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 14 results. Next

A258642 Suppose A103739(n) = (p^2 + q^2)/2 with p < q primes. The sequence gives values of p.

Original entry on oeis.org

3, 3, 5, 5, 3, 5, 7, 3, 5, 5, 13, 11, 3, 5, 13, 17, 5, 11, 19, 19, 7, 13, 17, 5, 3, 5, 13, 19, 17, 3, 5, 7, 17, 31, 3, 7, 13, 17, 37, 23, 5, 37, 11, 5, 43, 37, 29, 31, 7, 29, 31, 17, 5, 7, 13, 31, 43, 11, 19, 41, 3, 5, 7, 13, 29, 23, 47, 11, 37, 59, 31, 47, 7, 13, 41
Offset: 1

Views

Author

Zak Seidov, Jun 06 2015

Keywords

Examples

			A103739(1)=17=(3^3+5^2)/2 hence a(1)=3.
A103739(2)=29=(3^3+7^2)/2 hence a(2)=3.

Crossrefs

Cf. A103739, A258653 (values of q).

A258653 Suppose A103739(n)=(p^2 + q^2)/2 with p < q primes. The sequence gives values of q.

Original entry on oeis.org

5, 7, 7, 11, 13, 13, 13, 17, 17, 19, 17, 19, 23, 23, 23, 23, 29, 31, 29, 31, 37, 37, 37, 41, 43, 43, 43, 41, 43, 47, 47, 47, 47, 41, 53, 53, 53, 53, 43, 53, 59, 47, 59, 61, 47, 53, 59, 59, 67, 61, 61, 67, 73, 73, 73, 71, 67, 79, 79, 71, 83, 83, 83, 83, 79, 83
Offset: 1

Views

Author

Zak Seidov, Jun 06 2015

Keywords

Examples

			A103739(1)=17=(3^3+5^2)/2 hence a(1)=5;
A103739(2)=29=(3^3+7^2)/2 hence a(2)=7.

Crossrefs

Cf. A103739, A258642 (values of p).

A257816 Suppose A103739(n) = x^2 + y^2 with x < y. The sequence gives values of y.

Original entry on oeis.org

4, 5, 6, 8, 8, 9, 10, 10, 11, 12, 15, 15, 13, 14, 18, 20, 17, 21, 24, 25, 22, 25, 27, 23, 23, 24, 28, 30, 30, 25, 26, 27, 32, 36, 28, 30, 33, 35, 40, 38, 32, 42, 35, 33, 45, 45, 44, 45, 37, 45, 46, 42, 39, 40, 43, 51, 55, 45, 49, 56, 43, 44, 45, 48, 54, 53, 60
Offset: 1

Views

Author

Zak Seidov, Jun 06 2015

Keywords

Comments

See first comment in A103739.

Examples

			a(1) = (A258653(1) + A258642(1))/2 = (5+3)/2 = 4;
a(2) = (A258653(2) + A258642(2))/2 = (7+3)/2 = 5.

Crossrefs

Cf. A103739, A258642, A258653, A258654 (values of x).

Formula

a(n) = (A258653(n)+A258642(n))/2.

A258654 Suppose A103739(n) = x^2 + y^2 with x < y. The sequence gives values of x.

Original entry on oeis.org

1, 2, 1, 3, 5, 4, 3, 7, 6, 7, 2, 4, 10, 9, 5, 3, 12, 10, 5, 6, 15, 12, 10, 18, 20, 19, 15, 11, 13, 22, 21, 20, 15, 5, 25, 23, 20, 18, 3, 15, 27, 5, 24, 28, 2, 8, 15, 14, 30, 16, 15, 25, 34, 33, 30, 20, 12, 34, 30, 15, 40, 39, 38, 35, 25, 30, 13, 39, 23, 6, 29, 18
Offset: 1

Views

Author

Zak Seidov, Jun 06 2015

Keywords

Comments

See first comment in A103739.

Examples

			a(1)=(A258653(1) - A258642(1))/2=(5-3)/2=1;
a(2)=(A258653(2) - A258642(2))/2=(7-3)/2=2.

Crossrefs

Cf. A103739, A258642, A258653, A257816 (values of y).

Formula

a(n) = (A258653(n) - A258642(n))/2.

A258641 First differences of A103739.

