Michael B. Porter - cp's OEIS frontend

A211397 Number of Sophie Germain primes less than 2^n.

0, 2, 3, 4, 6, 8, 11, 18, 26, 39, 62, 103, 170, 281, 474, 834, 1464, 2555, 4493, 8051, 14499, 26375, 48024, 88175, 161833, 297544, 549330, 1018008, 1893255, 3527324, 6588118, 12334363, 23140567, 43497488, 81930886, 154587025, 292149120, 552997218, 1048340476, 1990145943, 3783145017

Offset: 1

Michael B. Porter, Feb 08 2013

nonn

Paul D. Beale, A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, arXiv preprint arXiv:1411.2484 [physics.comp-ph], 2014-2015.
Jetanat Datephanyawat and Paul D. Beale, Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers, arXiv:1811.11629 [cs.CR], 2018-2021.

Cf. A211395, A005385.

nmax = 37; stable = Table[0, {nmax}];
Do[s = 0;
  Do[If[And[PrimeQ[i], PrimeQ[2 i + 1]], s++], {i, 1, 2^n - 1}];
  Print[n, " ", s]; stable[[n]] = s, {n, 1, nmax}];
stable (* Paul D. Beale, Sep 19 2014 *)

a211397(n) = {local(r,i); r=0; for(i=1, 2^n-1, if(isprime(i)&&isprime(2*i+1), r=r+1)); r}

a(30)-a(37) from Paul D. Beale, Sep 19 2014

a(38)-a(41) from Amiram Eldar, Jul 25 2025

A166376 Triangle in which n-th row (n>1) gives prime factors of n^2 + 1 with repetition.

Original entry on oeis.org

2, 5, 2, 5, 17, 2, 13, 37, 2, 5, 5, 5, 13, 2, 41, 101, 2, 61, 5, 29, 2, 5, 17, 197, 2, 113, 257, 2, 5, 29, 5, 5, 13, 2, 181, 401, 2, 13, 17, 5, 97, 2, 5, 53, 577, 2, 313, 677, 2, 5, 73, 5, 157, 2, 421, 17, 53, 2, 13, 37, 5, 5, 41

Offset: 1

Michael B. Porter, Oct 13 2009

			Since 13^2+1 = 170 = 2*5*17, the three terms 2, 5, 17 appear in the sequence.

Cf. A027746 A002522

row(n)={m=n^2+1;while(m!=1,p=factor(m)[1,1];print(p);m=m/p)}

A167189 Larger prime power associated with record gap in A167186.

Original entry on oeis.org

4, 8, 16, 25, 49, 121, 169, 243, 512, 1331, 1681, 2809, 4913, 6241, 7921, 9409, 14641, 22201, 36481, 44521, 49729, 94249, 185761, 226981, 271441, 292681, 452929, 619369, 769129, 822649, 1181569, 1852321, 5442889, 6265009, 8994001, 10883401, 17648401, 18722929

Offset: 1

Michael B. Porter, Nov 01 2009

nonn

			49 is in the sequence since the gap between it the previous prime power, 32, is greater than any previous gap.

Donovan Johnson, Table of n, a(n) for n = 1..200

Smaller prime power is in A167188.
Record gaps between nonprime prime powers: A167186.
Gaps between nonprime prime powers: A053707.
List of nonprime prime powers: A025475.

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;if(d>d_max,print(n);d_max=d);n_prev=n))

A167188 Smaller prime power associated with record gap in A167186.

Original entry on oeis.org

1, 4, 9, 16, 32, 81, 128, 169, 361, 1024, 1369, 2401, 4489, 5329, 6889, 8192, 12769, 19683, 32768, 39601, 44521, 85849, 177241, 218089, 262144, 279841, 436921, 597529, 744769, 786769, 1142761, 1771561, 5340721, 6135529, 8826841, 10699441, 17447329, 18464209

Offset: 1

Michael B. Porter, Nov 01 2009

nonn

			32 is in the sequence since the gap between 32 and the next prime power, 49, is greater than any previous gap.

