A037074 -id:A037074 - cp's OEIS frontend

A123754 Positive numbers of the form 4*n^2 - 1 which are not semiprimes.

3, 63, 99, 195, 255, 399, 483, 575, 675, 783, 1023, 1155, 1295, 1443, 1599, 1935, 2115, 2303, 2499, 2703, 2915, 3135, 3363, 3843, 4095, 4355, 4623, 4899, 5475, 5775, 6083, 6399, 6723, 7055, 7395, 7743, 8099, 8463, 8835, 9215, 9603, 9999, 10815

Offset: 1

Roger L. Bagula, Nov 16 2006

nonn

Positive numbers of the form 4*n^2 - 1 which are semiprimes can be found in A037074.

Or, all positive products of the form A014076(i)*[A014076(i)+-2], duplicates removed. - R. J. Mathar, Aug 08 2007

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000466, A037074.

Select[4*(Range[54])^2-1, Not[PrimeQ[Sqrt[(#+ 1)]-1] && PrimeQ[Sqrt[(#+1)]+1]]&]
Select[4*Range[100]^2-1,PrimeOmega[#]!=2&] (* Harvey P. Dale, Jul 24 2016 *)

Equals ( A000466 - {-1} ) - A001358. - R. J. Mathar, Aug 08 2007

Edited by N. J. A. Sloane, Aug 03 2007

A126248 p(p+1)(p+2) where (p,p+2) are twin primes.

Original entry on oeis.org

60, 210, 1716, 5814, 26970, 74046, 215940, 373176, 1061106, 1259604, 2627934, 3374850, 5831820, 7077696, 7762194, 11852124, 13823760, 19682730, 22425486, 30371016, 42143844, 74087580, 80621136, 98610666, 142236126, 185192430

Offset: 1

Lekraj Beedassy, Dec 21 2006

nonn

Cf. A126249.

a(n) = A001359(n)*A014574(n)*A006512(n) = A037074(n)*A014574(n) = A007531(A006512(n)). a(n)=6*A126249(n).

A134020 Numbers that are one less than a square and have exactly 4 divisors.

Original entry on oeis.org

8, 15, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163

Offset: 1

Giovanni Teofilatto, Jan 11 2008

nonn

A037074 is a subsequence. - R. J. Mathar, Jun 08 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..5400

with(numtheory): a:=proc(n) if type(sqrt(n+1),integer)=true and tau(n)=4 then n else end if end proc: seq(a(n),n=1..800000); # Emeric Deutsch, Jan 27 2008

t={}; Do[If[DivisorSigma[0,n]==4&&IntegerQ[Sqrt[n+1]],AppendTo[t,n]],{n,10^6}]; t (* Jayanta Basu, Jun 03 2013 *)
Select[Range[1000]^2-1,DivisorSigma[0,#]==4&] (* Harvey P. Dale, Jul 28 2023 *)

Edited and extended by Emeric Deutsch, Jan 27 2008

Name reworded by Jon E. Schoenfield, Jan 15 2019

A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

Original entry on oeis.org

1, 4, 9, 10, 16, 18, 20, 25, 26, 27, 28, 33, 34, 36, 40, 42, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 95, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 120, 121, 122, 123, 124, 125

Offset: 1

Jonas Wallgren, Aug 10 2009, Aug 17 2009

nonn

From Franklin T. Adams-Watters, Aug 29 2009: (Start)
The k_i must all be positive integers.
Note that every integer > 33 is the sum of 5 positive squares, and for n > 5, every integer > n+13 is the sum of n positive squares. (End)
The complement of this sequence includes: A000040, A037074, A143206, 2 * A002145, and 3 * A094712. - Robert Israel, Jan 27 2025

			34 = 2*(4^2 + 1^2), 42 = 3*(3^2 + 2^2 + 1^2), thus 34 and 42 are in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290, A000404, A000408, A000414, A047700, A111178. [From Franklin T. Adams-Watters, Aug 29 2009]

Cf. A000040, A002145, A037074, A094712, A143206.

g:= proc(y,m)
  # can we write y as sum of m positive squares?
   option remember;
   local x;
   if y < m then return false fi;
   if m = 1 then return issqr(y) fi;
   if issqr(y-m+1) then return true fi;
   for x from 1 while x^2 + m-1 < y do
     if procname(y-x^2,m-1) then return true fi
   od;
   false
end proc:
filter:= proc(n)
  ormap(t -> g(n/t, t), numtheory:-divisors(n))
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 26 2025

issumsqs(n,k) = if(n<=0||k<=0,return(k==0&&n==0)); forstep(j=sqrtint(n),max(sqrtint(n\k),1),-1,if(issumsqs(n-j^2,k-1),return(1)));0
isa(n)=local(ds);ds=divisors(n);for(k=1,(#ds+1)\2,if(issumsqs(n\ds[k],ds[k]),return(1)));0
for(n=1,200,if(isa(n),print1(n","))) \\ Franklin T. Adams-Watters, Aug 29 2009

More terms from Franklin T. Adams-Watters, Aug 29 2009

A165280 If p and q are twin primes then pq + 1 is always divisible by 3, except for (p,q)=(3,5). Sequence gives values of (pq + 1)/3.

