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 1-10 of 13 results. Next

A244520 a(n) = A080715(n+1) / 2.

Original entry on oeis.org

1, 3, 5, 11, 15, 21, 29, 35, 39, 41, 51, 65, 95, 105, 155, 165, 179, 191, 221, 231, 239, 281, 329, 371, 419, 431, 485, 519, 611, 641, 659, 809, 905, 935, 989, 1019, 1031, 1049, 1121, 1199, 1229, 1289, 1451, 1469, 1481, 1509, 1541, 1661, 1821, 1931, 2109, 2129, 2141, 2339, 2549, 2795, 2969, 3021, 3039, 3189, 3299, 3329
Offset: 1

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Author

Joerg Arndt, Jul 10 2014

Keywords

Comments

Numbers k such that 2d + k/d is prime for every d|k. Such k must be an odd squarefree number. Primes in the sequence are A045536. - Thomas Ordowski, Nov 16 2017

Links

Crossrefs

Cf. A045536, A080715, A088627.

Programs

  • Mathematica
    Select[Range[1, 3400, 2], Function[n, AllTrue[Divisors@ n, PrimeQ[2 # + n/#] &]]] (* Michael De Vlieger, Nov 18 2017 *)
  • PARI
    is_ok(n)=n=2*n;fordiv(n,d,if(!isprime(d+n/d),return(0)));return(1);
for(n=1,10^4,if(is_ok(n),print1(n,", ")));

Formula

A088627(a(n)) = A000005(a(n)) = 2^m. - Thomas Ordowski, Nov 16 2017

A268403 Partial sums of A080715.

Original entry on oeis.org

1, 3, 9, 19, 41, 71, 113, 171, 241, 319, 401, 503, 633, 823, 1033, 1343, 1673, 2031, 2413, 2855, 3317, 3795, 4357, 5015, 5757, 6595, 7457, 8427, 9465, 10687, 11969, 13287, 14905, 16715, 18585, 20563, 22601, 24663, 26761, 29003, 31401, 33859, 36437, 39339, 42277
Offset: 1

Views

Author

Peter Kagey, Feb 03 2016

Keywords

Links

A145832 Numbers k such that for each divisor d of k, d + k/d is "round" ("square-root smooth").

Original entry on oeis.org

3, 7, 11, 15, 17, 23, 29, 31, 35, 39, 47, 53, 55, 59, 63, 71, 79, 83, 89, 95, 97, 107, 111, 119, 125, 127, 131, 139, 143, 146, 149, 159, 161, 164, 167, 175, 179, 181, 191, 197, 199, 207, 209, 215, 223, 233, 239, 241, 251, 263, 269, 279, 287, 293, 299, 307, 311
Offset: 1

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Author

Dan Sonnenschein (dans(AT)portal.ca), Oct 20 2008

Keywords

Comments

A necessary condition is that the number be one less than a round number; if this number is prime it's in the sequence.
Even composites in this sequence seem rare (see examples below for more details).

Examples

			The first term is a prime one less than the round number 4.
The first composite number in this sequence is 15, with divisor-pair sum 3+5 = 8.
Another such composite is 63, with divisor-pair sums: 3+21 = 24, 7+9 = 16.
There are only five even composites among the first 100 terms of this sequence.
The first such is 146, with divisor-pair sum 2+73 = 75. The second is 164, with divisor-pair sums 2+82 = 84 and 4+41 = 45. The remaining three are 458, 524 and 584.

Links

Crossrefs

Cf. A004809, A048098, A080715.

Programs

  • Magma
    [ n: n in [1..310] | forall{ k: k in [ Integers()!(d+n/d): d in [ D[j]: j in [1..a] ] ] | k ge (IsEmpty(T) select 1 else Max(T) where T is [ x[1]: x in Factorization(k) ])^2 } where a is IsOdd(#D) select (#D+1)/2 else #D/2 where D is Divisors(n) ]; // Klaus Brockhaus, Oct 24 2008
  • Mathematica
    smQ[n_] := FactorInteger[n][[-1, 1]]^2 <= n; seqQ[n_] := AllTrue[Divisors[n], smQ[# + n/#] &]; Select[Range[320], seqQ] (* Amiram Eldar, Jun 13 2020 *)

Extensions

Wrong term 305 removed by Amiram Eldar, Jun 13 2020

A179993 Numbers m with the property that, when a and b are positive integers such that a*b = m, |a-b| is prime.

