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 1-6 of 6 results.

A071695 Lesser members of twin prime pairs of form (4*k+1, 4*k+3), k > 0.

Original entry on oeis.org

5, 17, 29, 41, 101, 137, 149, 197, 269, 281, 461, 521, 569, 617, 641, 809, 821, 857, 881, 1049, 1061, 1229, 1277, 1289, 1301, 1481, 1697, 1721, 1877, 1949, 1997, 2081, 2129, 2141, 2237, 2309, 2381, 2549, 2657, 2729, 2789, 2801, 2969
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 04 2002

Keywords

Comments

Corresponding greater members: A071696(n).
Or, lesser members of twin prime pairs (A001359) which are also Pythagorean primes (A002144). Intersection of A001359 and A002144. - Zak Seidov, Apr 25 2008
A010051(a(n)) * A010051(a(n)+2) = 1. - Reinhard Zumkeller, Nov 10 2013

Links

Crossrefs

Cf. A001359, A002144, A071698, A071697.

Programs

  • Haskell
    a071695 n = a071695_list !! (n-1)
a071695_list = [p | p <- a002144_list, a010051' (p + 2) == 1]
-- Reinhard Zumkeller, Nov 10 2013
  • Mathematica
    Select[ Prime@ Range@ 1000, Mod[#, 4] == 1 && PrimeQ[# + 2] &] (* Robert G. Wilson v, Jan 22 2012 *)
  • PARI
    p=2;forprime(q=3,1e4,if(q-p==2 && p%4==1, print1(p", "));p=q) \\ Charles R Greathouse IV, Mar 20 2013

A071696 Greater members of twin prime pairs of form (4*k+1,4*k+3), k>0.

Original entry on oeis.org

7, 19, 31, 43, 103, 139, 151, 199, 271, 283, 463, 523, 571, 619, 643, 811, 823, 859, 883, 1051, 1063, 1231, 1279, 1291, 1303, 1483, 1699, 1723, 1879, 1951, 1999, 2083, 2131, 2143, 2239, 2311, 2383, 2551, 2659, 2731, 2791, 2803, 2971
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 04 2002

Keywords

Comments

Corresponding lesser members: A071695(n).
A010051(a(n)) * A010051(a(n)-2) = 1. - Reinhard Zumkeller, Nov 10 2013

Links

Crossrefs

Cf. A006512, A071699, A071697.
Cf. Subsequence of A002145.

Programs

  • Haskell
    a071696 n = a071696_list !! (n-1)
a071696_list = [p | p <- tail a002145_list, a010051' (p - 2) == 1]
-- Reinhard Zumkeller, Nov 10 2013
  • Mathematica
    Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[ (#[[1]]-1)/4]&][[All,2]] (* Harvey P. Dale, Aug 27 2021 *)
Select[Table[4k+{1,3},{k,750}],AllTrue[#,PrimeQ]&][[;;,2]] (* Harvey P. Dale, Sep 10 2024 *)

A071700 Product of twin primes of form (4*k+3,4*(k+1)+1), k>=0.

Original entry on oeis.org

15, 143, 3599, 5183, 11663, 32399, 36863, 51983, 57599, 97343, 121103, 176399, 186623, 359999, 435599, 685583, 1040399, 1065023, 1192463, 1327103, 1742399, 2039183, 2108303, 2214143, 2585663, 2624399, 2782223
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 04 2002

Keywords

Links

Crossrefs

Subsequence of A037074, A071700 and of A182140.
Cf. A071697, a(n) = A071698(n)*A071699(n).

Programs

  • Haskell
    a071700 n = a071700_list !! (n-1)
a071700_list = [x * y | x <- [3, 7 ..], a010051' x == 1,
                        let y = x + 2, a010051' y == 1]
-- Reinhard Zumkeller, Aug 05 2014
  • PARI
    for(k=0,1e3,if(isprime(4*k+3)&&isprime(4*k+5),print1(16*k^2+32*k +15", "))) \\ Charles R Greathouse IV, Jul 03 2013
  • PARI
    is(n)=my(k=sqrtint(n\16)); n==16*k^2+32*k+15 && isprime(4*k+3) && isprime(4*k+5) \\ Charles R Greathouse IV, Jul 03 2013
  • PARI
    is(n)=my(t); n%16==15 && issquare(n+1,&t) && isprime(t-1) && isprime(t+1) \\ Charles R Greathouse IV, Dec 12 2016
  • PARI
    list(lim)=my(v=List(),p=3); forprime(q=5,sqrtint(1+lim\1)+1, if(q-p==2 && p%4==3, listput(v,p*q)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 12 2016

Formula

a(n) >> n^2 log^4 n. - Charles R Greathouse IV, Jul 03 2013

A072010 In prime factorization of n replace all primes of form k*4+1 with k*4+3 and primes of form k*4+3 with k*4+1.

