A219528 -id:A219528 - cp's OEIS frontend

A382546 Positive integers whose prime factors are all in A219528.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Steven Lu, Mar 31 2025

nonn

Steven Lu, Table of n, a(n) for n = 1..10000

Cf. A219528.

Select[Range[75], And @@ Table[SubsetQ[{2, 3}, First /@ FactorInteger[p + 1]] || SubsetQ[{2, 3}, First /@ FactorInteger[p - 1]], {p, First /@ FactorInteger[#]}] &]

Sum_{n>=1} 1/a(n) = 1/Product_{p in A219528} (1-1/p) = 7.52982574262479641306... . - Amiram Eldar, Apr 14 2025

A219697 Primes neighboring a 7-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97, 101, 107, 109, 113, 127, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 239, 241, 251, 257, 269, 271, 281, 293, 337, 349, 359, 379, 383, 401, 419, 421, 431

Offset: 1

Jonathan Vos Post, Nov 25 2012

This is to the 7-smooth numbers A002473 as A219528 is to the 3-smooth numbers A003586 and as A219669 is to the 5-smooth numbers A051037. The first primes NOT within one of a 7-smooth number are 103, 131, 137, 157, 173, ...

			23 is in the sequence as one of 23-1 = 22 = 2 * 11 and 23+1 = 24 = 2^3 * 3 is 7-smooth and 23 is prime. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10765 (terms <= 10^16)

Cf. A000040, A002473, A051037, A080194, A219528.

mx = 2^10; t7 = Select[Sort[Flatten[Table[2^i * 3^j * 5^k * 7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}, {l, 0, Log[7, mx]}]]], # <= mx &]; Union[Select[t7 + 1, PrimeQ], Select[t7 - 1, PrimeQ]] (* T. D. Noe, Nov 26 2012 *)
Select[Prime[Range[90]],Max[FactorInteger[#-1][[;;,1]]]<11||Max[FactorInteger[#+1][[;;,1]]]<11&] (* Harvey P. Dale, Nov 03 2024 *)

is7smooth(n) = forprime(p = 2, 7, n /= p^valuation(n, p)); n==1
is(n) = isprime(n) && (is7smooth(n - 1) || is7smooth(n + 1)) \\ David A. Corneth, Apr 19 2021

Primes INTERSECTION {2^h 3^i 5^j 7^k +/-1 for h,i,j,k >= 0}.

A219669 Primes neighboring a 5-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 71, 73, 79, 89, 97, 101, 107, 109, 127, 149, 151, 163, 179, 181, 191, 193, 199, 239, 241, 251, 257, 269, 271, 359, 383, 401, 431, 433, 449, 479, 487, 499, 541, 577, 599, 601, 641, 647, 719, 751

Offset: 1

Jonathan Vos Post, Nov 24 2012

This is to A219528 as 3-smooth numbers A003586 are to 5-smooth numbers A051037.

			a(1) = 2 = 2^0 * 3^0 * 5^0 + 1 = 2^0 * 3^1 * 5^0 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A051037, A219528.

mx = 2^10; t5 = Select[Sort[Flatten[Table[2^i * 3^j * 5^k, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}]]], # <= mx &]; Union[Select[t5 + 1, PrimeQ], Select[t5 - 1, PrimeQ]] (* T. D. Noe, Nov 25 2012 *)

Primes INTERSECTION {2^i 3^j 5^k +/-1 for i,j,k >= 0}.

Corrected by T. D. Noe, Nov 25 2012

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

A219556 Semiprimes neighboring a 3-smooth number.

Original entry on oeis.org

4, 9, 10, 15, 17, 25, 26, 28, 33, 35, 37, 49, 55, 65, 82, 95, 97, 129, 143, 145, 161, 163, 215, 217, 287, 289, 323, 325, 485, 487, 511, 513, 649, 767, 769, 865, 973, 1457, 1459, 1535, 1537, 1727, 1729, 1943, 1945, 2047, 2049, 2186, 2188, 2305, 3071, 3073, 3455, 3457

Offset: 1

Jonathan Vos Post, Nov 22 2012

This is to A219528 as semiprime A001358 are to primes A000040.

Semiprime numbers of the form of 2^j*3^k +/- 1.

			a(1) = (2^0)*(3^1) + 1 = (2^2)*(3^0) - 1 = 4 = 2*2, a semiprime.
a(2) = (2^3)*(3^0) + 1 = 9 = 3*3.
a(3) = (2^0)*(3^2) + 1 = 10 = 2*5.
a(4) = (2^4)*(3^0) - 1 = 15 = 3*5.