Original entry on oeis.org

3, 8, 14, 18, 38, 62, 98, 138, 230, 258, 278, 318, 338, 390, 398, 402, 458, 542, 678, 710, 770, 798, 822, 938, 1022, 1118, 1202, 1238, 1298, 1322, 1490, 1622, 1658, 2030, 2222, 2238, 2378, 2438, 2522, 2558, 2618, 2858, 2910, 3002, 3218, 3230, 3698, 4058, 4178
Offset: 1

Views

Author

Emmanuel Vantieghem, Aug 05 2010, Aug 06 2010

Keywords

Comments

All numbers in this sequence are congruent to 0 or 2 mod 3.
It is not known if this sequence is infinite. For n > 1 all terms are even.
The intersection with A080715 seems to be empty. Is this provable ?
From Amiram Eldar, Nov 15 2021: (Start)
The nonsquarefree terms of this sequence, 8, 18, 98, 338, ..., are numbers of the form 2*p^2, where p is in A349327.
The least terms with 1, 2, 3, 4 and 5 distinct prime divisors are 3, 14, 138, 390 and 13576178, respectively. Are there terms with more than 5 distinct prime divisors? (End)
All terms have either 6 (for a(n) = 2*A349327^2) or 2^k (for a(n) in A005117) divisors. - Samuel Harkness, Mar 02 2023

Examples

			Example : For n = 5, the possible values of |a-b| are 17 = 19-2 and 37 = 38-1.

Links

Crossrefs

Cf. A080715, A349327, A005117.

Programs

  • Mathematica
    m=1;While[m < 10000, m++; If[Mod[m, 3] == 1, , V = Divisors[m]; L = Length[V]; j = 0; While[j < L/2, j++; x = (m/V[[j]]) - V[[j]]; If[PrimeQ[x], , j = L]]; If[j == L/2, X = Append[X, m],]]]; X
q[n_] := AllTrue[Divisors[n], #^2 > n || PrimeQ[Abs[# - n/#]] &]; Select[Range[4000], q] (* Amiram Eldar, Nov 15 2021 *)
  • Python
    from itertools import islice, takewhile, count
from sympy import isprime, divisors
def A179993(): # generator of terms
    for m in count(1):
        if all(isprime(m//a-a) for a in takewhile(lambda x: x*x <= m, divisors(m))):
            yield m
A179993_list = list(islice(A179993(),20)) # Chai Wah Wu, Nov 15 2021

A236423 Numbers k such that m^2 + k^2/m^2 is prime for every m|k.

Original entry on oeis.org

1, 2, 6, 10, 14, 26, 74, 94, 130, 134, 146, 170, 206, 326, 386, 466, 470, 634, 1094, 1354, 1570, 1654, 1766, 1966, 2174, 2766, 3046, 3254, 3274, 3446, 4006, 4174, 4666, 4754, 4954, 5086, 5774, 5834, 6046, 6866, 6926, 7114, 7466, 8854, 9046, 9494, 10006, 10126
Offset: 1

Views

Author

Thomas Ordowski, Jan 25 2014

Keywords

Comments

If n = x*y then x^2 + y^2 is a prime.
These n > 1 must be even and squarefree.
Conjecture: the set of such n is infinite.
The conjecture follows from, e.g., Schinzel's hypothesis H. - Charles R Greathouse IV, Jan 28 2014

Links

Crossrefs

Cf. A080715.
Subsequence of A005574. - Michel Marcus, Jun 03 2017

Programs

  • Mathematica
    Select[Range[10^4], (d = Divisors[#]^2; And @@ PrimeQ[d + #^2/d]) &] (* Giovanni Resta, Jan 26 2014 *)
  • PARI
    isok(n) = sumdiv(n, d, isprime(d^2 + n^2/d^2)) == numdiv(n); \\ Michel Marcus, Jan 25 2014
  • PARI
    is(n)=if(n%4!=2, return(n==1)); my(f=factor(n)); if(vecmax(f[,2])>1,return(0)); fordiv(f,m,if(!isprime(m^2+(n/m)^2),return(0)); if(m^2>n,break));1 \\ Charles R Greathouse IV, Jan 28 2014

Extensions

More terms from Michel Marcus, Jan 25 2014

A293756 a(n) = smallest number k with n prime factors such that d + k/d is prime for every d | k.

