A077065 -id:A077065 - cp's OEIS frontend

A232221 a(n) = Sum_{i=1..n} (A077068(i) - A077065(i)).

0, 0, 4, 20, 36, 52, 128, 216, 328, 464, 636, 796, 908, 1092, 1324, 1520, 1716, 1948, 2144, 2436, 2716, 2972, 3264, 3580, 3812, 4032, 4168, 4268, 4416, 4720, 5012, 5328, 5716, 6128, 6504, 6700, 6932, 7248, 7684, 8180, 8676, 9268, 9680, 10140, 10624, 11024, 11400

Offset: 1

N. J. A. Sloane, Nov 22 2013, based on a posting by V.J. Pohjola to the Sequence Fans Mailing List, Nov 22 2013

The sequence has wild fluctuations - see the successive plots in the links. This is typical behavior for a particle whose movement is governed by an arc-sine law (cf. Feller, Chap. III). - N. J. A. Sloane, Nov 23 2013
Negative stretches: terms 941-1031 and 13197-1431205. - Hans Havermann, Nov 23 2013
After reaching a local maximum of 21957005755012 at term 24118371, the sequence again descends with the first negative of a third such stretch at term 32437583. - Hans Havermann, Nov 28 2013
All terms are multiples of 4, cf. A008586. - Reinhard Zumkeller, Nov 22 2013
See A232361 and A232359 for peak values and where they occur: max{a(A232359(n)-1), a(A232359(n)+1)} < a(A232359(n)) = A232361(n). - Reinhard Zumkeller, Nov 24 2013

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS "graph" command, Pin plot of 200 terms, scatter plot of 10000 terms
Hans Havermann, Four graphs with an increasing number of terms, personal web page.
Veikko Pohjola, Re: Sister sequences, SeqFan list, Nov 22 2013. (Early observation on how the sequence evolves.)

Cf. A077065, A077068, A232359, A232361.

Partial sums of A232342.

a232221 n = a232221_list !! (n-1)
a232221_list = scanl1 (+) a232342_list
-- Reinhard Zumkeller, Dec 16 2013, Nov 22 2013

A232342 A077068(n) minus A077065(n).

Original entry on oeis.org

0, 0, 4, 16, 16, 16, 76, 88, 112, 136, 172, 160, 112, 184, 232, 196, 196, 232, 196, 292, 280, 256, 292, 316, 232, 220, 136, 100, 148, 304, 292, 316, 388, 412, 376, 196, 232, 316, 436, 496, 496, 592, 412, 460, 484, 400, 376, 412, 556, 736, 1000, 940, 1012

Offset: 1

Reinhard Zumkeller, Dec 16 2013

sign

All terms are multiples of 4, cf. A008586;
a(n) = A077068(n) - A077065(n).
First term < 0: a(426) = -104.

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

Cf. A232221 (partial sums).

a232342 n = a232342_list !! (n-1)
a232342_list = zipWith (-) a077068_list a077065_list

A179882 a(n) is the corresponding value of contraharmonic mean B(h) of numbers k such that gcd(k, h) = 1 (k < h) for numbers h from A179877(n) and A179878(n).

Original entry on oeis.org

1, 7, 15, 31, 39, 55, 71, 111, 119, 151, 175, 177, 231, 239, 255, 303, 311, 313, 319, 329, 335, 337, 345, 375, 391, 393, 479, 521, 559, 575, 591, 593, 601, 607, 623, 655, 657, 679, 777, 785, 791, 823, 855, 863, 871, 879, 889, 905, 911, 929, 937, 959, 961, 991

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Subsequence of A179873 and A179874.
It appears that for n >= 3, (4*A005384(n)+1)/3 is a subsequence. - Hilko Koning, Jul 27 2018
This happens for this subsequence of A179877: 10, 22, 46, 58, 82, 106, 166, 178, ... apparently "Semiprimes of form prime - 1" >= 10 (see A077065). - Michel Marcus, Jul 27 2018

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

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

{1}~Join~Select[Partition[Table[ContraharmonicMean@ Select[Range[n - 1], GCD[#, n] == 1 &], {n, 2, 1500}], 2, 1], And[IntegerQ@ First@ #, SameQ @@ #] &][[All, 1]] (* Michael De Vlieger, Jul 30 2018 *)

lista(nn) = {vch = vector(nn, k, ch(k)); for (i=1, nn-1, if ((vch[i] == vch[i+1]) && !frac(vch[i]), print1(vch[i], ", ")););} \\ Michel Marcus, Jul 27 2018

a(n) = A175505(A179877(n)) / A175506(A179877(n)).

a(n) = A175505(A179878(n)) / A175506(A179878(n)).

