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 10 results.

A130006 Prime numbers arising from A050704.

Original entry on oeis.org

2, 3, 3, 5, 5, 7, 11, 11, 11, 17, 19, 23, 17, 23, 29, 29, 37, 31, 43, 43, 29, 47, 47, 43, 53, 59, 67, 41, 71, 71, 59, 71, 83, 71, 97, 83, 59, 79, 89, 83, 107, 113, 71, 107, 127, 103, 131, 157, 137, 173, 179, 131, 167, 101, 179, 193, 139, 167, 179, 107, 191, 197, 173
Offset: 1

Views

Author

Tomas Xordan, Jun 15 2007

Keywords

Examples

			a(10)=17 because A050704(10) - prime factors of A050704(10) = 28-(7+2+2)=28-11=17.

Crossrefs

Cf. A050704.

Formula

a(n)=A050704(n) - (prime factors of A050704(n)).

A050703 Numbers that when added to the sum of their prime factors (with multiplicity) become prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 26, 33, 34, 35, 38, 44, 46, 48, 51, 55, 57, 58, 65, 68, 74, 85, 86, 90, 93, 96, 111, 112, 116, 118, 123, 135, 141, 143, 145, 155, 158, 161, 166, 177, 178, 185, 188, 194, 201, 203, 205, 206, 208, 209, 210, 212, 215, 221, 224, 225, 252
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Comments

No term of this sequence can be prime, since for a prime p, A075254(p)=2*p, hence not prime. - Michel Marcus, Jul 24 2015
From Robert Israel, Jul 24 2015: (Start)
Similarly, no term of the sequence can be a prime power.
Contains 2*n for n in A023208 and 3*n for n in A023213. (End)

Examples

			252 = 2*2*3*3*7; 252 + (2 + 2 + 3 + 3 + 7) = 252 + 17 = 269, which is prime.

Links

Crossrefs

Cf. A050704-A050710, A075254, A023208, A023213.

Programs

  • Maple
    filter:= n ->isprime(convert(map(convert,ifactors(n)[2],`*`),`+`)+n):
select(filter, [$1..1000]); # Robert Israel, Jul 24 2015
  • Mathematica
    upto=300;Rest[Select[Complement[Range[upto], Prime[Range[ PrimePi[upto]]]], PrimeQ[#+ Total[Times@@@FactorInteger[#]]]&]] (* Harvey P. Dale, Apr 20 2011 *)
Select[Range[500], PrimeQ[# + Total [Times @@@ FactorInteger[#]] && PrimeOmega[#] > 1] &]  (* K. D. Bajpai, Sep 12 2014 *)
  • PARI
    sopfr(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1]*f[i,2])
is(n)=!isprime(n)&&isprime(n+sopfr(n)) \\ Charles R Greathouse IV, Jul 19 2011

Formula

{n: A075254(n) in A000040}. - R. J. Mathar, Jul 27 2015

Extensions

Name clarified by Michel Marcus, Jul 24 2015

A050710 Smallest composite that when added to sum of prime factors reaches a prime after n iterations.

Original entry on oeis.org

6, 8, 4, 32, 49, 45, 60, 125, 82, 66, 150, 129, 559, 417, 358, 378, 314, 279, 247, 183, 1152, 1102, 2265, 1929, 1658, 1524, 1414, 5708, 8047, 6033, 8430, 8020, 7852, 11805, 11715, 9388, 12622, 13471, 13146, 12562, 12512, 20830, 16869, 13492, 58832
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			n = 2 gives a(2) = 8 -> 8 = 2*2*2 so 8 + (2+2+2) = 14 and composite (iteration 1); 14 = 2*7 so 14 + (2+7) = 23 and already prime after the second iteration.

Links

Crossrefs

Cf. A050703, A050704, A050705, A050706, A050707, A050708, A050709.
See also A050767.

Programs

  • Mathematica
    a[n_]:=Length[NestWhileList[#+Total[Times@@@FactorInteger[#]]&,n,!PrimeQ[#] &]]-1; t={}; Do[i=2; While[a[i]!=n,i++]; AppendTo[t,i], {n,45}]; t (* Jayanta Basu, May 25 2013 *)

Extensions

More terms from Michel ten Voorde

A050705 Composite number such that when sum of its prime factors is added or subtracted becomes prime.

Original entry on oeis.org

10, 12, 14, 15, 20, 21, 26, 33, 35, 38, 44, 48, 51, 65, 68, 86, 93, 96, 111, 112, 116, 123, 161, 188, 201, 203, 206, 209, 210, 215, 221, 278, 297, 300, 304, 306, 321, 352, 356, 371, 384, 395, 398, 413, 420, 441, 471, 485, 524, 533, 543, 545, 546, 551, 570, 626
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Comments

Prime factors counted with multiplicity, e.g., 44 = 2*2*11 so the sum of its prime factors is 15 (not 13). - Harvey P. Dale, May 30 2012

Examples

			E.g., 545 = 5*109 so 545 +- (5+109) = 545 +- 114 = 659 and 431 and both are primes.