Original entry on oeis.org

12, 48, 108, 300, 588, 1200, 1728, 3468, 3888, 6348, 7500, 10800, 12288, 13068, 17328, 19200, 24300, 26508, 32448, 40368, 58800, 62208, 71148, 90828, 108300, 120000, 127308, 137388, 145200, 218700, 225228, 228528, 245388, 259308, 346800

Offset: 1

Tanin (Mirza Sabbir Hossain Beg) (mirzasabbirhossainbeg(AT)yahoo.com), Sep 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001097, A001359, A006512, A037074, A075369.

[(p+1)^2 div 3: p in PrimesInInterval(5,1100) |IsPrime(p+2)]; // Marius A. Burtea, Jan 02 2020

p = Rest@ Select[ Prime@ Range@ 175, PrimeQ[ # + 2] & ]; (p (p + 2) + 1)/3 (* Robert G. Wilson v, Sep 13 2009 *)
(Times@@#+1)/3&/@Select[Partition[Prime[Range[3,200]],2,1],#[[2]]-#[[1]] == 2&] (* Harvey P. Dale, Aug 28 2022 *)

a(n) = (A001359(n+1)*A006512(n+1)+1)/3 = (A037074(n+1)+1)/3 = A075369(n+1)/3. - Robert G. Wilson v, Sep 13 2009

More terms from Robert G. Wilson v, Sep 13 2009

Edited by N. J. A. Sloane, Sep 15 2009

A229108 Increasing a(n)is the smallest number of the form p^a*q^b, where a,b are positive integers and p < q are odd primes such that max( p^a, q^b)/min( p^a, q^b) <= 1 + 2/prime(n).

Original entry on oeis.org

15, 35, 63, 143, 323, 575, 675, 783, 899, 1763, 2303, 3599, 5183, 6399, 6723, 10403, 11663, 15875, 19043, 22499, 27221, 28223, 32399, 36863, 39203, 50621, 51983, 53357, 57599, 58563, 72899, 77837, 79523, 95477, 97343, 119021, 121103, 123197, 129599

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Sep 13 2013

nonn

			15 is the least number of considered form, and 5/3 = 1 + 2/prime(2). So a(2)=15; in case of n=23, not only 28223 but also 29237 satisfies required inequality and we choose the smallest from them.

Peter J. C. Moses, Table of n, a(n) for n = 2..1001

Cf. A037074.

tmp={1}; Do[test=1+2/Prime[n]; AppendTo[tmp, NestWhile[#+2&, Last[tmp]+2, !((Max[#]/Min[#]&[Map[#[[1]]^#[[2]]&, FactorInteger[#]]] <= test) && (Length[FactorInteger[#]]==2))&]], {n,2,30}]; Rest[tmp]

factorPP(n)=my(f=factor(n)); vecsort(vector(#f~,i,f[i,1]^f[i,2]))
list(n)=my(v=primes(n),t=1,f);for(i=1,n,while(1, f=factorPP(t += 2); if(#f==2 && f[2]/f[1] <= 1+2/v[i], v[i]=t; break))); v \\ Charles R Greathouse IV, Sep 13 2013

If [prime(n), prime(n+1)] is a twin pair, then a(n) <= prime(n)*prime(n+1).

A234542 Positive integer solutions to sigma(sigma(k) - k - 3)/phi(k - phi(k) + 3) = 3.

Original entry on oeis.org

15, 35, 68, 95, 119, 143, 155, 188, 203, 280, 289, 299, 323, 371, 395, 611, 695, 731, 779, 791, 803, 851, 899, 923, 959, 995, 1055, 1139, 1146, 1199, 1355, 1369, 1379, 1391, 1403, 1424, 1643, 1703, 1739, 1751, 1763, 1883, 1895, 1919, 2051, 2123, 2159, 2231, 2471, 2483, 2723, 2759, 2809

Offset: 1

Wesley Ivan Hurt, Dec 27 2013

nonn

If n is the product of a pair of twin primes (A037074), then n is in the sequence (The first few terms of A037074 are: 15, 35, 143, 323, 899, 1763, 3599, ..). For these numbers, the numerator is equal to 3*sqrt(n + 1) and the denominator (A186749) is equal to sqrt(n + 1), giving 3 as a result.