More terms from Michel Marcus, Jul 27 2018

A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

Original entry on oeis.org

3, 4, 5, 6, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282

Offset: 1

Alois P. Heinz, Sep 03 2009

Numbers m such that the triangular number T(m) = m*(m+1)/2 has exactly two divisors >= m.
Also numbers m such that m*(m+1)/2 is the product of two primes.
Contains all numbers in A005383. - Harry Richman, Jan 09 2025
Contains all numbers in A077065. - Alois P. Heinz, Jan 19 2025

			10 is in the sequence, because there is only one nontrivial decomposition of {1..10} into subsets with equal element sum: {1,10}, {2,9}, {3,8}, {4,7}, {5,6}; 11|55.
13 is in the sequence with decomposition of {1..13}: {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}, {13}; 13|91.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A001222, A164978, A035470, A068443, A069904, A001358.

Cf. A005383, A077065 (distinct subsequences).

a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 2, a(n-1))
      while not (andmap(isprime, [k, (k+1)/2]) or
                 andmap(isprime, [k+1, k/2]))
      do od; k
    end:
seq(a(n), n=1..100);

Select[Range@1304, PrimeOmega[#] + PrimeOmega[# + 1] == 3 &] (* Robert G. Wilson v, Jun 28 2010 and updated Sep 21 2018 *)

is(n)=if(isprime(n),bigomega(n+1)==2, isprime(n+1) && bigomega(n)==2) \\ Charles R Greathouse IV, Sep 08 2015

is(n)=if(n%2, isprime((n+1)/2) && isprime(n), isprime(n/2) && isprime(n+1)) \\ Charles R Greathouse IV, Mar 16 2022

list(lim)=my(v=List()); forprime(p=3,lim, if(isprime((p+1)/2), listput(v,p))); forprime(p=5,lim+1, if(isprime(p\2), listput(v,p-1))); Set(v) \\ Charles R Greathouse IV, Mar 16 2022

{ m :  A035470(m) = 2 }.
{ m :  A164978(m) = 2 }.
{ m : |{d|m*(m+1)/2 : d>=m}| = 2 }.
{ m :  m*(m+1)/2 in {A068443} }.
{ m :  m*(m+1)/2 in {A001358} }.
{ m :  A069904(m) = 2 }.
{ m :  A001222(n) + A001222(n+1) = 3 }. - Alois P. Heinz, Jan 08 2022
{ A005383 } union { A077065 }. - Alois P. Heinz, Jan 19 2025

A077068 Semiprimes of the form prime + 1.

Original entry on oeis.org

4, 6, 14, 38, 62, 74, 158, 194, 278, 314, 398, 422, 458, 542, 614, 662, 674, 734, 758, 878, 998, 1094, 1154, 1202, 1214, 1238, 1322, 1382, 1454, 1622, 1658, 1754, 1874, 1934, 1994, 2018, 2138, 2342, 2474, 2558, 2594, 2798, 2858, 2918, 3062, 3218, 3254

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

a(n) = A005383(n)+1 = 2*A005382(n).
There are 672 semiprimes of form prime+1 below 100000.
a(n) = A232342(n) + A077065(n). - Reinhard Zumkeller, Dec 16 2013

			A001358(25)=74=2*37 is a term as 74=A000040(21)+1=73+1.

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

Cf. A008864, A001358, A000040, A077065, A077067.

a077068 n = a077068_list !! (n-1)
a077068_list = filter ((== 1) . a010051 . (`div` 2)) a008864_list
-- Reinhard Zumkeller, Nov 22 2013

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#-1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[ Prime@ Range@ 460 +1,PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x+1|x<-primes(10^5),bigomega(x+1)==2] \\ Charles R Greathouse IV, Nov 22 2013

is(n)=isprime(n-1) && isprime(n\2) \\ Charles R Greathouse IV, Mar 20 2016

A010051(A008864(n)/2) = A064911(A008864(n)) = 1. - Reinhard Zumkeller, Nov 22 2013

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

Original entry on oeis.org

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

A350593 Numbers k such that tau(k) + tau(k+1) = 6, where tau is the number of divisors function A000005.