Original entry on oeis.org

12, 8, 36, 16, 8, 12, 40, 8, 36, 36, 12, 28, 8, 72, 60, 24, 108, 60, 60, 48, 60, 60, 24, 76, 8, 72, 12, 48, 40, 8, 12, 120, 72, 88, 20, 60, 60, 60, 60, 84, 36, 12, 72, 156, 60, 72, 60, 48, 12, 60, 48, 288, 12, 60, 252, 168, 12, 120, 60, 88, 8, 12, 60, 12
Offset: 1

Views

Author

Zak Seidov, Jun 06 2015

Keywords

Comments

All terms are divisible by 4. Minimal value is 8.

Crossrefs

Cf. A103739.

Formula

a(n) = A103739(n+1) - A103739(n).

A093343 Primes of form (prime(n)^2 + prime(n+1)^2)/2.

Original entry on oeis.org

17, 37, 229, 2029, 14449, 22501, 25609, 28909, 32401, 42061, 57601, 72901, 116989, 176401, 181501, 265261, 304729, 324901, 378229, 462409, 497041, 695581, 804709, 1089961, 1299721, 1416109, 1664101, 1742401, 1932181, 1971241, 2712709, 2873029, 3062509, 3186229
Offset: 1

Views

Author

Giovanni Teofilatto, Apr 26 2004

Keywords

Comments

Except for the first term, all terms == 1 mod 6. - Zak Seidov, Dec 02 2009
Except 17, all terms == 1 mod 12. Primes of the form A028334(n+1)^2 + A024675(n)^2. - Thomas Ordowski, Jun 28 2013

Links

Crossrefs

Cf. A103739.

Programs

  • Mathematica
    Select[Mean/@Partition[Prime[Range[500]]^2,2,1],PrimeQ] (* Harvey P. Dale, Jun 16 2021 *)

Formula

Conjecture: a(n) ~ A224888(n). - Thomas Ordowski, Jul 25 2013

Extensions

Corrected and extended by Rick L. Shepherd, Nov 24 2004

A143850 Numbers of the form (p^2 + q^2)/2, for odd primes p and q.

Original entry on oeis.org

9, 17, 25, 29, 37, 49, 65, 73, 85, 89, 97, 109, 121, 145, 149, 157, 169, 185, 193, 205, 229, 241, 265, 269, 277, 289, 325, 349, 361, 409, 425, 433, 445, 481, 485, 493, 505, 529, 541, 565, 601, 625, 661, 685, 689, 697, 709, 745, 769, 829, 841, 845, 853, 865
Offset: 1

Views

Author

T. D. Noe, Sep 03 2008

Keywords

Comments

The primes in this sequence are listed in A103739.
a(n) mod 4 = 1. See A227697 for related sequence. - Richard R. Forberg, Sep 22 2013
The squares of primes in this sequence form the subsequence A001248 \ {4}. - Bernard Schott, Jul 09 2022

Links

Crossrefs

Cf. A001248, A045636, A103739, A227697.
Cf. A075892 (a subsequence).

Programs

  • Mathematica
    Take[Union[Total[#]/2&/@(Tuples[Prime[Range[2,20]],2]^2)],60] (* Harvey P. Dale, Dec 28 2014 *)
  • PARI
    list(lim)=my(v=List(), p2); lim\=1; if(lim<9, lim=8); forprime(p=3, sqrtint(2*lim-9), p2=p^2; forprime(q=3, min(sqrtint(2*lim-p2), p), listput(v, (p2+q^2)/2))); Set(v) \\ Charles R Greathouse IV, Feb 14 2017

A103558 Semiprimes of the form p^2 + q^2, where p and q are primes.

Original entry on oeis.org

34, 58, 74, 146, 178, 194, 218, 298, 314, 365, 386, 458, 482, 533, 538, 554, 698, 818, 866, 965, 1082, 1202, 1322, 1418, 1538, 1658, 1685, 1706, 1853, 1858, 1874, 2018, 2042, 2138, 2218, 2234, 2258, 2498, 2642, 2813, 2818, 2858, 2978, 3098, 3218, 3338
Offset: 1

Views

Author

Giovanni Teofilatto, Mar 23 2005

Keywords

Comments

p and q must be distinct, otherwise p^2 + q^2 = 2*p*p has three prime factors. - Klaus Brockhaus
Even terms are 2*A103739. - Robert Israel, Nov 03 2017

Examples

			34 is a term because 3^2 + 5^2 = 34 = 2*17; 58 is a term because 3^2 + 7^2 = 58 = 2*29; 74 is a term because 5^2 + 7^2 = 74 = 2*37.

Links

Crossrefs

Cf. A001358, A006881, A103739.

