Walter Carlini - cp's OEIS frontend

A346150 Alternating runs of primes and composites, with the runs of primes being of composite length and the runs of composites being of prime length.

2, 4, 6, 3, 5, 7, 11, 8, 9, 10, 13, 17, 19, 23, 29, 31, 12, 14, 15, 16, 18, 37, 41, 43, 47, 53, 59, 61, 67, 20, 21, 22, 24, 25, 26, 27, 71, 73, 79, 83, 89, 97, 101, 103, 107, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42

Offset: 1

Walter Carlini, Jul 07 2021

In other words, use sequence A073846 to list alternating runs of primes and composites, with the number of elements in each run given by successive terms in A073846 - with each even-indexed term of A073846 (being itself prime) denoting the length of each run of composites and each odd-indexed term of A073846 (being itself composite) denoting the length of each run of primes.

			a(1) = 2, this being a length 1 (1 is initial index) run of primes.
a(2) = 4 & a(3) = 6, 4 and 6 being a length 2 (2 is first prime) run of composites.
a(4) = 3, a(5) = 5, a(6) = 7, and a(7) = 11 being a length 4 (4 is first composite) run of primes.
a(8) = 8, a(9) = 9, and a(10) = 10, being a length 3 (3 is 2nd prime) run of composites.

Cf. A000040 (primes), A002808 (composites), A073846.

m=10;c1=Select[Range@m,!PrimeQ@#&];p1=Prime@Range@Total@c1;p2=Prime@Range@m;c2=Select[Range[2,2Total@p2],!PrimeQ@#&][[;;Total@p2]];t1=TakeList[p1,c1];t2=TakeList[c2,p2];min=Min[Length/@{t1,t2}];Flatten@Riffle[t1[[;;min]],t2[[;;min]]] (* Giorgos Kalogeropoulos, Jul 30 2021 *)

A277186 Sum of primes within 2n-wide closed interval centered upon prime(n).

Original entry on oeis.org

5, 10, 17, 26, 31, 67, 83, 83, 119, 139, 161, 228, 281, 281, 341, 408, 474, 553, 546, 635, 635, 780, 824, 1092, 954, 1008, 1008, 1139, 1197, 1336, 1621, 1687, 1650, 1823, 1854, 1854, 2238, 2634, 2507, 2587, 2450, 2673, 3223, 3223, 3403, 3403, 3591, 4054, 4054, 4331, 4535, 4535, 4828, 4444, 4666

Offset: 1

Walter Carlini, Oct 04 2016

a(n) is the sum of primes within the closed interval [prime(n)-n, prime(n)+n], where prime(n) is the n-th prime.

			a(3) = 2 + 3 + 5 + 7 = 17; starting at prime(3) = 5, subtract 3 and add 3 to obtain the interval 2 through 8, and then add up the primes within that interval, inclusive of the endpoints of the interval.

Cf. A000040, A014688, A014689, A061397.

Table[Total@ Select[Range[Prime@ n - n, Prime@ n + n], PrimeQ], {n, 55}] (* Michael De Vlieger, Oct 04 2016 *)

a(n) = sum(k=prime(n)-n, prime(n)+n, isprime(k)*k); \\ Michel Marcus, Nov 01 2016

More terms from Michael De Vlieger, Oct 04 2016

A277005 Least prime greater than n-th compositorial.

Original entry on oeis.org

2, 5, 29, 193, 1733, 17291, 207367, 2903041, 43545611, 696729629, 12541132817, 250822656001, 5267275776047, 115880067072017, 2781121609728037, 69528040243200079, 1807729046323200001, 48808684250726400031, 1366643159020339200397

Offset: 0

Walter Carlini, Sep 25 2016

			a(0) = A151800(A036691(0)) = A151800(1) = 2; where the zeroth compositorial, A036691(0), is the empty product = 1.
a(3) = 193, which is the least prime number greater than the third compositorial number, 192 = 4 * 6 * 8.

Cf. A036691, A038710, A151800.

findComp[n_] := FixedPoint[n + PrimePi@ # + 1 &, n + PrimePi@ n + 1] ; Table[NextPrime@ Product[findComp@ k, {k, n}], {n, 0, 18}] (* Michael De Vlieger, Sep 25 2016, after Robert G. Wilson v at A036691 *)

a(n) = A151800(A036691(n)). - Michel Marcus, Sep 25 2016

a(18) corrected by Sean A. Irvine, Sep 26 2023

A255383 Compositorial mod sum-of-composites.

Original entry on oeis.org

0, 4, 12, 0, 1, 41, 0, 72, 2, 0, 48, 126, 0, 20, 0, 0, 90, 95, 115, 4, 0, 140, 161, 90, 261, 138, 208, 512, 72, 420, 51, 0, 0, 924, 899, 29, 893, 72, 840, 727, 129, 1185, 194, 732, 1080, 1612, 566, 175, 1352, 1192, 1204, 1360, 428, 957, 2170, 0, 0, 513, 2240

Offset: 1

Walter Carlini, May 14 2015

			For n = 5, a(5) = (4*6*8*9*10) mod (4+6+8+9+10) = 17280 mod 37 = 1.