Amiram Eldar, Table of n, a(n) for n = 1..4377 (terms below 10^60)

Cf. A001358, A003586, A219528.

mx = 4000; A003586 = Flatten@ Table[2^i*3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}]; Union@ Join[ Select[A003586, PrimeOmega[# - 1] == 2 &] - 1, Select[A003586, PrimeOmega[# - 1] == 2 || PrimeOmega[# + 1] == 2 &] + 1] (* Robert G. Wilson v, Nov 22 2012 *)

A219785 Primes not neighboring an 11-smooth number.

Original entry on oeis.org

103, 137, 157, 173, 227, 229, 233, 277, 283, 311, 313, 317, 347, 367, 373, 389, 409, 443, 457, 467, 509, 521, 523, 547, 557, 563, 569, 571, 607, 613, 619, 643, 653, 677, 683, 691, 709, 733, 739, 743, 761, 773, 787, 797, 821, 823, 827, 829, 853, 857, 859, 877

Offset: 1

Jonathan Vos Post, Nov 27 2012

			103 is in the sequence because it is prime and the closest 11-smooth numbers are 100 and 105, which differ from 103 by 3 and -2 respectively, neither being -1 or +1.
137 is in the sequence because it is prime and neither 137 - 1 = 136 = 2^3 * 17 nor 137 + 1 = 138 = 2 * 3 * 23 are 11-smooth.

Cf. A000040, A002473, A051037, A051038, A080194, A219528, A219697.

mx = 2^10; t11 = Select[Sort[Flatten[Table[2^i 3^j 5^k 7^l 11^m, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}, {l, 0, Log[7, mx]}, {m, 0, Log[11, mx]}]]], # <= mx &]; Complement[Prime[Range[PrimePi[mx]]], Union[Select[t11 + 1, PrimeQ], Select[t11 - 1, PrimeQ]]] (* T. D. Noe, Nov 27 2012 *)

Numbers k such that k is prime and k is neither (2^i * 3^j * 5^k * 7^l * 11^m) - 1 nor (2^i * 3^j * 5^k * 7^l * 11^m) + 1 for any i, j, k, l, m >= 0.

A219790 Smallest prime not neighboring a prime(n)-smooth number.

Original entry on oeis.org

11, 29, 43, 67, 103, 137, 173, 173, 173, 283, 283, 283, 283, 283, 317, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 787, 787, 787, 907, 907, 907, 907, 1433, 1433, 1433, 1433, 1433, 1447, 1543, 1543, 1867, 1867, 1867, 1867, 1867, 1867

Offset: 1

Jonathan Vos Post, Nov 27 2012

			a(2) = 29, the smallest prime not neighboring a 3-smooth number, since 3 is the 2nd prime; i.e., not of the form 2^j*3^k +/- 1. 43-1 = 2*3*7, 43+1 = 2*2*11, so neither are 5-smooth.
a(3) = 43, the smallest prime not neighboring a 5-smooth number, since 5 is the 3rd prime, and 43-1 = 42 = 2 * 3 * 7 is not 5 smooth, and 43+1 = 44 = 2^2 * 11 is not 5 smooth. - corrected by _Jason Kimberley_, Nov 29 2012
a(4) = 67, the smallest prime not neighboring a 7-smooth number, since 7 is the 4th prime, and 67-1 = 66 = 2 * 3 * 11 is not 7 smooth, and 67+1 = 68 = 2^2 * 17 is not 7 smooth. - corrected by _Jason Kimberley_, Nov 29 2012
a(5) = 103, the smallest prime not neighboring a 11-smooth number, since 11 is the 5th prime, and 103-1 = 102 = 2 * 3 * 17 is not 11 smooth, and 103+1 = 104 = 2^3 * 13 is not 11 smooth.
a(6) = 137, the smallest prime not neighboring a 13-smooth number, since 13 is the 6th prime, and 137-1 = 136 = 2^3 * 17 is not 13 smooth, and 137+1 = 138 = 2 * 3 * 23 is not 13 smooth.

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

Cf. A000040, A002473, A051037, A051038, A080194, A219528, A219697, A219785.

a(n)=my(p=prime(n));forprime(q=6*p-1,,if(vecmax(factor(q-1)[,1])>p && vecmax(factor(q+1)[,1])>p,return(q))) \\ Charles R Greathouse IV, Nov 28 2012

a(n) > 6p for n > 1, where p is the n-th prime. - Charles R Greathouse IV, Nov 28 2012

a(3) and a(4) corrected by Charles R Greathouse IV, Nov 28 2012

a(1) and a(7)-a(53) from Charles R Greathouse IV, Nov 28 2012

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A382546 Positive integers whose prime factors are all in A219528.

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A219697 Primes neighboring a 7-smooth number.

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A219669 Primes neighboring a 5-smooth number.

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A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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A219556 Semiprimes neighboring a 3-smooth number.

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A219785 Primes not neighboring an 11-smooth number.

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A219790 Smallest prime not neighboring a prime(n)-smooth number.

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