A077068 -id:A077068 - 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

A077065 Semiprimes of form prime - 1.

Original entry on oeis.org

4, 6, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578, 2818, 2878, 2902

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

There are 670 semiprimes of form prime-1 below 10^5.

			A001358(16) = 46 = 2*23 is a term as 46 = A000040(15) - 1 = 47 - 1.

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

Intersection of A006093 and A001358.
Intersection of A006093 and A100484.
Cf. A077068, A232342, A064911.
Cf. A005384, A005385.
Cf. A000010, A023900, A194593.

a077065 n = a077065_list !! (n-1)
a077065_list = filter ((== 1) . a010051' . (`div` 2)) a006093_list
-- Reinhard Zumkeller, Nov 22 2013, Oct 27 2012

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is NthPrime(n)-1]; // Vincenzo Librandi, Oct 17 2012

q:= n-> (n::even) and andmap(isprime, [n+1, n/2]):
select(q, [$1..5000])[];  # Alois P. Heinz, Jul 19 2023

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#+1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[Range[3000],PrimeOmega[#]==2&&PrimeQ[#+1]&] (* Harvey P. Dale, Oct 16 2012 *)
Select[ Prime@ Range@ 430 - 1, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x-1|x<-primes(10^4),bigomega(x-1)==2] \\ Charles R Greathouse IV, Nov 22 2013

a(n) = A005385(n) - 1 = 2*A005384(n).
A010051(A006093(a(n))/2) = A064911(A006093(a(n))) = 1. - Reinhard Zumkeller, Nov 22 2013
a(n) = A077068(n) - A232342(n). - Reinhard Zumkeller, Dec 16 2013
a(n) = A000010(A194593(n+1)). - Torlach Rush, Aug 23 2018
A000010((a(n)*2)+2) = A023900((a(n)*2)+2). - Torlach Rush, Aug 23 2018

A121885 Excess of n-th prime over previous semiprime.

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 1, 3, 5, 2, 2, 4, 1, 2, 1, 3, 2, 2, 4, 2, 1, 2, 2, 6, 8, 1, 3, 2, 4, 2, 3, 5, 3, 5, 2, 2, 1, 4, 1, 3, 4, 6, 3, 5, 2, 2, 1, 3, 7, 2, 4, 2, 3, 1, 2, 4, 3, 3, 5, 2, 2, 2, 4, 3, 2, 2, 1, 3, 7, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 4, 4, 6, 2, 6, 2, 3, 3

Offset: 3

Jonathan Vos Post, Aug 31 2006

See: A102415 Greatest semiprime less than n-th prime. See: A102414 Smallest semiprime greater than n-th prime.

T. D. Noe, Table of n, a(n) for n = 3..10000

Cf. A000040, A001358, A062721, A077068, A102414, A102415.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Table[i = Prime[n] - 1; While[! SemiPrimeQ[i], i--]; Prime[n] - i, {n, 3, 100}] (* T. D. Noe, Oct 08 2012 *)
eps[n_]:=Module[{c=n-1},While[PrimeOmega[c]!=2,c--];n-c]; Table[eps[n],{n,Prime[Range[3,90]]}] (* Harvey P. Dale, Aug 12 2014 *)

dsemi(n)= { local(k=0); if(isprime(n),k=0;while(bigomega(n-k)<>2&&kAntonio Roldán, Oct 08 2012

a(n) = Min{A000040(n)-s for s < A000040(n) and s in A001358(k)}. a(n) = A000040(n) - A102415(n).

Extended by T. D. Noe, Oct 08 2012

A121884 Excess of n-th semiprime over previous prime.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 4, 1, 2, 3, 2, 4, 2, 4, 5, 1, 4, 2, 1, 4, 3, 2, 3, 4, 2, 4, 5, 6, 3, 2, 2, 5, 6, 8, 9, 10, 2, 2, 3, 2, 3, 4, 6, 7, 4, 1, 2, 4, 3, 2, 4, 5, 2, 4, 6, 1, 2, 3, 4, 6, 7, 10, 2, 3, 4, 6, 7, 8, 10, 3, 2, 4, 6, 8, 2, 3, 2, 5, 2, 4, 3, 1, 4, 6, 8, 2, 5, 6, 8, 9, 10, 12, 2, 1, 2, 4, 6

Offset: 1

Jonathan Vos Post, Aug 31 2006

a(n) = 1 iff n is in A077068. a(n) = 2 iff n is in A062721.

Robert G. Wilson v, Table of n, a(n) for n=1..1000

Cf. A000040, A001358, A062721, A077068.

