A115103 -id:A115103

Search: a115103 -id:a115103

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A067386

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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 *)

PROG

(PARI) is(n)=omega(n-1)==omega(n+1) && isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Feb 23 2002

STATUS

approved

A323536

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..36.

FORMULA

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

EXAMPLE

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.

KEYWORD

nonn

AUTHOR

Zak Seidov, Jan 17 2019

STATUS

approved

A371622

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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.

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Zak Seidov and Robert Israel, Apr 01 2024

STATUS

approved

A323498

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..27.

EXAMPLE

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.

PROG

(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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Zak Seidov, Jan 16 2019

EXTENSIONS

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

STATUS

approved

A323537

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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?

LINKS

Table of n, a(n) for n=1..23.

FORMULA

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

CROSSREFS

Subsequence of A323536.

Cf. A115103.

KEYWORD

nonn

AUTHOR

Zak Seidov, Jan 17 2019

STATUS

approved

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