Cf. A002808, A036691, A053767, A255217.

comp=Select[Range[2,83],!PrimeQ[#]&];Mod[Rest[FoldList[Times,1,comp]],Accumulate[comp]] (* Ivan N. Ianakiev, May 22 2015 *)

a(n) = A036691(n) mod A053767(n).

More terms from Alois P. Heinz, May 21 2015

A255217 Primorial mod sum-of-primes.

Original entry on oeis.org

0, 1, 0, 6, 14, 18, 52, 0, 70, 90, 50, 98, 0, 148, 82, 150, 110, 453, 450, 213, 262, 637, 0, 69, 530, 129, 1106, 339, 1110, 1416, 1290, 1443, 994, 30, 2274, 933, 646, 0, 0, 168, 0, 3234, 0, 786, 2014, 3270, 1680, 0, 1222, 0, 1070, 690, 0, 2934, 416, 0, 0, 0, 708

Offset: 1

Walter Carlini, Apr 25 2015

Does 0 appear infinitely often in this sequence? See A051838.

			For n = 4, a(4) = (2*3*5*7) mod (2+3+5+7) = 210 mod 17 = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002110 (Primorial numbers), A007504 (Sum of first n primes)

Table[Mod[Product[Prime[i],{i,n}],Sum[Prime[i],{i,n}]],{n,60}] (* Ivan N. Ianakiev, Apr 25 2015 *)
With[{pr=Prime[Range[60]]},Mod[#[[1]],#[[2]]]&/@Thread[{FoldList[ Times, pr], Accumulate[pr]}]] (* Harvey P. Dale, Jan 22 2016 *)

a(n) = my(vp=primes(n)); vecprod(vp) % vecsum(vp); \\ Michel Marcus, Dec 05 2021

lista(nn) = {my(s=0, p=1); forprime(q=2, nn, s += q; p *= q; print1(p%s, ", "););} \\ Michel Marcus, Dec 05 2021

a(n) = prime(n)# mod A007504(n).

More terms from Michel Marcus, Apr 25 2015

A153284 a(n) = n + Sum_{j=1..n-1} (-1)^j * a(j) for n >= 2, a(1) = 1.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1

Offset: 1

Walter Carlini, Dec 23 2008

Row sums of triangle A153860. - Gary W. Adamson, Jan 03 2009

1 followed by interleaving of A000012 and A010701. - Klaus Brockhaus, Jan 04 2009

			a(1)=1, a(2)=2-a(1)=2-1=1, a(3)=3+a(2)-a(1)=3+1-1=3, a(4)=4-a(3)+a(2)-a(1)=4-3+1-1=1, a(5)=5+1-3+1-1=3, a(6)=6-3+1-3+1-1=1, a(7)=7+1-3+1-3+1-1, etc.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Equals A010684 with the addition of the leading term of 1
The first sequence of a family that includes A153285 and A153286
Cf. A153860.
Cf. A000012 (all 1's sequence), A010701 (all 3's sequence). - Klaus Brockhaus, Jan 04 2009

S:=[ 1 ]; for n in [2..105] do Append(~S, n + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

a(n)=1 if n is 1 or even; a(n)=3 if n is odd other than 1.

G.f.: x*(1 + x + 2*x^2)/((1+x)*(1-x)). - Klaus Brockhaus, Jan 04 2009 and Oct 15 2009

A153285 a(1)=1; for n > 1, a(n) = n^2 + Sum_{j=1..n-1} (-1)^j*a(j).

Original entry on oeis.org

1, 3, 11, 7, 23, 11, 35, 15, 47, 19, 59, 23, 71, 27, 83, 31, 95, 35, 107, 39, 119, 43, 131, 47, 143, 51, 155, 55, 167, 59, 179, 63, 191, 67, 203, 71, 215, 75, 227, 79, 239, 83, 251, 87, 263, 91, 275, 95, 287, 99, 299, 103, 311, 107, 323, 111, 335, 115, 347, 119, 359

Offset: 1

Walter Carlini, Dec 23 2008

1 followed by interleaving of A004767 and A017653. - Klaus Brockhaus, Jan 04 2009

			a(1) = 1;
a(2) = 2^2 - a(1) = 4 - 1 = 3;
a(3) = 3^2 + a(2) - a(1) = 9 + 3 - 1 = 11;
a(4) = 4^2 - 11 + 3 - 1 = 7;
a(5) = 25 + 7 - 11 + 3 - 1 = 23;
a(6) = 36 - 23 + 7 - 11 + 3 - 1 = 11; etc.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

The second of a family of sequences that includes A153284 and A153286

Cf. A004767 (4n+3), A017653 (12n+11). - Klaus Brockhaus, Jan 04 2009

S:=[ 1 ]; for n in [2..61] do Append(~S, n^2 + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

