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 1-5 of 5 results.

A067386 Primes p such that p+1 and p-1 have the same number of distinct prime factors.

Original entry on oeis.org

3, 11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 131, 139, 163, 181, 193, 229, 239, 281, 307, 311, 349, 373, 379, 383, 409, 439, 443, 487, 491, 521, 577, 599, 601, 617, 619, 643, 683, 701, 709, 727, 739, 743, 761, 811, 821, 827, 829, 853, 863, 881, 883, 919, 937
Offset: 1

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Author

Benoit Cloitre, Feb 23 2002

Keywords

Links

Crossrefs

Cf. A115103 (same number of prime factors with multiplicity).

Programs

  • Maple
    q:= p-> isprime(p) and nops(ifactors(p+1)[2])=nops(ifactors(p-1)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, May 08 2022
  • Mathematica
    Select[Prime[Range[200]],PrimeNu[#-1]==PrimeNu[#+1]&] (* Harvey P. Dale, Jun 28 2020 *)
  • PARI
    is(n)=omega(n-1)==omega(n+1) && isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

A323536 Primes p such that p - k and p + k have the same number of prime factors (with multiplicity), for k = 1..7.

Original entry on oeis.org

91289867, 247780811, 350499731, 353523083, 394923913, 418273259, 441459853, 452876747, 645159257, 702723851, 718541749, 728741617, 729758423, 776424947, 791860151, 1191670069, 1289075413, 1457951063, 1508119211, 1527473449, 1563808777, 1568639509, 1611010391, 1662823523, 1705045429, 1801303463, 1856184949, 1869622537, 1973952949, 2003664181, 2185051189, 2204016173, 2310441383, 2331375133, 2439952297, 2448065387
Offset: 1

Views

Author

Zak Seidov, Jan 17 2019

Keywords

Examples

			p=91289867 is in the sequence because A001222(p-1)=A001222(p+1) = 4, A001222(p-2)=A001222(p+2)=3, A001222(p-3)=A001222(p+3)=5 etc, pairwise equal.

Crossrefs

Subsequence of A323498.
Cf. A115103.

Formula

{primes p: A001222(p-k)=A001222(p+k) for all k=1..7}.

A371622 Primes p such that p - 2 and p + 2 have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

5, 23, 37, 53, 67, 89, 113, 131, 157, 173, 211, 251, 277, 293, 307, 337, 379, 409, 449, 487, 491, 499, 503, 607, 631, 683, 701, 719, 751, 769, 787, 919, 929, 941, 953, 991, 1009, 1039, 1117, 1129, 1181, 1193, 1201, 1237, 1259, 1381, 1399, 1439, 1459, 1471, 1493, 1499, 1511, 1549, 1567, 1597, 1613
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Apr 01 2024

Keywords

Comments

Primes p such that A001222(p - 2) = A001222(p + 2).

Examples

			a(2) = 23 is a term because 23 is prime and 23 - 2 = 21 = 3 * 7 and 23 + 2 = 25 = 5^2 are both products of 2 primes, counted with multiplicity.

Links

Crossrefs

Cf. A001222, A115103. Contains A063643, A063645 and A371651. Contained in A371656.

Programs

  • Maple
    filter:= p -> isprime(p) and numtheory:-bigomega(p-2) = numtheory:-bigomega(p+2):
select(filter, [seq(i,i=3..10000,2)]);
  • Mathematica
    s = {}; p = 3; Do[While[PrimeOmega[p - 2] != PrimeOmega[p + 2], p =
NextPrime[p]]; Print[p]; AppendTo[s, p]; p = NextPrime[p], {100}]; s

A323498 Primes p such that p - k and p + k have the same number of prime factors (with multiplicity), for k = 1..6.

Original entry on oeis.org

2131991, 2917927, 3776273, 4742407, 6853409, 16850609, 21789233, 24095791, 24810251, 26316233, 27470537, 27667529, 28962127, 29896439, 30949327, 31289527, 36123853, 36443893, 38824913, 40941233, 41660009, 42533551, 44233193, 45868967, 48313567, 49265009, 51135991
Offset: 1

Views

Author

Zak Seidov, Jan 16 2019

Keywords

Comments

At least one of p - k and p + k must be composite for each k in for k = 1..5.
Proof: If k = 3 then p - k and p + k are even. If k isn't three then exactly one of p - k, p and p + k is divisible by 3. QED. - David A. Corneth, Jan 18 2019

Examples

			For p = 2131991 is in the sequence because for k=1, p - 1 = 2*5*7*7*9*229 and p + 1 = 2*2*2*3*3*29611 are both 6-almost primes, for k=2, p - 2 = 3*710663 and p + 2 = 29*73517 are both semiprimes, etc.

Crossrefs

Cf. A115103 (k=1), A323536 (k=7), A323537 (k=8).

Programs

  • PARI
    upto(n) = {my(res = List(), q = 5); forprime(p = 7, n, t = 1; for(m = 1, 2, for(i = 0, 2, if(bigomega(p + 2*i + m) != bigomega(p - 2*i - m), t = 0; next(2) ) ) ); if(t == 1, listput(res, p)); q = p; ); res } \\ David A. Corneth, Jan 17 2019
  • PARI
    is(n) = if(!isprime(n) || n < 7, return(0)); for(k = 1, 6, if(bigomega(n + k) != bigomega(n - k), return(0))); 1 \\ David A. Corneth, Jan 17 2019
  • Perl
    use ntheory ':all'; for (my($p,$k)=(2,6); $p <= 10**7; $p = next_prime($p)) { print "$p\n" if vecall {factor($p-$) == factor($p+$)} 1..$k } # Daniel Suteu, Jan 17 2019

Extensions

a(23)-a(27) from David A. Corneth, Jan 17 2019

A323537 Primes p such that p - k and p + k have the same number of prime factors (with multiplicity), for k = 1..8.

Original entry on oeis.org

409476689, 567234347, 626039111, 1072153139, 1496271467, 2076082213, 2624039507, 2727032857, 3211049893, 3735161737, 5378782091, 6126967991, 6945015541, 6976654453, 8002150391, 8363830667, 9010299827, 9238046989, 9559151653, 10108444091, 10673561207, 11220524747, 11755487027
Offset: 1

Views

Author

Zak Seidov, Jan 17 2019

Keywords

Comments

First six terms such that k_max = 9: 6945015541, 9010299827, 13680125387, 18434278453, 20011563589, 22661476973.
First case of k_max = 10: 47298912347.
Larger cases of k_max?

Crossrefs

Subsequence of A323536.
Cf. A115103.

Formula

{primes p: A001222(p+k)=A001222(p-k) for all k=1..8}
Showing 1-5 of 5 results.

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