A092307 -id:A092307 - cp's OEIS frontend

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

16, 36, 40, 56, 88, 135, 156, 184, 204, 210, 220, 248, 250, 260, 296, 306, 315, 328, 330, 340, 342, 372, 414, 459, 472, 490, 516, 536, 546, 580, 584, 636, 650, 686, 690, 708, 714, 732, 735, 738, 804, 808, 819, 836, 850, 852, 870, 872, 940, 950, 966, 975, 996

Offset: 1

Giovanni Teofilatto and Jonathan Vos Post, Aug 20 2005

4-almost primes of the form semiprime + 1.

			a(1) = 16 because (3*5+1)=(2^4) = 16.
a(2) = 36 because (5*7+1)=((2^2)*(3^2)) = 36.
a(3) = 40 because (3*13+1)=((2^3)*5) = 40.
a(4) = 56 because (5*11+1)=((2^3)*7) = 56.
a(5) = 88 because (3*29+1)=((2^3)*11) = 88.
a(6) = 135 because (2*67+1)=((3^3)*5) = 135.
a(7) = 156 because (5*31+1)=((2^2)*3*13) = 156.
a(8) = 184 because (3*61+1)=((2^3)*23) = 184.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes are in A000040. Semiprimes are in A001358. 4-almost primes are in A014613.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in this sequence.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Similar to A076153; after A076153(0)=3 next difference is A076153(100)=1792 whereas A109287(100)=1794.
Cf. A077065, A079148, A092307, A109288, A109289, A109290.

bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[1000], bo[ # ] == 4 && bo[ # - 1] == 2 &] (* Ray Chandler, Aug 27 2005 *)

is(n)=bigomega(n)==4 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Sep 16 2015

a(n) is in this sequence iff a(n) is in A014613 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Aug 27 2005

Edited by Ray Chandler, Mar 20 2007

A109373 Semiprimes of the form semiprime + 1.

Original entry on oeis.org

10, 15, 22, 26, 34, 35, 39, 58, 86, 87, 94, 95, 119, 122, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 362, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 10 because (3*3+1)=(2*5) = 10.
a(2) = 15 because (2*7+1)=(3*5) = 15.
a(3) = 22 because (3*7+1)=(2*11) = 22.
a(4) = 26 because (5*5+1)=(2*13) = 26.
a(5) = 34 because (3*11+1)=(2*17) = 34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Primes are in A000040. Semiprimes are in A001358.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in this sequence.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A056809, A077065, A079148, A086005, A086006, A092307.
Subsequence of A088707; A064911.

a109373 n = a109373_list !! (n-1)
a109373_list = filter ((== 1) . a064911) a088707_list
-- Reinhard Zumkeller, Feb 20 2012

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[ 700], fQ[ # - 1] && fQ[ # ] &] (* Robert G. Wilson v *)
With[{sps=Select[Range[700],PrimeOmega[#]==2&]},Transpose[Select[ Partition[ sps,2,1],#[[2]]-#[[1]]==1&]][[2]]] (* Harvey P. Dale, Sep 05 2012 *)

is(n)=bigomega(n)==2 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Jan 31 2017

a(n) is in this sequence iff a(n) is in A001358 and (a(n)-1) is in A001358.

a(n) = A070552(n) + 1.

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005

Edited by Ray Chandler, Mar 20 2007

A082539 Primes p such that there is no prime q, q < p with q+1 dividing p+1.

Original entry on oeis.org

2, 3, 13, 37, 61, 73, 109, 157, 193, 229, 241, 277, 313, 337, 373, 397, 409, 421, 457, 541, 577, 613, 661, 673, 709, 733, 757, 829, 877, 997, 1009, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297, 1321, 1381, 1453, 1489, 1597, 1621, 1657

Offset: 1

Benoit Cloitre, May 11 2003

nonn

Contains A005383, primes p such that (p+1)/2 is prime. - T. D. Noe, Apr 28 2004

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A005383, A049084, A084196, A084198, A084200.

Cf. A092307 (primes p such that there are no primes q, 3 < q < p, such that (q-1) divides (p-1)).

Select[Prime[Range[260]], AllTrue[Most[Divisors[# + 1]], !PrimeQ[#1 - 1] &] &] (* Amiram Eldar, Jun 06 2022 *)

A084196(A049084(a(n))) = 0.

More terms from Reinhard Zumkeller, May 18 2003

A109067 3-almost primes of the form semiprime + 1.