Programs

  • Maple
    N:= 10000: # to get all terms <= N
P:= select(isprime, [$1..floor(sqrt(N))]):
Res:= NULL:
for i from 1 to nops(P) do
  for j from 1 to i-1 do
    r:= P[i]^2 + P[j]^2;
    if r > N then break fi;
    if numtheory:-bigomega(r) = 2 then Res:= Res, r fi;
od od:
sort(convert({Res},list)); # Robert Israel, Nov 03 2017
  • Mathematica
    fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Sort[ Flatten[ Table[ Prime[p]^2 + Prime[q]^2, {p, 16}, {q, p - 1}]]], fQ[ # ] &] (* Robert G. Wilson v, Mar 23 2005 *)
  • PARI
    {m=53;v=[];forprime(p=2,m, forprime(q=nextprime(p+1),m,if(bigomega(k=p^2+q^2)==2, v=concat(v,k))));v=vecsort(v);stop=nextprime(m+1)^2;for(j=1,length(v),if(v[j]Klaus Brockhaus

Extensions

More terms from Klaus Brockhaus and Robert G. Wilson v, Mar 23 2005

A282378 Primes of the form (p^2 + q^2)/2 such that (p^4 + q^4)/2 is prime, where p and q are primes.

Original entry on oeis.org

17, 89, 97, 149, 157, 229, 241, 769, 937, 1109, 1117, 1129, 1249, 1549, 1753, 2161, 2221, 2389, 3301, 3769, 4129, 4261, 4801, 4909, 5113, 5449, 5569, 5869, 6121, 6469, 7369, 7621, 7681, 8089, 8329, 8389, 8761, 9649, 10009, 10429, 11161, 11941, 12241, 12409
Offset: 1

Views

Author

Thomas Ordowski and Altug Alkan, Feb 13 2017

Keywords

Links

Crossrefs

Subsequence of A103739.

Programs

  • PARI
    list(lim)=my(v=List(),p2,p4,t); lim\=1; if(lim<9,lim=9); forprime(p=3,sqrtint(2*lim-9), p2=p^2; p4=p2^2; forprime(q=3, min(sqrtint(2*lim-p2),p), if(isprime(t=(p2+q^2)/2) && isprime((p4+q^4)/2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 14 2017

Extensions

More terms from Arkadiusz Wesolowski, Feb 13 2017

A282997 Primes of the form (p^2 + q^2)/2 such that |q^2 - p^2| is square, where p and q are prime.

Original entry on oeis.org

17, 97, 16561, 89041, 2579199841, 3497992081, 5645806321, 21103207681, 428888025121, 686770904161, 2726023770721, 4017427557361, 6831989588161, 6933052766641, 10138513506001, 19387278797041, 23452359542401, 35287577206801, 40057354132561, 62093498771041, 64116963608881
Offset: 1

Views

Author

Thomas Ordowski and Altug Alkan, Feb 26 2017

Keywords

Comments

Primes of the form x^4 + y^4 such that q = x^2 + y^2 and p = |y^2 - x^2| are both primes.
Primes of the form n^4 + (n+1)^4 such that q = n^2 + (n+1)^2 and p = 2n+1 are both primes; so for n in A128780.
Primes of the form x^4 + y^4 such that |y^4 - x^4| is a semiprime.
From Robert G. Wilson v, Feb 26 2017: (Start)
{q, p, a(n) = (p^2+q^2)/2}
{5, 3, 17}
{13, 5, 97}
{181, 19, 16561}
{421, 29, 89041}
{71821, 379, 2579199841}
{83641, 409, 3497992081}
{106261, 461, 5645806321}
{205441, 641, 21103207681}
{926161, 1361, 428888025121}
{1171981, 1531, 686770904161}
(End)

Examples

			17 = (3^2 + 5^2)/2 and 5^2 - 3^2 = 4^2.

Links

Crossrefs

Subsequence of A002645 and of A094407.
Cf. A103739, A128780.

Programs

  • Mathematica
    lst = {}; a = 2; While[a < 2501, b = Mod[a, 2] + 1; While[b < a, If[ PrimeQ[a^4 + b^4] && PrimeOmega[a^4 - b^4] == 2, AppendTo[lst, (a^4 + b^4)]]; b += 2]; a++]; lst (* Robert G. Wilson v, Feb 27 2017 *)
  • PARI
    list(lim)=my(v=List(),t,n); while((t=n++^4+(n+1)^4)<=lim, if(isprime(t) && isprime(n^2+(n+1)^2) && isprime(2*n+1), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 26 2017

Formula

a(n) = A128780(n)^4 + (A128780(n)+1)^4.
a(n) == 1 (mod 16).

Extensions

a(11) onward from Robert G. Wilson v, Feb 26 2017
Showing 1-10 of 14 results. Next

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

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