Original entry on oeis.org

1, 2, 1, 4, 7, 2, 5, 8, 1, 14, 9, 4, 15, 10, 7, 16, 19, 2, 17, 28, 5, 18, 21, 8, 49, 30, 1, 20, 31, 14, 29, 32, 9, 38, 35, 4, 39, 34, 15, 56, 43, 10, 41, 36, 7, 42, 45, 16, 25, 98, 19, 60, 55, 2, 63, 40, 17, 62, 57, 28, 63, 58, 5, 64, 105, 18, 65, 76, 21, 70, 69
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Comments

a(3^n) = 1; a(2^n) = 2^n;
a(n)>2 is prime iff n=m*3^i (i>=0), a(n)=a(m) and (m,a(m)) or (a(m),m) is a twin prime pair of form ((4*k+1),(4*k+3)), a(m)*m=A071697(j)=A071695(j)*A071696(j) for some j.

Examples

			a(26928) = a(2^4*3^2*11*17) = a(2)^4 * a(3)^2 * a(11) * a(17)
= 2^4 * 1^2 * 9 * 19 = 2736.

Links

Crossrefs

Cf. A002144, A002145, A072012(n) = a(a(n)).
For a(n) = n see A072011.
Cf. A072014, A072015.
Cf. A027746.

Programs

  • Haskell
    a072010 1 = 1
a072010 n = product $ map f $ a027746_row n where
   f 2 = 2
   f p = p + 2 * (2 - p `mod` 4)
-- Reinhard Zumkeller, Apr 09 2012
  • Mathematica
    a[1] = 1; a[p_?PrimeQ] = p + 2*(2 - Mod[p, 4]); a[n_] := Times @@ (a[#[[1]]]^#[[2]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 04 2012 *)

Formula

Multiplicative with a(p) = p + 2*(2 - p mod 4), p prime.

A261073 Semiprimes whose prime factors are of equal binary length and which differ from each other in one bit position only.

Original entry on oeis.org

6, 35, 323, 437, 713, 899, 1763, 1961, 2021, 2537, 3233, 4757, 5561, 5609, 6497, 7313, 9797, 10403, 10961, 11009, 18209, 19043, 21353, 22499, 23393, 26969, 27221, 29177, 37001, 38021, 39203, 45113, 71273, 72899, 79523, 87953, 95477, 98201, 99221, 106793, 114857, 114929, 123353
Offset: 1

Views

Author

Antti Karttunen, Sep 22 2015

Keywords

Examples

			6 = 2*3 is present, as 2 in binary is "10" and 3 in binary is "11", so both have two (significant) bits and they differ only in one bit-position from each other.
35 = 5*7 is present, as 5 in binary is "101" and 7 in binary is "111", which both have three bits, differing only in the middle position from each other.

Links

Crossrefs

Cf. A000523, A001222, A006530, A020639, A101080.
Cf. also A261074, A261075.
Cf. A071697 (a subsequence).
Intersection of A085721 and A261077.

Programs

  • Mathematica
    Select[Range[10^6], And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #, Total@ BitXor[IntegerDigits[#1, 2], IntegerDigits[#2, 2]] == 1 & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)
  • PARI
    A000523 = n -> logint(n, 2);
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261073(n) = { my(a,b); if(bigomega(n)!=2, 0, a=A020639(n); b = (n/a); ((A000523(a) == A000523(b)) && (1 == norml2(binary(bitxor(a,b)))))); };
i=0; n=0; while(i < 5000, n++; if(isA261073(n), i++; write("b261073.txt", i, " ", n)));
  • Scheme
    ;; With Antti Karttunen's IntSeq-library.
(define A261073 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 1 (A101080bi (A020639 n) (A006530 n)))))))

A072011 Numbers k such that A072010(k) = k.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 35, 64, 70, 128, 140, 256, 280, 323, 512, 560, 646, 899, 1024, 1120, 1225, 1292, 1763, 1798, 2048, 2240, 2450, 2584, 3526, 3596, 4096, 4480, 4900, 5168, 7052, 7192, 8192, 8960, 9800, 10336, 10403, 11305
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Comments

If A072010(n) = n then also A072010(n*3^i) = n and A072010(n*2^j) = n*2^j.
For m=(4*k+1)*(4*k+3), product of twin prime pairs: A072010(m) = m, as well as for values m in the free monoid generated by the range of A071697.

Examples

			395675 is a term as f(395675) = f(323*1225) = f((17*19)*(5*7)^2) = f(17*19)*(f(5*7))^2 = f(17)*f(19)*(f(5)*f(7))^2 = 19*17*(7*5)^2 = 323*1225 = 395675 for f = A072010.

Links

Crossrefs

Cf. A002144, A002145, A071695, A071696, A071697, A072010.
Showing 1-6 of 6 results.

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

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