Donovan Johnson, Table of n, a(n) for n = 1..200

Larger prime power: A167189.
Record gaps between nonprime prime powers: A167186.
Gaps between nonprime prime powers: A053707.
List of nonprime prime powers: A025475.

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;if(d>d_max,print(n_prev);d_max=d);n_prev=n))

A167186 Record gaps between nonprime prime powers.

Original entry on oeis.org

3, 4, 7, 9, 17, 40, 41, 74, 151, 307, 312, 408, 424, 912, 1032, 1217, 1872, 2518, 3713, 4920, 5208, 8400, 8520, 8892, 9297, 12840, 16008, 21840, 24360, 35880, 38808, 80760, 102168, 129480, 167160, 183960, 201072, 258720, 290760, 301242, 358848, 375468, 415920

Offset: 1

Michael B. Porter, Oct 29 2009, Oct 31 2009, Nov 03 2009

nonn

			17 is in the sequence since A025475(9) - A025475(8) = 49 - 32 = 17, and no previous gap is larger.
A025475(10) - A025475(9) = 64 - 49 = 15, but the previous gap is larger, so 15 is not in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..200

List of nonprime prime powers: A025475.
Gaps between nonprime prime powers: A053707.
Record gaps between prime powers including primes: A121492.

Join[{3},DeleteDuplicates[Differences[Select[Range[10^6],PrimePowerQ[#] && !PrimeQ[ #]&]], GreaterEqual]] (* Harvey P. Dale, Feb 28 2023 *)

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
d_max=0;n_prev=1;for(n=2,32e6,if(isA025475(n),d=n-n_prev;n_prev=n;if(d>d_max,print(d);d_max=d)))

A167236 Larger prime power associated with gaps in A121492.

Original entry on oeis.org

2, 7, 16, 23, 37, 59, 97, 149, 211, 307, 907, 1151, 1361, 5623, 8501, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779

Offset: 1

Michael B. Porter, Nov 01 2009, Nov 03 2009

nonn

			59 is in the sequence since 53 and 59 are consecutive prime powers with a difference of 6 and no smaller pair of consecutive prime powers differ by 6 or more.

Jan Kristian Haugland, Table of n, a(n) for n = 1..87. The extra terms after a(33) are copied from A000101 as the associated prime gaps do not contain any prime powers.

Size of gap: A121492
Smaller prime power (start of gap): A002540
Gaps between prime powers: A057820
List of prime powers: A000961

isA000961(n) = (omega(n) == 1 || n == 1)
d_max=0;n_prev=1;for(n=2,1e6,if(isA000961(n),d=n-n_prev;if(d>d_max,print(n);d_max=d);n_prev=n))

a(34) onwards from Jan Kristian Haugland, Oct 18 2024

A167185 Largest prime power <= n that is not prime.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 8, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 27, 27, 27, 27, 27, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 1

Michael B. Porter, Oct 29 2009

			For a(14), 10, 12, and 14 are not prime powers, and 11 and 13 are prime powers but they are prime. Since 9 = 3^3 is a prime power, a(14) = 9.

Robert Price, Table of n, a(n) for n = 1..2000

List of nonprime prime powers: A025475.
Next nonprime prime power: A167184.
Previous prime power including primes: A031218.

Array[SelectFirst[Range[#, 1, -1], Or[And[! PrimeQ@ #, PrimePowerQ@ #], # == 1] &] &, 74] (* Michael De Vlieger, Jun 14 2017 *)

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
A167185(n) = {local(m);m=n;while(!isA025475(m),m--);m}

from sympy import factorint
def A167185(n): return next(filter(lambda m:len(f:=factorint(m))<=1 and max(f.values(),default=2)>1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

p = [n for n in (1..81) if (is_prime_power(n) or n == 1) and not is_prime(n)]
r = [[p[i]]*(p[i+1] - p[i]) for i in (0..9)]
print([y for x in r for y in x]) # Peter Luschny, Jun 14 2017

A167184 Smallest prime power >= n that is not prime.

