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 1-9 of 9 results.

A177791 Partial sums of A050705.

Original entry on oeis.org

10, 22, 36, 51, 71, 92, 118, 151, 186, 224, 268, 316, 367, 432, 500, 586, 679, 775, 886, 998, 1114, 1237, 1398, 1586, 1787, 1990, 2196, 2405, 2615, 2830, 3051, 3329, 3626, 3926, 4230, 4536, 4857, 5209, 5565, 5936, 6320, 6715, 7113, 7526, 7946, 8387, 8858
Offset: 1

Views

Author

Jonathan Vos Post, May 13 2010

Keywords

Comments

Partial sums of composite number such that when sum of its prime factors is added or subtracted becomes prime. The subsequence of primes in the partial sums begins: 71, 151, 367, 1237, 1787, 3329, 5209, 8387, 9343, 13781. The subsequence of partial sums which are themselves composite number such that when sum of their prime factors is added or subtracted becomes prime, begins: 10, 51, which other such fixed points are there?

Examples

			a(13) = 10 + 12 + 14 + 15 + 20 + 21 + 26 + 33 + 35 + 38 + 44 + 48 + 51 = 367 is prime.

Crossrefs

Cf. A000040, A001414, A050705, A050703-A050710.

Formula

a(n) = SUM[i=1..n] A050705(i) = SUM[i=1..n] {n such that n+A001414(n) is in A000040, and n-A001414(n) is in A000040}.

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

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

A337028 Numbers k such that k + A001414(k), k - A001414(k) and k mod A001414(k) are all prime.

Original entry on oeis.org

10, 12, 14, 15, 20, 26, 33, 35, 38, 51, 65, 68, 86, 96, 111, 112, 116, 161, 201, 203, 206, 209, 215, 221, 278, 297, 300, 304, 321, 371, 395, 398, 413, 420, 471, 533, 545, 551, 570, 626, 671, 698, 720, 755, 779, 803, 837, 858, 866, 910, 972, 1020, 1046, 1124, 1155, 1161, 1286, 1326, 1349, 1385
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Aug 11 2020

Keywords

Examples

			a(3)=14 is in the sequence because A001414(14)=9, and 14-9=5, 14+9=23 and 14 mod 9 = 5 are all prime.

Links

Crossrefs

Cf. A001414. Subset of A050705.

Programs

  • Maple
    A001414:= proc(n) local F; F:= ifactors(n)[2]; convert(map(convert,F,`*`),`+`) end proc:
filter:= proc(n) local s; s:= A001414(n); isprime(n+s) and isprime(n-s) and isprime(n mod s) end proc:
select(filter, [$1..2000]);

A343454 Numbers k such that k^2+2*A001414(k) and k^2-2*A001414(k) are primes.

Original entry on oeis.org

21, 33, 35, 39, 111, 339, 473, 629, 735, 779, 795, 801, 959, 1025, 1119, 1149, 1245, 1253, 1281, 1575, 1589, 1695, 1851, 1919, 1961, 1985, 2199, 2315, 2523, 2561, 2681, 2759, 3003, 3065, 3189, 3233, 3315, 3443, 3893, 3983, 4175, 4299, 4359, 4375, 4455, 4503, 4693, 4925, 5247, 5585, 5609, 5703
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Apr 15 2021

Keywords

Comments

Square roots of squares in A050705.
All terms are odd.
Includes 3*p if p, 9*p^2+2*p+6 and 9*p^2-2*p-6 are all primes; the generalized Bunyakovsky conjecture implies there are infinitely many of these.

Examples

			a(3) = 35 is a term because A001414(35) = 12 and 35^2-2*12 = 1201 and 35^2+2*12 = 1249 are primes.

Links

Crossrefs

Cf. A001414, A050705.

Programs

  • Maple
    spf:= n -> add(t[1]*t[2],t=ifactors(n)[2]):
filter:= proc(n) local s; s:= spf(n); isprime(n^2-2*s) and isprime(n^2+2*s) end proc:
select(filter, [seq(i,i=1..10000,2)]);
Showing 1-9 of 9 results.

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