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; lst1 = Select[ Range@ 325, semiPrimeQ@# &]; lst = Select[ Range@ 500, semiPrimeQ@# &]; lst - (NextPrime[ #, -1] & /@ lst) (* Robert G. Wilson v, Mar 16 2009 *)
#-NextPrime[#,-1]&/@Select[Range[400],PrimeOmega[#]==2&] (* Harvey P. Dale, Feb 15 2018 *)

a(n) = Min{A001358(n)-p for p < A001358(n) and p in A000040(k)}.

a(31)-a(103) from Robert G. Wilson v, Mar 16 2009

A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

Original entry on oeis.org

0, 0, 1, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Derek Orr, May 29 2014

nonn

It is unknown whether a(6) = 3 is the highest number in this sequence.
No terms higher than 3 among the first 10000 terms. - Antti Karttunen, Jan 20 2025
a(n) is the number of primes among n-1, n/2 and q, where q satisfies q*(q+1)=n. So a(n) <= 2 for n > 6, and a(n) = 2 iff n != 6 is in A053185 + 1 or A077068. - Jinyuan Wang, Jan 20 2025

			4*k/(4+k) has two solutions: k=4, p=2 and k=12, p=3. Thus a(4) = 2.
From _Antti Karttunen_, Jan 18 2025: (Start)
For n=3, the ratio (k*n)/(k+n) obtains for k=1..3*(3-1) the values 3/4, 6/5, 3/2, 12/7, 15/8, 2, and only the last one of these is prime, therefore a(3) = 1.
For n=26, the only k such that (k*n)/(k+n) is a prime, is k=26, with (26^2)/(2*26) = 13, therefore a(26) = 1. (End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A053185, A063647, A077068.

A243016(n) = { my(s); sum(k=1, n*(n-1), s = (k*n)/(k+n); (1==denominator(s) && isprime(s))); }; \\ Edited by Antti Karttunen, Jan 18 2025

a(n) <= A063647(n). - Antti Karttunen, Jan 18 2025

Data section extended up to a(105) and incorrect terms, that were caused by dropping of a(26) and a(27) (first discrepancies at n=26, 28, 30, 34, etc.) corrected by Antti Karttunen, Jan 18 2025

A276983 Semiprimes n such that n-1 or n+1 is prime.

Original entry on oeis.org

4, 6, 10, 14, 22, 38, 46, 58, 62, 74, 82, 106, 158, 166, 178, 194, 226, 262, 278, 314, 346, 358, 382, 398, 422, 458, 466, 478, 502, 542, 562, 586, 614, 662, 674, 718, 734, 758, 838, 862, 878, 886, 982, 998, 1018, 1094, 1154, 1186, 1202, 1214, 1238, 1282, 1306, 1318, 1322

Offset: 1

Gary E. Davis, Sep 24 2016

nonn

Union of A077065 and A077068.

			a(3) = 10 = 2*5 is a product of 2 primes and 10+1 = 11 is prime.
a(4) = 14 = 2*7 is a product of 2 primes and 14-1 = 13 is prime.

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

Cf. A001358, A077065, A077068, A120628.

select(t -> isprime(t/2) and (isprime(t-1) or isprime(t+1)), [seq(i,i=2..10000,2)]); # Robert Israel, Sep 30 2016

func[n_] := PrimeOmega[n] == 2 && (PrimeQ[n + 1] || PrimeQ[n - 1])
Select[Range[1000], func[#] &]

isok(n) = (bigomega(n)==2) && (isprime(n-1) || isprime(n+1)); \\ Michel Marcus, Sep 24 2016

lista(nn) = forprime(p=2, nn, if(isprime(2*p+1) || isprime(2*p-1), print1(2*p, ", "))); \\ Altug Alkan, Sep 30 2016

from sympy import isprime, primerange
def aupto(N): return [t for t in (2*p for p in primerange(2, N//2+1)) if isprime(t-1) or isprime(t+1)]
print(aupto(1322)) # Michael S. Branicky, Aug 21 2022

a(n) = 2*A120628(n).

A233973 a(n) = A232221(n)/4.

Original entry on oeis.org

0, 0, 1, 5, 9, 13, 32, 54, 82, 116, 159, 199, 227, 273, 331, 380, 429, 487, 536, 609, 679, 743, 816, 895, 953, 1008, 1042, 1067, 1104, 1180, 1253, 1332, 1429, 1532, 1626, 1675, 1733, 1812, 1921, 2045, 2169, 2317, 2420, 2535, 2656, 2756, 2850, 2953

Offset: 1

Omar E. Pol, Dec 18 2013

nonn

Cf. A001358, A077065, A077068, A232221.

a(n) = (Sum_{i=1..n} (A077068(i)-A077065(i)))/4.

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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A077065 Semiprimes of form prime - 1.

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A121885 Excess of n-th prime over previous semiprime.

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A121884 Excess of n-th semiprime over previous prime.

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Mathematica

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A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

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PARI

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Extensions

A276983 Semiprimes n such that n-1 or n+1 is prime.

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Maple