Links

Crossrefs

Cf. A050703, A050704, A050706, A050707, A050708, A050709, A050710.

Programs

  • Mathematica
    spfQ[n_]:=Module[{s=Total[Times@@@FactorInteger[n]]},!PrimeQ[n] && PrimeQ[ n+s]&&PrimeQ[n-s]]; Select[Range[700],spfQ] (* Harvey P. Dale, May 30 2012 *)
  • PARI
    lista(nn) = {forcomposite(n=2, nn, f = factor(n); sopfr = sum(j=1, #f~, f[j, 1]*f[j, 2]); if (isprime(n+sopfr) && isprime(n-sopfr), print1(n, ", ")););} \\ Michel Marcus, Jul 03 2017

A050707 Composites c that reach a prime after 3 iterations of c -> c + sum of prime factors of c.

Original entry on oeis.org

4, 16, 27, 28, 30, 42, 76, 87, 92, 95, 108, 114, 120, 124, 128, 133, 136, 147, 148, 154, 172, 196, 202, 204, 216, 222, 238, 242, 243, 244, 245, 255, 256, 260, 285, 286, 292, 308, 310, 325, 338, 340, 342, 350, 386, 412, 418, 422, 423, 426, 435, 440, 458, 464
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			204 is a term:
Iteration 1: 204 = 2*2*3*17 so 204 + (2+2+3+17) = 204 + 24 = 228 and composite.
Iteration 2: 228 = 2*2*3*19 so 228 + (2+2+3+19) = 228 + 26 = 254 and composite.
Iteration 3: 254 = 2*127 so 254 + (2+127) = 254 + 129 = 383 and prime.

Links

Crossrefs

Cf. A050703, A050704, A050705, A050706, A050708, A050709, A050710.

Programs

  • Magma
    f:=func; a:=[]; for k in [4..500] do if not IsPrime(k) and not IsPrime(f(k)) and not IsPrime(f(f(k))) and IsPrime(f(f(f(k)))) then Append(~a,k); end if; end for; a; // Marius A. Burtea, Oct 17 2019
  • Mathematica
    nxt[n_]:=Total[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[n]]]+n; Select[ Range[500],PrimeQ[NestList[nxt,#,3]]=={False,False,False,True}&] (* Harvey P. Dale, Feb 23 2014 *)

Extensions

Name edited by Michel Marcus, Oct 17 2019

A050706 Composites c that reach a prime after 2 iterations of c-> c+sum of prime factors of c.

Original entry on oeis.org

8, 9, 18, 22, 24, 25, 36, 39, 40, 54, 78, 80, 81, 91, 94, 99, 104, 106, 115, 119, 121, 122, 126, 134, 138, 142, 144, 146, 152, 159, 164, 170, 174, 187, 189, 214, 218, 219, 226, 228, 231, 232, 237, 250, 258, 262, 264, 265, 266, 272, 274, 276, 280, 282, 288, 289
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			226 is a term:
Iteration 1: 226 = 2*113 so 226 + (2+113) = 226 + 115 = 341 and composite.
Iteration 2: 341 = 11*31 so 341 + (11+31) = 341 + 42 = 383 and prime.

Crossrefs

Cf. A050703, A050704, A050705, A050707, A050708, A050709, A050710.

Programs

  • Magma
    f:=func; a:=[]; for k in [4..300] do if not IsPrime(k) and not IsPrime(f(k)) and IsPrime(f(f(k))) then Append(~a, k); end if; end for; a; // Marius A. Burtea, Oct 18 2019
  • Mathematica
    aQ[n_]:=PrimeQ[Nest[#+Total[Times@@@FactorInteger[#]]&,n,2]]; Select[Range[289],!PrimeQ[#]&&aQ[#]&] (* Jayanta Basu, May 31 2013 *)

Extensions

Name edited by Michel Marcus, Oct 18 2019

A050708 Composites c that reach a prime after 4 iterations of c-> c+sum of prime factors of c.

Original entry on oeis.org

32, 62, 63, 64, 69, 77, 98, 100, 102, 105, 110, 117, 171, 182, 186, 190, 195, 200, 217, 230, 234, 240, 246, 248, 270, 324, 354, 381, 388, 392, 400, 405, 410, 430, 436, 438, 444, 455, 456, 474, 481, 482, 483, 490, 528, 540, 568, 576, 582, 584, 598, 605, 625
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			182 is a term:
Iteration 1: 182 = 2*7*13 so 182 + (2+7+13) = 182 + 22 = 204 and composite.
Iteration 2: 204 = 2*2*3*17 so 204 + (2+2+3+17) = 204 + 24 = 228 and composite.
Iteration 3: 228 = 2*2*3*19 so 228 + (2+2+3+19) = 228 + 26 = 254 and composite.
Iteration 4: 254 = 2*127 so 254 + (2+127) = 254 + 129 = 383 and prime.