Original entry on oeis.org

27, 50, 52, 63, 66, 70, 75, 78, 92, 116, 124, 130, 147, 170, 186, 188, 195, 207, 222, 236, 238, 255, 266, 268, 275, 279, 290, 292, 310, 322, 356, 363, 366, 387, 399, 404, 412, 418, 423, 428, 438, 452, 455, 470, 474, 483, 494, 498, 506, 518, 530, 534, 539, 555

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 27 because (2*13+1)=(3^3) = 27.
a(2) = 50 because (7*7+1)=(2*5^2) = 50.
a(3) = 52 because (3*17+1)=(2^2*13) = 52.
a(4) = 63 because (2*31+1)=(3^2*7) = 63.
a(5) = 66 because (5*13+1)=(2*3*11) = 66.
a(6) = 70 because (3*23+1)=(2*5*7) = 70.
a(7) = 75 because (2*37+1)=(3*5^2) = 75.
a(8) = 78 because (7*11+1)=(2*3*13) = 78.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes are in A000040. Semiprimes are in A001358. 3-almost primes are in A014612.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in this sequence.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[600], f[ # ] == 3 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)
Select[Select[Range[600],PrimeOmega[#]==2&]+1,PrimeOmega[#]==3&] (* Harvey P. Dale, Nov 24 2013 *)

list(lim)=my(v=List(),t); forprime(p=2,lim, forprime(q=2,min(p,lim\p), if(bigomega(t=p*q+1)==3, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

a(n) is in this sequence iff a(n) is in A014612 and a(n)-1 is in A001358.

Edited and extended by Ray Chandler, Mar 20 2007

A109383 5-almost primes of the form semiprime + 1.

Original entry on oeis.org

112, 120, 162, 300, 304, 378, 392, 396, 408, 520, 552, 567, 592, 612, 630, 656, 675, 680, 688, 696, 700, 750, 780, 918, 924, 944, 952, 980, 990, 1044, 1100, 1116, 1136, 1140, 1160, 1168, 1170, 1242, 1264, 1272, 1300, 1323, 1352, 1372, 1380, 1386, 1416, 1470

Offset: 1

Jonathan Vos Post, Aug 25 2005

			a(1) = 112 because (3*37)+1 = (2^4) * 7 = 112.
a(2) = 120 because (7*17)+1 = (2^3) * 3 * 5 = 120.
a(3) = 162 because (7*23)+1 = 2 * (3^4) = 162.

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

Primes are in A000040. Semiprimes are in A001358. 5-almost primes are in A014614.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in this sequence.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[1500], f[ # ] == 5 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)

v=vector(10000);i=0; for(n=1,9e99, if(issemi(n)&bigomega(n+1)==5, v[i++]=n+1;if(i==#v, return))); v \\ Charles R Greathouse IV, Feb 14 2011

a(n) is in this sequence iff a(n) is in A014614 and (a(n)-1) is in A001358.

Extended by Ray Chandler, Mar 20 2007

A152951 Complementary von Staudt prime numbers.

Original entry on oeis.org

71, 131, 191, 251, 311, 419, 431, 491, 599, 683, 743, 911, 947, 971, 1031, 1091, 1103, 1151, 1163, 1427, 1451, 1511, 1559, 1571, 1583, 1607, 1667, 1811, 1871, 1931, 1979, 2003, 2111, 2267, 2351, 2399, 2411, 2423, 2531, 2543, 2591, 2663, 2711, 2843, 2927, 2939

Offset: 0

Peter Luschny, Dec 24 2008

A prime number in the arithmetic progression 12n-1 which is not a von Staudt prime number, i.e., 12p <> denominator(B(p-1)/(p-1)), where B(n) is the Bernoulli number.

Dana Jacobsen, Table of n, a(n) for n = 0..10883
Peter Luschny, Von Staudt prime number, definition and computation.

Cf. A092307.

select(j->(denom(bernoulli(j-1)/(j-1))<>12*j),select(isprime,[seq(12*k-1,k=1..100)]));

Select[ 12*Range[200] - 1, PrimeQ[#] && 12 # != Denominator[ BernoulliB[# - 1]/(# - 1)]& ] (* Jean-François Alcover, Jul 29 2013 *)

use ntheory ":all"; forprimes { my $p=$; say if $ % 12 == 11 && vecany { $ > 3 && $ < $p-1 && is_prime($+1) } divisors($p-1); } 10000; # _Dana Jacobsen, Dec 29 2015

use ntheory ":all"; forprimes { say if $ % 12 == 11 && (bernfrac($-1))[1] != 6*$; } 10000; # _Dana Jacobsen, Dec 29 2015

More terms from Dana Jacobsen, Dec 29 2015

A092308 For p=prime(n), a(n) = the number of primes q such that q-1 divides p-1.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 4, 4, 3, 4, 5, 7, 5, 4, 3, 4, 3, 8, 5, 4, 8, 4, 3, 5, 7, 5, 4, 3, 8, 6, 6, 4, 4, 5, 4, 6, 8, 5, 3, 4, 3, 11, 4, 8, 5, 7, 8, 4, 3, 6, 5, 3, 11, 4, 5, 3, 4, 7, 8, 8, 4, 4, 6, 4, 9, 4, 8, 10, 3, 7, 7, 3, 4, 6, 7, 3, 4, 11, 8, 8, 4, 13, 4, 11, 4, 3, 7, 7, 6, 7, 3, 3, 6, 5, 5, 3, 4, 8, 6, 14, 6, 4

Offset: 1

T. D. Noe, Feb 12 2004

nonn

For many primes p, there are only 3 primes (2,3,p) such that q-1|p-1. See A092307 for a list of those primes.

			a(12)=7 because for prime(12)=37 there are seven primes q={2, 3, 5, 7, 13, 19, 37} such that q-1 divides 36.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A092307 (primes for which a(n)=3).