Original entry on oeis.org

1, 2, 6, 30, 210, 186162
Offset: 0

Views

Author

Thomas Ordowski, Nov 11 2017

Keywords

Comments

For n > 0, a(n) is even and squarefree.
For n > 0, a(n) gives 2^(n-1) distinct primes.
If the k-tuple conjecture is true, then this sequence is infinite. - Carl Pomerance, Nov 12 2017
a(n) is the least integer k with n prime divisors such that A282849(k) = A000005(k). - Michel Marcus, Nov 13 2017
a(n) is the smallest k with n prime factors such that A282849(k) = 2^n. - Thomas Ordowski, Nov 13 2017
a(6), if it exists, has a prime divisor greater than 10^3. - Arkadiusz Wesolowski, Nov 14 2017

Examples

			a(2) = 2*3 = 6 because k = 6 is the smallest number with 2 prime factors such that for d = {1, 2, 3, 6} we have 1 + 6/1 = 6 + 6/6 = 7 is prime and 2 + 6/2 = 3 + 6/3 = 5 is prime.
From _Michael De Vlieger_, Nov 14 2017: (Start)
First differences of prime indices of a(n):
n       a(n)   A287352(a(n))
-----------------------------
1         2    1
2         6    1, 1
3        30    1, 1, 1
4       210    1, 1, 1, 1
5    186162    1, 1, 6, 1, 11
(End)

Links

Crossrefs

Subsequence of A080715 (d + k/d is prime for every d|k).
Cf. A103787, A282849, A294925, A295124, A295169.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,j,k,n,ok; print(1);for n from 1 to q do for k from 2 to q do a:=ifactors(k)[2]; a:=add(a[j][2],j=1..nops(a)); if a=n then b:=divisors(k); ok:=1;
for j from 1 to nops(b) do if not isprime(b[j]+k/b[j]) then ok:=0; break; fi; od; if ok=1 then print(k); break; fi; fi; od; od; end: P(10^8); # Paolo P. Lava, Nov 16 2017
  • PARI
    isok(k, n)=if (!issquarefree(k), return (0)); if (omega(k) != n, return (0)); fordiv(k, d, if (!isprime(d+k/d), return(0))); 1;
a(n) = {my(k=1); while( !isok(k, n), k++); k;} \\ Michel Marcus, Nov 11 2017

Formula

a(n) = 2*A295124(n-1) for n > 0. - Thomas Ordowski, Nov 15 2017

Extensions

a(5) from Michel Marcus, Nov 11 2017

A233514 Numbers n such that all numbers of the form Fib(n)/d + d are prime (or nonexistent), where Fib(n) is the n-th Fibonacci number and d is a nontrivial divisor of Fib(n).

Original entry on oeis.org

3, 4, 5, 7, 9, 11, 13, 15, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677
Offset: 1

Views

Author

T. D. Noe, Jan 31 2014

Keywords

Comments

Damir et al. conjecture that this sequence is finite.

Links

Crossrefs

Not a subsequence of A037917.
Cf. A001605, A080715, A233515.

Programs

  • Mathematica
    f2[n_] := Module[{d = Rest[Most[Divisors[n]]]}, n/d + d]; Select[Range[3, 200], And @@ PrimeQ[f2[Fibonacci[#]]] &]
  • PARI
    is(n)=my(F=fibonacci(n)); if(n%6==0 || n%25==0 || n%56==0 || n%91==0 || n%110==0 || n%153==0 || !issquarefree(F), return(0)); fordiv(F,d,if(d>1 && dCharles R Greathouse IV, Feb 04 2014

Extensions

a(18)-a(24) from Charles R Greathouse IV, Feb 04 2014
Terms 2971 to 9677 from Don Reble. - N. J. A. Sloane, Nov 04 2022

A233515 Fibonacci numbers whose index is in A233514.

