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 1-7 of 7 results.

A190680 Primes p such that sopfr(p-1) = sopfr(p+1) is also prime, where sopfr is A001414.

Original entry on oeis.org

11, 251, 1429, 906949, 1050449, 1058389, 3728113, 9665329, 13623667, 14320489, 30668003, 30910391, 45717377, 49437001, 55544959, 57510911, 58206653, 58772257, 69490901, 72191321, 73625789, 75235973, 79396433, 99673891, 103821169, 104662139, 121322449, 125938889, 147210257, 164810311, 169844879, 170650169, 201991721
Offset: 1

Views

Author

Charles R Greathouse IV, May 16 2011

Keywords

Comments

The first three terms were computed by J. M. Bergot (personal communication from J. M. Bergot to N. J. A. Sloane, May 16 2011).
The number of terms < 10^n: 0, 1, 2, 3, 3, 4, 8, 24, 70, 253, 839, ..., . - Robert G. Wilson v, May 31 2011

Examples

			sopfr(250) = sopfr(2*5^3) = 2 + 5*3 = 17 = 2*2 + 3*2 + 7 = sopfr(2^2*3^2*7) = sopfr(252), and 17 and 251 are prime, so 251 is in this sequence.

Links

Crossrefs

Subsequence of A086711. Cf. A190722.

Programs

  • Mathematica
    f[n_] := Plus @@ Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]; fQ[n_] := Block[{pn = f[n - 1], pp = f[n + 1]}, pn == pp && PrimeQ@ pn]; p = 2; lst = {}; While[p < 216000000, If[ fQ@ p, AppendTo[lst, p]]; p = NextPrime@ p]; lst (* Robert G. Wilson v, May 18 2011 *)

A339180 Primes p such that p mod A001414(p-1) = p mod A001414(p+1).

Original entry on oeis.org

11, 17, 31, 151, 241, 251, 577, 727, 991, 1429, 1567, 1597, 1741, 2243, 2887, 3001, 3041, 3571, 3739, 4003, 4049, 4129, 4271, 4513, 4801, 5407, 6673, 6733, 6833, 7873, 8951, 9539, 9631, 10487, 10639, 11789, 12097, 14627, 14629, 14947, 16561, 16927, 18617, 18749, 18797, 19081, 19457, 20551, 21121
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Nov 26 2020

Keywords

Examples

			a(4) = 151 is in the sequence because 151 is prime, A001414(150)=2+3+5+5=15, A001414(152)=2+2+2+19=25, and 151 mod 15 = 151 mod 25 = 1.

Links

Crossrefs

Includes A086711, A339181 and A339182.
Cf. A001414.

Programs

  • Maple
    spf:= n -> add(t[1]*t[2],t=ifactors(n)[2]):
select(p -> isprime(p) and p mod spf(p-1) = p mod spf(p+1), [seq(i,i=3..100000,2)]);

A190722 Primes p such that A008472(p-1) = A008472(p+1) and is a prime.

Original entry on oeis.org

3, 45751, 149351, 171529, 223099, 434237, 678077, 706841, 1996297, 3993037, 6340457, 7199113, 7419761, 9000317, 13129271, 15052777, 17193217, 18436879, 18749881, 18998519, 23353469, 23689423, 33746663, 40985411, 41437751, 43547797, 51198097, 53773651, 56825687, 60207809, 62190113, 79778899, 81708353, 83019421
Offset: 1

Views

Author

Robert G. Wilson v, May 17 2011

Keywords

Comments

A008472 is the sum of the distinct primes dividing n.

Examples

			For p = 45751, p-1 = 2*3*5^3*61; 2+3+5+61=71 and p+1 = 2^3*7*19*43; 2+7+19+43 = 71.

Links

Crossrefs

Cf. A190680, A008472, A086711.
Subsequence of A203182.

Programs

  • Magma
    [p:p in PrimesInInterval(3,10^8)|(&+PrimeDivisors(p-1) eq &+PrimeDivisors(p+1)) and IsPrime(&+PrimeDivisors(p-1))]; // Marius A. Burtea, Nov 14 2019
  • Mathematica
    fQ[n_] := Block[{pn = Plus @@ (First@# & /@ FactorInteger[n - 1]), pp = Plus@@ (First@# & /@ FactorInteger[n + 1])}, pn == pp && PrimeQ[pn]];
p = 2; lst = {}; While[p < 10^8, If[fQ@p, AppendTo[lst, p]; Print@p]; p =
NextPrime@p]; lst
pQ[n_]:=Module[{p1=Total[FactorInteger[n-1][[All,1]]],p2=Total[ FactorInteger[ n+1][[All,1]]]},p1==p2&&PrimeQ[p1]]; Select[ Prime[ Range[5*10^6]],pQ] (* Harvey P. Dale, Jun 18 2017 *)

A203182 Primes p such that A008472(p-1) = A008472(p+1), where A008472 = sum of distinct primes dividing n.