Table[p=Prime[n]; Length[Select[Divisors[p-1]+1, PrimeQ]], {n, 150}]

A119660 Prime factor of the distinct numbers appearing as denominators of Bernoulli numbers A090801 that is greater than all previous a(n). a(1) = 2.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 239, 263, 347, 359, 383, 443, 467, 479, 503, 563, 587, 647, 659, 719, 827, 839, 863, 887, 983, 1019, 1187, 1223, 1259, 1283, 1307, 1319, 1367, 1439, 1487, 1499, 1523, 1619, 1787

Offset: 1

Alexander Adamchuk, Jul 28 2006

nonn

a(n) is identical to A079148[n] up to a(14)=227. Most a(n) except 2,3,239,443,647,659,827,1223,1259,1499,1787... belong to A005385[n]: Safe primes p: (p-1)/2 is also prime.

Except for 2 and 3, the same as A092307. - T. D. Noe, Sep 25 2006

			A090801[n] begins {1, 2, 6, 30, 42, 66, 138, 282, 330, 354, 498, 510, 642, 690, ...} = {1, {2,1}, {2,3}, {2,3,5}, {2,3,7}, {2,3,11}, {2,3,23}, {2,3,47}, {2,3,5,11}, {2,3,59}, {2,3,83}, {2,3,5,17}, {2,3,107}, {2,3,5,23}, ...}.
a(1) = 2, a(2) = 3, a(3) = 5, a(4) = 7, a(5) = 11, a(6) = 23, a(7) = 47, a(8) = 59, a(9) = 83, a(10) = 107.

Cf. A090801, A079148, A005385.

A152952 Von Staudt primes which are not safe primes (A005385).

Original entry on oeis.org

239, 443, 647, 659, 827, 1223, 1259, 1499, 1787, 1847, 2087, 2243, 2339, 2687, 2699, 3299, 3659, 3767, 4943, 5903, 6263, 6287, 6299, 6563, 6863, 6959, 7043, 7487, 7583, 7883, 7907, 7919, 8087, 8219, 8243, 8387, 8627, 8663

Offset: 1

Peter Luschny, Dec 25 2008

nonn

			239 is a von Staudt prime because the denominator(B(239-1)/(239-1))=239*12, where B(n) is the Bernoulli number, but (239-1)/2=119=7*17 is not a prime.

Jean-François Alcover, Table of n, a(n) for n = 1..100
P. Luschny, Von Staudt prime number, definition and computation.

Cf. A092307, A005385 and A152951.

a := proc(n) local k,L; L:= []; for k from 11 by 12 to n do map(i->i+1,divisors(k-1)); select(isprime,%) minus {2,3}; if % = {k} then L := [op(L),k] fi; od; select(isprime,map(i->i+i+1,select(isprime,[$1..iquo(n,2)]))): sort(convert(convert(L,set) minus convert(%,set),list)): end:

vonStaudtPrimeQ[p_?PrimeQ] := Denominator[BernoulliB[p-1]/(p-1)] == 12*p; safePrimeQ[p_?PrimeQ] := PrimeQ[(p-1)/2]; Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[vonStaudtPrimeQ[p] && !safePrimeQ[p], Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Jan 27 2014 *)

A219742 Bernoulli denominators with 8 divisors in increasing order (without repetitions).

Original entry on oeis.org

30, 42, 66, 138, 282, 354, 498, 642, 1002, 1074, 1362, 1434, 1578, 2082, 2154, 2298, 2658, 2802, 2874, 3018, 3378, 3522, 3882, 3954, 4314, 4962, 5034, 5178, 5322, 5898, 6114, 7122, 7338, 7554, 7698, 7842, 7914, 8202, 8634, 8922, 8994, 9138, 9714, 10722

Offset: 1

Arkadiusz Wesolowski, Nov 29 2012

nonn

Let m, n >= 1 and let f(m) denote number of Bernoulli numbers less than or equal to 10^m having denominator divisible by a(n). For any n, f(m) = floor(10^m/(a(n)/6 - 1)). It appears that the fraction of even Bernoulli numbers with denominator 6 is not so close to 1/6.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Paul Erdős and Samuel S. Wagstaff, Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), pp. 104-112.
Eric Weisstein's World of Mathematics, Bernoulli Number
Index entries for sequences related to Bernoulli numbers

Cf. A002445, A092307, A114648, A138636.

6*Prime@Flatten@Position[Table[p = Prime[n]; Length@Select[Divisors[p - 1] + 1, PrimeQ], {n, 277}], 3]

a(n) = 6*A092307(n).

A002445 INTERSECT A138636.

cp's OEIS Frontend

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

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A109373 Semiprimes of the form semiprime + 1.

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A082539 Primes p such that there is no prime q, q < p with q+1 dividing p+1.

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A109067 3-almost primes of the form semiprime + 1.

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A109383 5-almost primes of the form semiprime + 1.

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A152951 Complementary von Staudt prime numbers.

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A092308 For p=prime(n), a(n) = the number of primes q such that q-1 divides p-1.

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