A050703 -id:A050703 - cp's OEIS frontend

A050769 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 2 skipped primes.

10, 12, 14, 15, 21, 44, 90, 135, 210, 252, 294, 384, 468, 504, 513, 686, 704, 720, 768, 832, 864, 972, 1155, 1368, 1701, 1890, 2176, 2184, 2352, 2400, 2880, 3080, 3400, 3640, 3888, 4032, 4536, 4725, 5200, 6174, 6384, 8750, 9548, 10350, 10400, 10500

Offset: 0

Patrick De Geest, Sep 15 1999

nonn

			a(6) = 44 + (2 + 2 + 11) = ending prime 59. Between 44 and 59 there are 2 primes: 47 and 53.

Harvey P. Dale, Table of n, a(n) for n = 0..333 (all terms up to 5*10^6)

Cf. A050703, A050710.

ckpgQ[n_]:=Module[{c=n+Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]]]},CompositeQ[n]&&PrimeQ[c]&&PrimePi[c]-PrimePi[n] == 3]; Select[Range[11000],ckpgQ] (* Harvey P. Dale, Nov 29 2014 *)

A050772 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 5 skipped primes.

Original entry on oeis.org

18, 24, 25, 46, 57, 161, 203, 209, 288, 319, 323, 391, 736, 798, 837, 858, 928, 930, 1035, 1088, 1089, 1218, 1300, 1376, 1690, 2254, 2418, 2478, 2673, 2842, 2871, 3045, 3220, 3325, 3458, 3510, 3588, 4186, 4508, 4617, 4824, 5054, 5180, 5248, 5472, 6069

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			18 is a term because 18 + (2+3+3) = 26 + (2+13) = ending prime 41. Between 18 and 41 one finds 5 primes 19, 23, 29, 31 and 37.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A050703, A050710.

filter:= proc(n) local r, s, t;
  if isprime(n) then return false fi;
  t:= 0: s:= n;
  do
   r:= s;
   s:= s + add(p[1]*p[2],p=ifactors(s)[2]);
   t:= t + numtheory:-pi(s-1) - numtheory:-pi(r);
   if isprime(s) then return t=5 fi;
   if t > 5 then return false fi;
  od;
end proc:
select(filter, [$2..10000]); # Robert Israel, May 08 2020

ok[n_] := CompositeQ[n] && Block[{k=n, p = NextPrime[n, 6]}, While[k < p, k += Total[ Times @@@ FactorInteger[k]]]; k == p]; Select[Range@ 6069, ok] (* Giovanni Resta, May 08 2020 *)

Offset changed to 1 by Robert Israel, May 08 2020

A050776 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 9 skipped primes.

Original entry on oeis.org

30, 42, 99, 174, 188, 212, 216, 295, 329, 348, 371, 620, 627, 629, 649, 851, 901, 925, 957, 1081, 1189, 1248, 1353, 1363, 1696, 1830, 1880, 2133, 2173, 2491, 2664, 3660, 3843, 4020, 4420, 5056, 5688, 6327, 6700, 7540, 7626, 7808, 7888, 7900, 8295, 8778

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			a(3)=99 + (3+3+11) = 116 + (2+2+29) = ending prime 149. Between 99 and 149 one finds 9 primes 101, 103, 107, 109, 113, 127, 131, 137 and 139.

Cf. A050703, A050710.

Offset corrected by Sean A. Irvine, Aug 18 2021

A050779 Primes that are not ending primes after the iterated procedure of 'composite added to the sum of its prime factors reaches a prime'.

Original entry on oeis.org

2, 3, 5, 7, 13, 37, 43, 61, 67, 73, 97, 101, 137, 139, 157, 163, 173, 181, 193, 197, 199, 211, 223, 233, 257, 277, 281, 283, 307, 347, 349, 353, 367, 379, 389, 397, 409, 421, 433, 457, 463, 487, 499, 547, 557, 563, 577, 601, 613, 617, 641, 643, 661, 673, 677

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Cf. A050778, A050703-A050710.

a[n_]:=NestWhile[#+Total[Times@@@FactorInteger[#]]&,n,!PrimeQ[#]&]; t={}; Do[If[!PrimeQ[n],AppendTo[t,a[n]]],{n,4,nn=678}]; Complement[Prime[Range[PrimePi[nn]]],Select[Union[t],#Jayanta Basu, Jun 01 2013 *)

A281995 Squarefree numbers that, when added to the sum of their prime factors, remain squarefree.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 115, 118

Offset: 1

K. D. Bajpai, Feb 04 2017

nonn

			a(6) = 10 = 2*5 that is squarefree. 10 + 2 + 5 = 17 = 1*17, which is also squarefree.
a(14) = 22 = 2*11 that is squarefree. 22 + 2 + 11 = 35 = 5*7, which is also squarefree.
a(219) = 434 = 2*7*31 that is squarefree. 434 + 2 + 7 + 31 = 474 = 2*3*79, which is also squarefree.