Crossrefs

Cf. A050703, A050704, A050705, A050706, A050707, A050709, A050710.

Extensions

Name edited by Michel Marcus, Oct 18 2019

A050709 Composites c that reach a prime after 5 iterations of c-> c+sum of prime factors of c.

Original entry on oeis.org

49, 50, 52, 56, 84, 88, 169, 176, 180, 198, 220, 302, 322, 336, 339, 363, 364, 372, 387, 402, 406, 407, 424, 429, 434, 442, 492, 494, 508, 552, 553, 562, 589, 612, 628, 633, 650, 708, 724, 744, 760, 788, 816, 819, 843, 848, 860, 861, 862, 870, 872, 896, 918
Offset: 1

Views

Author

Patrick De Geest, Aug 15 1999

Keywords

Examples

			49 is a term:
Iteration 1: 49 = 7*7 so 49 + (7+7) = 49 + 14 = 63 and composite.
Iteration 2: 63 = 3*3*7 so 63 + (3+3+7) = 63 + 13 = 76 and composite.
Iteration 3: 76 = 2*2*19 so 76 + (2+2+19) = 76 + 23 = 99 and composite.
Iteration 4: 99 = 3*3*11 so 99 + (3+3+11) = 99 + 17 = 116 and composite.
Iteration 5: 116 = 2*2*29 so 116 + (2+2+29) = 116 + 33 = 149 and prime.

Crossrefs

Cf. A050703, A050704, A050705, A050706, A050707, A050708, A050710.

Extensions

Name edited by Michel Marcus, Oct 18 2019

A050780 Numbers k such that sopfr(k) = sopfr(k + sopfr(k)).

Original entry on oeis.org

39, 60, 70, 95, 119, 240, 2079, 2130, 2183, 3000, 3125, 3431, 4250, 6293, 6468, 9310, 10164, 10241, 10679, 13433, 14039, 14111, 15561, 16199, 16799, 23552, 24601, 27004, 28116, 28560, 31416, 32883, 42112, 44268, 52193, 52969, 53754, 59072
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Comments

sopfr(k) = sum of the prime factors of k (with multiplicity).

Examples

			sopfr(39) = 3 + 13 = 16 = 5 + 11 = sopfr(39 + sopfr(39)), so 39 is in the sequence.

Links

Crossrefs

Cf. A001414, A050781, A050703, A050704, A050705, A050706, A050707, A050708, A050709, A050710.

Programs

  • Magma
    f:=func; [k:k in [2..60000]| f(k) eq f(k+f(k))]; // Marius A. Burtea, Oct 17 2019
  • Mathematica
    sopf[n_] := Total[Apply[Times, FactorInteger[n], {1}]]; ok[n_] := n + sopf[n] - sopf[n + sopf[n]] == n; Select[Range[59200], ok] (* Jean-François Alcover, Apr 18 2011 *)

Extensions

Edited by Jon E. Schoenfield, Dec 25 2016
Offset changed to 1 by Jon E. Schoenfield, Oct 17 2019

A358002 Numbers k such that one of k-A001414(k) and k+A001414(k) is a prime and the other is the square of a prime.

Original entry on oeis.org

135, 936, 1431, 3510, 5005, 5106, 5278, 9471, 10648, 10659, 22126, 26724, 27420, 27840, 37014, 37149, 39321, 40311, 54730, 59031, 62830, 87186, 124914, 128616, 129411, 133494, 187705, 196078, 208285, 209451, 212695, 309885, 322191, 325465, 375513, 410515, 412476, 433041, 459844, 466620, 595833, 622083
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Oct 23 2022

Keywords

Comments

The Generalized Bunyakovsky conjecture implies that there are, for example, infinitely many primes q == 11 (mod 26) such that p = (q^2+9)/26 and 28*p+9 are prime, and then 27*p is in the sequence.

Examples

			a(4) = 3510 is a term because 3510 = 2*3^3*5*13 so A001414(3510) = 2+3*3+5+13 = 29 and 3510-29 = 3481 = 29^2 is the square of a prime, while 3510+29 = 3539 is prime.

Links

Crossrefs

Cf. A001414, A050703, A050704, A075254.

Programs

  • Maple
    filter:= proc(n) local t,s,x,y;
   s:= add(t[1]*t[2], t = ifactors(n)[2]);
   x:= s+n; y:= n-s;
   if issqr(x) then isprime(sqrt(x)) and isprime(y)
   else issqr(y) and isprime(sqrt(y)) and isprime(x)
   fi
end proc:
select(filter, [$1..10^6]);
Showing 1-10 of 10 results.

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

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