Original entry on oeis.org

2, 3, 5, 13, 34, 89, 233, 610, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241
Offset: 1

Views

Author

T. D. Noe, Jan 31 2014

Keywords

Comments

Damir et al. conjecture that this sequence is finite. Note that it is A005478 (prime Fibonacci numbers) and two additional terms (34 and 610) so far.

Links

Crossrefs

Cf. A005478, A080715, A233514.

A235334 Numbers n such that for any positive integers (a, b), if a * b = n then a + b is a square.

Original entry on oeis.org

3, 323, 5183, 777923, 1327103, 6718463, 12446783, 16402499, 229159043, 432972863, 1214383103, 2191925123, 4787532863, 6927565823, 10809345023, 12619826243, 22218287363, 31123310723, 32399999999, 42469790723, 79101562499, 131734154303, 151291437443
Offset: 1

Views

Author

Michel Lagneau, Jan 06 2014

Keywords

Comments

It seems that n is the product of twin primes of A232878 for n > 3.
Conjecture: the numbers n such that for any positive integers (a, b), a * b = n and a + b is a square are the product of twin primes, and a*b+1 is also a perfect square.

Examples

			323 is the product of two positive integers in 2 ways: 1 * 323 and 17 * 19. The sums of the pairs of multiplicands are 323+1 = 18^2 and 17+19 = 6^2 respectively. All are squares.

Links

Crossrefs

Cf. A001097, A080715, A232878.

Programs

  • Mathematica
    t={}; Do[ds=Divisors[n]; If[EvenQ[Length[ds]], ok=True; k=1; While[k<=Length[ds]/2 && (ok=IntegerQ[Sqrt[ds[[k]]+ds[[ -k]]]]), k++ ]; If[ok, AppendTo[t, n]]], {n, 2, 10^8}]; t ***[Program from T.D. Noe adapted for this sequence. See A080715]***
  • PARI
    isok(n) = {d = divisors(n); if (#d % 2, return (0)); for (i = 1, #d/2, if (! issquare(d[i]+n/d[i]), return (0));); return (1);} \\ Michel Marcus, Jan 06 2014

Extensions

a(21)-a(23) from Hiroaki Yamanouchi, Oct 02 2014

A280590 Numbers k such that for any positive integers (a, b), if a * b = k then sigma(a) + sigma(b) is a prime.

Original entry on oeis.org

1, 3, 5, 6, 11, 17, 24, 26, 27, 29, 38, 41, 59, 71, 101, 107, 125, 137, 149, 158, 179, 191, 197, 206, 218, 227, 239, 269, 281, 311, 344, 347, 419, 431, 446, 458, 461, 521, 536, 569, 599, 617, 641, 659, 698, 809, 821, 827, 857, 878, 881, 1019, 1031, 1049, 1061
Offset: 1

Views

Author

Michel Lagneau, Jan 06 2017

Keywords

Comments

The subsequence of primes {3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, ... is exactly A001359 (lesser of twin primes).

Examples

			1 is in the sequence because 1 = 1*1 and sigma(1) + sigma(1) = 1 + 1 = 2 is prime.
24 is in the sequence because A038548(24) = 4 => four decompositions of 24 = 1*24 = 2*12 = 3*8 = 4*6 and
sigma(1) + sigma(24) =  1 + 60 = 61 is prime;
sigma(2) + sigma(12) =  3 + 28 = 31 is prime;
sigma(3) + sigma(8)  =  4 + 15 = 19 is prime;
sigma(4) + sigma(6)  =  7 + 12 = 19 is prime.

Links

Crossrefs

Cf. A000203, A001359, A038548, A080715.

Programs

  • Mathematica
    t={};Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=PrimeQ[DivisorSigma[1,ds[[k]]]+DivisorSigma[1,ds[[-k]]]]),k++];If[ok,AppendTo[t,n]]],{n,2,4000}];t
  • PARI
    isok(n) = {fordiv(n, d, if (d^2 <= n, if (! isprime(sigma(d) + sigma(n/d)), return (0)););); return(1);} \\ Michel Marcus, Jan 06 2017
Showing 1-10 of 13 results. Next

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