Original entry on oeis.org

3, 18913, 24733, 29633, 32429, 42719, 45751, 46103, 61409, 117991, 149351, 171529, 174019, 176017, 223099, 294893, 326369, 363691, 421727, 423503, 434237, 472631, 658579, 678077, 686423, 706841, 735901, 770059, 771629, 906949, 936827, 937571, 1073447, 1256029
Offset: 1

Views

Author

Michel Lagneau, Dec 30 2011

Keywords

Comments

Conjecture: the sequence is infinite.

Examples

			18913 is in the sequence because:
sum of the distinct prime divisors of 18912 = 2+3+197 = 202;
sum of the distinct prime divisors of 18914 = 2+7+193 = 202.

Links

Crossrefs

Cf. A008472, A190680, A086711.

Programs

  • Maple
    with(numtheory):for n from 1 to 100000 do:p:=ithprime(n):p1:=p-1: p2:=p+1:t1:=ifactors(p1)[2]; t11 := sum(t1[i][1], i=1..nops(t1)):t2:=ifactors(p2)[2]; t22 := sum(t2[i][1], i=1..nops(t2)):if t11=t22 then printf(`%d, `,p):else fi:od:
  • Mathematica
    Select[Prime[Range[100000]],Total[Transpose[FactorInteger[#-1]][[1]]] == Total[Transpose[FactorInteger[#+1]][[1]]]&] (* Harvey P. Dale, Sep 22 2013 *)

A342738 Primes p such that A001414(p+1) = A001414(p-1) + 1.

Original entry on oeis.org

5, 7, 19, 41, 197, 2549, 4159, 8467, 9433, 26701, 27551, 46817, 57037, 91097, 130859, 153281, 157049, 197683, 351727, 423103, 466181, 517991, 526291, 567181, 575231, 652903, 663167, 772339, 1055231, 1062013, 1088239, 1171199, 1232461, 1551871, 1603297, 1662833, 2782469, 2920531, 2957917, 3226159
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Mar 22 2021

Keywords

Examples

			a(3) = 19 is a term because it is prime and A001414(20) = 9 = 1 + A001414(18).

Links

Crossrefs

Cf. A001414, A086711, A342823.

Programs

  • Maple
    spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
select(p -> spf(p+1)=spf(p-1)+1, [seq(ithprime(i),i=1..10^5)]);

A342823 Primes p such that A001414(p+1) = A001414(p-1) - 1.

Original entry on oeis.org

29, 127, 449, 571, 727, 1721, 4027, 11969, 16987, 18913, 26449, 37139, 43609, 48871, 48953, 63799, 64781, 114479, 180847, 220021, 400031, 400597, 476911, 607549, 679969, 705883, 706841, 770059, 776449, 807539, 912367, 932177, 964793, 1007959, 1052237, 1095851, 1356227, 1444567, 1573339, 1664633
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Mar 22 2021

Keywords

Examples

			a(3) = 449 is a term because it is prime and A001414(450) = 18 = A001414(448) - 1.

Links

Crossrefs

Cf. A001414, A086711, A342738.

Programs

  • Maple
    spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
select(p -> spf(p+1)=spf(p-1)-1, [seq(ithprime(i),i=1..10^5)]);

A126975 Primes p with property that, if q is the next prime, then the sum of the prime factors of p+q, taken with multiplicity, is a prime.

Original entry on oeis.org

2, 5, 23, 43, 83, 97, 103, 131, 149, 157, 179, 191, 193, 229, 251, 293, 337, 383, 397, 401, 431, 443, 463, 541, 569, 601, 643, 709, 739, 857, 859, 863, 887, 907, 911, 967, 971, 983, 1019, 1039, 1069, 1091, 1093, 1223, 1229, 1249, 1279, 1283, 1321, 1373
Offset: 1

Views

Author

J. M. Bergot, Mar 20 2007

Keywords

Examples

			97 is a member: 97 + 101 = 198. Its factors with multiplicity are 2*3*3*11 and their sum is 2+3+3+11=19, which is a prime.

Crossrefs

Cf. A086711.

Programs

  • Magma
    [ p: p in PrimesUpTo(1400) | IsPrime(&+[ k[1]*k[2]: k in Factorization(p+NextPrime(p)) ] ) ]; /* Klaus Brockhaus, Mar 25 2007 */
  • Mathematica
    sopfr[n_] := Plus @@ Times @@@ FactorInteger[n];Prime@Select[Range[240], PrimeQ[sopfr[Prime[ # ] + Prime[ # + 1]]] &] (* Ray Chandler, Mar 25 2007 *)
  • PARI
    {m=1400; p=2; while(p

Extensions

Corrected and extended by Ray Chandler and Klaus Brockhaus, Mar 25 2007
Showing 1-7 of 7 results.

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

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