Original entry on oeis.org

1, 4, 4, 4, 8, 8, 8, 8, 9, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27, 27, 32, 32, 32, 32, 32, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81

Offset: 1

Michael B. Porter, Oct 29 2009

			For a(12), 12, 14, and 15 are not prime powers, and 13 is a prime power but it is prime. Since 16 = 2^4 is a prime power, a(12) = 16.

Robert Price, Table of n, a(n) for n = 1..2000

List of nonprime prime powers: A025475.
Previous nonprime prime power: A167185.
Next prime power including primes: A000015.

Module[{ppwrs=Join[{1},Sort[Flatten[Table[Prime[Range[5]]^p,{p,2,10}]]]]}, Flatten[ Table[Select[ppwrs,#>=n&,1],{n,90}]]] (* Harvey P. Dale, Oct 06 2014 *)

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
A167184(n) = {local(m);m=n;while(!isA025475(m),m++);m}

from itertools import count
from sympy import factorint
def A167184(n): return next(filter(lambda m:len(f:=factorint(m))<=1 and max(f.values(),default=2)>1, count(n))) # Chai Wah Wu, Oct 25 2024

A166061 19-rough numbers: positive integers that have no prime factors less than 19.

Original entry on oeis.org

1, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Michael B. Porter, Oct 05 2009

Or, positive integers relatively prime to 510510 = 2*3*5*7*11*13*17.

			437 = 19 * 23 is in the sequence since the two prime factors, 19 and 23, are not less than 19.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rough Number
Index entries for sequences related to smooth numbers [From _Michael B. Porter_, Oct 10 2009]

k-rough numbers: A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.

Join[{1},Select[Range[300],Min[Transpose[FactorInteger[#]][[1]]]>18&]] (* Harvey P. Dale, Feb 18 2015 *)

isA166061(n) = gcd(n,510510)==1 \\ Michael B. Porter, Oct 10 2009

a(n) = k*n + O(1) where k = 17017/3072 = 5.539388.... In particular, k*n - 31 < a(n) < k*n + 25. - Charles R Greathouse IV, Sep 24 2018

A166063 23-rough numbers: positive integers that have no prime factors less than 23.

Original entry on oeis.org

1, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

Offset: 1

Michael B. Porter, Oct 05 2009

Or, positive integers relatively prime to 9699690 = 2*3*5*7*11*13*17*19.

First composite term is 529 = 23^2.

			667 = 23 * 29 is in the sequence since the two prime factors, 23 and 29, are not less than 23.

G. C. Greubel, Table of n, a(n) for n = 1..1200
Eric Weisstein's World of Mathematics, Rough Number
Index entries for sequences related to smooth numbers

k-rough numbers: A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063.

Cf. A332797 (subsequence).

A166063 := proc(n)
    option remember;
    local a;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            numtheory[factorset](a) ;
            if min(op(%)) >= 23 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A166063(n),n=1..80) ; # R. J. Mathar, Nov 05 2024

Select[Range[500],FactorInteger[#][[1,1]]>22&] (* Harvey P. Dale, Nov 22 2010 *)

isA166063(n) = gcd(n,9699690)==1 \\ Michael B. Porter, Oct 10 2009

a(n) = k*n + O(1) where k = 323323/55296 = 5.8471.... In particular, k*n - 51 < a(n) < k*n + 45. - Charles R Greathouse IV, Sep 21 2018

A166061 SETMINUS A332798 - R. J. Mathar, Nov 05 2024

Additional terms provided provided by Harvey P. Dale, Nov 22 2010

cp's OEIS Frontend

User: Michael B. Porter

A211397 Number of Sophie Germain primes less than 2^n.

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A166376 Triangle in which n-th row (n>1) gives prime factors of n^2 + 1 with repetition.

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A167189 Larger prime power associated with record gap in A167186.

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A167188 Smaller prime power associated with record gap in A167186.

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A167186 Record gaps between nonprime prime powers.

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A167236 Larger prime power associated with gaps in A121492.

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A167185 Largest prime power <= n that is not prime.

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A167184 Smallest prime power >= n that is not prime.

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