefs

Cf. A124936, A105571, A256382.
Cf. A256387 (no prime), A256388 (a single prime).

A284014 Numbers k such that {k + 2, k + 4} and {k^2 + 2, k^2 + 4} are both twin prime pairs.

Original entry on oeis.org

1, 3, 15, 57, 147, 2085, 6687, 6957, 11055, 15267, 17385, 17577, 20505, 20637, 23667, 26247, 31077, 31317, 32115, 32967, 34497, 39225, 47775, 52065, 53715, 55335, 56205, 58365, 62187, 63585, 66567, 67215, 70875, 77235, 77475, 82005, 85827, 89595, 89817, 107505
Offset: 1

Views

Author

K. D. Bajpai, Mar 18 2017

Keywords

Comments

After a(1), all the terms are multiples of 3.
After a(2), all the terms are congruent to 5 or 7 (mod 10).

Examples

			a(2) = 3, {3 + 2 = 5, 3 + 4 = 7} and {3^2 + 2 = 11, 3^2 + 4 = 13} are twin prime pairs.
a(3) = 15, {15 + 2 = 17, 15 + 4 = 19} and {15^2 + 2 = 227, 15^2 + 4 = 229} are twin prime pairs.

Links

Crossrefs

Appears to be the intersection of A086381 and A256388, but that may be unproven.
Cf. A000040, A001359, A007591, A010051, A067201, A178336, A178337.

Programs

  • Magma
    [n: n in [0..100000] | IsPrime(n+2) and IsPrime(n+4) and IsPrime(n^2+2) and IsPrime(n^2+4)];
  • Mathematica
    Select[Range[1000000], PrimeQ[# + 2] && PrimeQ[# + 4] && PrimeQ[#^2 + 2] && PrimeQ[#^2 + 4] &]
  • PARI
    for(n=1, 100000,2; if(isprime(n+2) && isprime(n+4) && isprime(n^2+2) &&isprime(n^2+4), print1(n, ", ")))
  • Scheme
    ;; With Antti Karttunen's IntSeq-library.
(define A284014 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 (+ n 2))) (= 1 (A010051 (+ n 4))) (= 1 (A010051 (+ (* n n) 2))) (= 1 (A010051 (+ (* n n) 4)))))))
;; Antti Karttunen, Apr 15 2017

A284058 Numbers k such that {k + 2, k + 4} and {k^3 + 2, k^3 + 4} are twin prime pairs.

Original entry on oeis.org

1, 3, 69, 1719, 3555, 8535, 8625, 9765, 10065, 17955, 27939, 32319, 34209, 35445, 39159, 44769, 47415, 55329, 56235, 75615, 85929, 91965, 96219, 97545, 98895, 122385, 122595, 138075, 142695, 143649, 145719, 152025, 191829, 192975, 197955, 200379, 201819, 202059
Offset: 1

Views

Author

K. D. Bajpai, Mar 19 2017

Keywords

Comments

After a(1), all the terms are multiples of 3.
After a(2), all the terms are congruent to 5 or 9 (mod 10).
a(n) == {9 or 15} (mod 30) for n>2. - Robert G. Wilson v, Mar 19 2017

Examples

			a(2) = 3, {3 + 2 = 5, 3 + 4 = 7} and {3^3 + 2 = 29, 3^3 + 4 = 31} are twin prime pairs.
a(3) = 69, {69 + 2 = 71, 69 + 4 = 73} and {69^3 + 2 = 328511, 69^3 + 4 = 328513} are twin prime pairs.

Links

Crossrefs

Intersection of A256388 and A178337.
Cf. A000040, A001359, A086381, A144953, A178336.

Programs

  • Mathematica
    Select[Range[1000000], PrimeQ[# + 2] && PrimeQ[# + 4] && PrimeQ[#^3 + 2] && PrimeQ[#^3 + 4] &]
  • PARI
    for(n=1, 100000,2; if(isprime(n+2) && isprime(n+4) && isprime(n^3+2) && isprime(n^3+4), print1(n, ", ")))

A306353 Number of composites among the first n composite numbers whose least prime factor p is that of the n-th composite number.