Original entry on oeis.org

5, 6, 7, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282

Offset: 1

Jon E. Schoenfield, Jan 08 2022

nonn

Since tau(k) + tau(k+1) = 6, (tau(k), tau(k+1)) must be (1,5), (2,4), (3,3), (4,2), or (5,1); of these, (1,5) and (5,1) are impossible (tau(m) = 1 only for m=1, but then neither m+1 nor m-1 would have 5 divisors), and (3,3) is also impossible (both k and k+1 would have to be squares of primes), so (tau(k), tau(k+1)) must be either (2,4) or (4,2).
For every prime p, tau(p) = 2. For every semiprime s, tau(s) = 4, with the exception of the squares of primes; for p prime, tau(p^2) = 3, since the divisors of p^2 are 1, p, and p^2.
The only numbers that have exactly 4 divisors but are not semiprimes are the cubes of primes; for prime p, the divisors of p^3 are 1, p, p^2, and p^3.
As a result, this sequence consists of:
(1) the primes p such that (p+1)/2 is prime (A005383), with the exception of p=3 (since p+1 = 4 has 3 divisors, not 4),
(2) semiprimes of the form prime - 1 (A077065), with the exception of the semiprime 4 (since it does not have 4 divisors), and
(3) the special case k = 7, since it is the unique prime p such that p+1 has 4 divisors but is not a semiprime.
For all k > 4, tau(k) + tau(k+1) >= 6; for k = 1..4, tau(k) + tau(k+1) = 3, 4, 5, 5.

			   k  tau(k)  tau(k+1)  tau(k) + tau(k+1)
  --  ------  --------  -----------------
   1     1        2         1 + 2 = 3
   2     2        2         2 + 2 = 4
   3     2        3         2 + 3 = 5
   4     3        2         3 + 2 = 5
   5     2        4         2 + 4 = 6   so   5 = a(1)
   6     4        2         4 + 2 = 6   so   6 = a(2)
   7     2        4         2 + 4 = 6   so   7 = a(3)
   8     4        3         4 + 3 = 7
   9     3        4         3 + 4 = 7
  10     4        2         4 + 2 = 6   so  10 = a(4)
  11     2        6         2 + 6 = 8
  12     6        2         6 + 2 = 8
  13     2        4         2 + 4 = 6   so  13 = a(5)

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000005, A005383, A077065, A092405, A164977.

Numbers k such that Sum_{j=0..N-1} tau(k+j) = 2*Sum_{k=1..N} tau(k): A000040 (N=1), (this sequence) (N=2), A350675 (N=3), A350686 (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).

Select[Range[1300], Plus @@ DivisorSigma[0, # + {0, 1}] == 6 &] (* Amiram Eldar, Jan 08 2022 *)
Position[Total/@Partition[DivisorSigma[0,Range[1300]],2,1],6]//Flatten (* Harvey P. Dale, Sep 03 2022 *)

isok(k) = numdiv(k) + numdiv(k+1) == 6; \\ Michel Marcus, Jan 08 2022

from itertools import count, islice
from sympy import divisor_count
def A350093_gen(): # generator of terms
    a, b = divisor_count(1), divisor_count(2)
    for k in count(1):
        if a + b == 6:
            yield k
        a, b = b, divisor_count(k+2)
A350093_list = list(islice(A350093_gen(),12)) # Chai Wah Wu, Jan 11 2022

{ k : tau(k) + tau(k+1) = 6 }.
UNION(A005383 \ {3}, A077065 \ {4}, {7}).
a(n) = A164977(n+1) for n>=4. - Hugo Pfoertner, Jan 08 2022

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

cp's OEIS Frontend

A232221 a(n) = Sum_{i=1..n} (A077068(i) - A077065(i)).

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A232342 A077068(n) minus A077065(n).

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A179882 a(n) is the corresponding value of contraharmonic mean B(h) of numbers k such that gcd(k, h) = 1 (k < h) for numbers h from A179877(n) and A179878(n).

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A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

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A077068 Semiprimes of the form prime + 1.

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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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