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

Cf. A001414, A005117, A050703.

filter:= n -> numtheory:-issqrfree(n) and numtheory:-issqrfree(n+convert(numtheory:-factorset(n),`+`)):
select(filter, [$1..1000]); # Robert Israel, Feb 15 2017

Select[Range[500], SquareFreeQ[#] && SquareFreeQ[# + Total[Times @@@ FactorInteger[#]]] &]

isok(n) = issquarefree(n) && issquarefree(n + vecsum(factor(n)[, 1])); \\ Michel Marcus, Feb 05 2017

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

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

nonn

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.

			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.

Robert Israel, Table of n, a(n) for n = 1..900

Cf. A001414, A050703, A050704, A075254.

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]);

A050770 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 3 skipped primes.

Original entry on oeis.org

26, 33, 35, 55, 208, 297, 300, 351, 352, 420, 432, 441, 486, 660, 735, 910, 1000, 1140, 1225, 1452, 1584, 1715, 1938, 2016, 2250, 2548, 2816, 3003, 3724, 4056, 4180, 4455, 4459, 4600, 4736, 4845, 5280, 5625, 5746, 5888, 5915, 6422, 6475, 6864, 6916, 6930

Offset: 0

Patrick De Geest, Sep 15 1999

nonn

			a(2)=33 + (3+11) = ending prime 47. Between 33 and 47 one finds 3 primes 37, 41 and 43.

Cf. A050703, A050710.

A050771 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 4 skipped primes.

Original entry on oeis.org

8, 9, 34, 38, 51, 65, 68, 85, 96, 116, 143, 224, 225, 304, 306, 416, 448, 544, 546, 570, 600, 621, 640, 714, 783, 810, 1020, 1134, 1197, 1254, 1280, 1408, 1521, 1540, 1666, 1716, 1806, 1900, 1953, 2088, 2646, 2760, 3159, 3185, 3255, 3392, 3920, 3968, 4176

Offset: 0

Patrick De Geest, Sep 15 1999

nonn

			a(2)=9 + (3+3) = 15 + (3+5) = ending prime 23. Between 9 and 23 one finds 4 primes 11, 13, 17 and 19.

Cf. A050703, A050710.

A050773 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 6 skipped primes.

Original entry on oeis.org

4, 22, 54, 93, 111, 145, 155, 185, 221, 231, 377, 450, 667, 688, 992, 1122, 1161, 1428, 1638, 1995, 2144, 2220, 2624, 2820, 2838, 3627, 3774, 3876, 4225, 4761, 5106, 5434, 5590, 5676, 5850, 6188, 6216, 6405, 6734, 8364, 8554, 9196, 9471, 9558, 10660

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			a(2)=22 + (2+11) = 35 + (5+7) = ending prime 47. Between 22 and 47 one finds 6 primes 23, 29, 31, 37, 41 and 43.

Cf. A050703, A050710.

Offset corrected by Sean A. Irvine, Aug 18 2021

A050774 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 7 skipped primes.

Original entry on oeis.org

39, 40, 58, 78, 205, 272, 330, 341, 357, 437, 475, 476, 520, 533, 551, 616, 738, 833, 1105, 1184, 1274, 1984, 2262, 2312, 2652, 2964, 3008, 3010, 3198, 3444, 3776, 4386, 5530, 5831, 6020, 6204, 6360, 6768, 7056, 7380, 7420, 7930, 8322, 8835, 8844, 8991

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

			a(1)=39 + (3+13) = 55 + (5+11) = ending prime 71. Between 39 and 71 one finds 7 primes 41, 43, 47, 53, 59, 61 and 67.

Cf. A050703, A050710.

Offset corrected by Sean A. Irvine, Aug 18 2021

cp's OEIS Frontend

A050769 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 2 skipped primes.

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A050772 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 5 skipped primes.

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A050776 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 9 skipped primes.

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A050779 Primes that are not ending primes after the iterated procedure of 'composite added to the sum of its prime factors reaches a prime'.

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A281995 Squarefree numbers that, when added to the sum of their prime factors, remain squarefree.

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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.

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A050770 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 3 skipped primes.

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A050771 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 4 skipped primes.

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A050773 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 6 skipped primes.

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A050774 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 7 skipped primes.

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