Original entry on oeis.org

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

Views

Author

Jamie Morken and Vincenzo Librandi, Feb 09 2019

Keywords

Comments

Composites with least prime factor p are on that row of A083140 which begins with p
Sequence with similar values: A122005.
Sequence written as a jagged array A with new row when a(n) > a(n+1):
1, 2, 3,
1, 4, 5, 6,
2, 7, 8, 9,
3, 10, 11,
1, 12,
4, 13, 14, 15,
5, 16,
2, 17, 18,
6, 19, 20, 21,
7, 22, 23,
1, 24,
8, 25, 26,
3, 27,
9, 28, 29, 30.
A153196 is the list B of the first values in successive rows with length 4.
B is given by the formula for A002808(x)=A256388(n+3), an(x)=A153196(n+2)
For example: A002808(26)=A256388(3+3), an(26)=A153196(3+2).
A243811 is the list of the second values in successive rows with length 4.
A047845 is the list of values in the second column and A104279 is the list of values in the third column of the jagged array starting on the second row.
Sequence written as an irregular triangle C with new row when a(n)=1:
1,2,3,
1,4,5,6,2,7,8,9,3,10,11,
1,12,4,13,14,15,5,16,2,17,18,6,19,20,21,7,22,23,
1,24,8,25,26,3,27,9,28,29,30,10,31,4,32,33,11,34,35,36,12,37,2,38,39,13,40,41,5,42,14,43,44,3,45,15,46,6,47,48,16,49,50,51,17,52,53,54,18,55,56,7,57,19,58,4,59.
A243887 is the last value in each row of C.
The second value D on the row n > 1 of the irregular triangle C is a(A053683(n)) or equivalently A084921(n). For example for row 3 of the irregular triangle:
D = a(A053683(3)) = a(16) = 12 or D = A084921(3) = 12. This is the number of composites < A066872(3) with the same least prime factor p as the A053683(3) = 16th composite, A066872(3) = 26.
The number of values in each row of the irregular triangle C begins: 3,11,18,57,39,98,61,141,265,104,351,268,...
The second row of the irregular triangle C is A117385(b) for 3 < b < 15.
The third row of the irregular triangle C has similar values as A117385 in different order.

Examples

			First composite 4, least prime factor is 2, first case for 2 so a(1)=1.
Next composite 6, least prime factor is 2, second case for 2 so a(2)=2.
Next composite 8, least prime factor is 2, third case for 2 so a(3)=3.
Next composite 9, least prime factor is 3, first case for 3 so a(4)=1.
Next composite 10, least prime factor is 2, fourth case for 2 so a(5)=4.

Links

Crossrefs

Cf. A002808, A256388, A056608, A083140, A122005, A153196, A243811, A047845, A104279, A243887, A117385, A216244, A084921, A066872, A053683, A038110, A038111, A020639.

Programs

  • Mathematica
    counts = {}
values = {}
For[i = 2, i < 130, i = i + 1,
If[PrimeQ[i], ,
x = PrimePi[FactorInteger[i][[1, 1]]];
  If[Length[counts] >= x,
   counts[[x]] = counts[[x]] + 1;
   AppendTo[values, counts[[x]]], AppendTo[counts, 1];
   AppendTo[values, 1]]]]
   (* Print[counts] *)
   Print[values]
  • PARI
    c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = my(c=c(n), lpf = vecmin(factor(c)[,1]), nb=0); for(k=2, c, if (!isprime(k) && vecmin(factor(k)[,1])==lpf, nb++)); nb; \\ Michel Marcus, Feb 10 2019

Formula

a(n) is approximately equal to A002808(n)*(A038110(x)/A038111(x)), with A000040(x)=A020639(A002808(n)).
For example if n=325, a(325)~= A002808(325)*(A038110(2)/A038111(2)) with A000040(2)=A020639(A002808(325)).
This gives an estimate of 67.499... and the actual value of a(n)=67.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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