			119 appears in the sequence since sigma(sigma(119) - 119 - 3)/phi(119 - phi(119) + 3) = sigma(22)/phi(26) = 36/12 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001065, A037074, A051953, A186749.

with(numtheory); A234542:=n->`if`(sigma(sigma(n)-n-3)/phi(n-phi(n)+3)=3,n,NULL); seq(A234542(n), n=1..5000);

Select[Range[1000], DivisorSigma[1, DivisorSigma[1, #] - # - 3]/EulerPhi[# - EulerPhi[#] + 3] == 3 &] (* Alonso del Arte, Jan 01 2014 *)

Solutions to A000203(A001065(k) - 3)/A000010(A051953(k) + 3) = 3.

A263927 Least solution x to the equation x' = (x+1)/n, where x' is the arithmetic derivative of x; -1 if no solution exists.

Original entry on oeis.org

30, 15, 2, 3, 119, 5, 209, 7, 323, 559, 527, 11, 779, 13, 899, 1919, 1189, 17, 2507, 19, 1763, 42455401, 2759, 23, 5249, 2911, 3239, 23519, 3827, 29, 5207, 31

Offset: 1

Paolo P. Lava, Oct 30 2015

For n <= 100 unknown terms are a(33), a(57), a(73), a(88) and a(92) (tested up to 10^8).
All solutions for n = 1 are Giuga numbers. - Paolo P. Lava, Jul 06 2018
a(33) and a(57) are greater than 10^12, if they exist. - Giovanni Resta, Jul 09 2018

			30' = 31 and 31 = (30 + 1) / 1;
15' = 8 and 8 =(15 + 1) / 2;
2' = 1 and 1 = (2 + 1) / 3.

Paolo P. Lava, Values for n<=100

Cf. A003415, A007850 (Giuga numbers), A014574, A037074.

with(numtheory): P:=proc(q) local a,k,n,p;
for k from 1 to q do for n from 1 to q do
if n*add(op(2,p)/op(1,p),p=ifactors(n)[2])=(n+1)/k then
print(n); break; fi; od; od; end: P(10^9);

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Table[k = 0; While[f@ k != (k + 1)/n, k++]; k, {n, 21}] (* Michael De Vlieger, Nov 05 2015, after Michael Somos at A003415 *)

d(x) = {local(fac); if(x<1, 0, fac=factor(x); sum(i=1, matsize(fac)[1], x*fac[i, 2]/fac[i, 1]))}
a(n) = {x=2; while(k, if(d(x) == (x+1)/n, return(x)); x++)} \\ Altug Alkan, Nov 05 2015

a(n) = n-1 if n-1 is a prime.

a(A014574(n)/2) = A037074(n) if n-1 is not a prime. - Paolo P. Lava, Jul 06 2018

Added a(52) and a(58) in a-file by Giovanni Resta, Jul 09 2018

A334141 Numbers that are the product of distinct twin primes.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 29, 31, 33, 35, 39, 41, 43, 51, 55, 57, 59, 61, 65, 71, 73, 77, 85, 87, 91, 93, 95, 101, 103, 105, 107, 109, 119, 123, 129, 133, 137, 139, 143, 145, 149, 151, 155, 165, 177, 179, 181, 183, 187, 191, 193, 195, 197, 199

Offset: 1

Ilya Gutkovskiy, Apr 15 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Subsequence of A005117.

Cf. A001097, A037074, A048599, A074480, A073485.

N:= 1000: # for terms <= N
P:= select(isprime, {seq(i,i=3..N+2,2)}):
TP:= P intersect map(`+`,P,2):
TP:= map(t -> (t-2,t), TP):
TP:= sort(convert(TP,list)):
S:= {1}:
for i from 1 to nops(TP) do
  S0:= S;
  S:= S union map(`*`, select(`<=`,S,N/TP[i]),TP[i]);
od:
sort(convert(S,list)); # Robert Israel, Oct 28 2020

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, ", ")))}

cp's OEIS Frontend

A123754 Positive numbers of the form 4*n^2 - 1 which are not semiprimes.

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A126248 p*(p+1)*(p+2) where (p,p+2) are twin primes.

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A134020 Numbers that are one less than a square and have exactly 4 divisors.

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A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

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A165280 If p and q are twin primes then pq + 1 is always divisible by 3, except for (p,q)=(3,5). Sequence gives values of (pq + 1)/3.

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Magma

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A229108 Increasing a(n)is the smallest number of the form p^a*q^b, where a,b are positive integers and p < q are odd primes such that max( p^a, q^b)/min( p^a, q^b) <= 1 + 2/prime(n).

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Mathematica

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A234542 Positive integer solutions to sigma(sigma(k) - k - 3)/phi(k - phi(k) + 3) = 3.

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Maple

Mathematica

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A263927 Least solution x to the equation x' = (x+1)/n, where x' is the arithmetic derivative of x; -1 if no solution exists.

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A126248 p(p+1)(p+2) where (p,p+2) are twin primes.

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.