(define (A153285 n) (cond ((= 1 n) n) ((even? n) (+ n n -1)) (else (+ (* 6 n) -7)))) ;; Antti Karttunen, Aug 10 2017

a(n) = 2n-1 if n is 1 or an even number;
a(n) = 6n-7 if n is an odd number other than 1.
G.f.: x*(1 + 3*x + 9*x^2 + x^3 + 2*x^4)/((1+x)^2*(1-x)^2). - Klaus Brockhaus, Oct 15 2009
a(n) = 4*(n-1) - (2*n-3)*(-1)^n for n>1, a(1)=1. - Bruno Berselli, Sep 14 2011

Extended beyond a(30) by Klaus Brockhaus, Jan 04 2009

A153286 a(n) = n^3 + sum((-1)^j*a(j)); for j=1 to n-1; a(1)=1.

Original entry on oeis.org

1, 7, 33, 37, 135, 91, 309, 169, 555, 271, 873, 397, 1263, 547, 1725, 721, 2259, 919, 2865, 1141, 3543, 1387, 4293, 1657, 5115, 1951, 6009, 2269, 6975, 2611, 8013, 2977, 9123, 3367, 10305, 3781, 11559, 4219, 12885, 4681, 14283, 5167, 15753, 5677, 17295

Offset: 1

Walter Carlini, Dec 23 2008, Jan 01 2009

1 followed by interleaving of A154105 and 3*A154106. - Klaus Brockhaus, Jan 04 2009

			a(1)=1, a(2)=2^3-a(1)=8-1=7, a(3)=3^3+a(2)-a(1)=27+7-1=33, a(4)=64-33+7-1=37, a(5)=125+37-33+7-1=135, a(6)=216-135+37-33+7-1=91, etc.

The third of a family of sequences that includes A153284 and A153285.

Cf. A154105 (12*n^2 + 18*n + 7), A154106 (12*n^2 + 22*n + 11). - Klaus Brockhaus, Jan 04 2009

S:=[ 1 ]; for n in [2..45] do Append(~S, n^3 + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

G.f.: x*(1 + 7*x + 30*x^2 + 16*x^3 + 39*x^4 + x^5 + 2*x^6)/((1+x)^3*(1-x)^3). - Klaus Brockhaus, Jan 04 2009
From Walter Carlini, Jan 12 2009: (Start)
a(n) = 3n^2 - 3n + 1 if n is 1 or an even number;
a(n) = 9n^2 - 21n + 15 if n is any odd number other than 1. (End)

Extended beyond a(30) by Klaus Brockhaus, Jan 04 2009

G.f. corrected by Klaus Brockhaus, Oct 15 2009

A134694 a(0) = 2; a(n) = least prime p such that p >= a(n-1) + 2^n.

Original entry on oeis.org

2, 5, 11, 19, 37, 71, 137, 269, 541, 1061, 2087, 4139, 8237, 16433, 32831, 65599, 131143, 262217, 524369, 1048661, 2097257, 4194409, 8388733, 16777381, 33554639, 67109071, 134217943, 268435697, 536871157, 1073742073, 2147483929

Offset: 0

Walter Carlini, Jan 27 2008

Primes separated by at least successive powers of 2.

			a(0) = 2 (by definition).
a(1) = 5 because 5 is the least prime >= 4 = 2 + 2^1.
a(2) = 11 because 11 is the least prime >= 9 = 5 + 2^2.
a(3) = 19 because 19 is the least prime >= 19 = 11 + 2^3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000040.

a = {2}; Do[i = a[[ -1]]+2^n; While[ !PrimeQ[i], i++ ]; AppendTo[a, i], {n,1,50}]; a (* Stefan Steinerberger, Jan 28 2008 *)
nxt[{n_,a_}]:={n+1,NextPrime[a+2^(n+1)-1]}; NestList[nxt,{0,2},30][[All,2]] (* Harvey P. Dale, Jan 04 2017 *)

More terms from Stefan Steinerberger, Jan 28 2008

A119411 Product of the first prime(n) primes.

Original entry on oeis.org

6, 30, 2310, 510510, 200560490130, 304250263527210, 1922760350154212639070, 7858321551080267055879090, 267064515689275851355624017992790, 279734996817854936178276161872067809674997230

Offset: 1

Walter Carlini, Jul 26 2006

			a(1) = p(p(1))# = p(2)# (because p(1) = 2 is the first prime number) = 2* 3 = 6 (by the definition of primorial, see A002110); that is, the product of the first 2 prime numbers.
a(2) = p(p(2))# = p(3)# = 2 * 3 * 5 = 30 = the product of the first 3 primes.
a(3) = 2 * 3 * 5 * 7 * 11 = 2310 = the product of the first 5 primes.
a(4) = 2 * 3 * 5 * 7 * 11 * 13 * 17 = 510510 = product of first 7 primes.

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.

Cf. A002110.

Array[Times @@ Array[Prime, Prime@# ] &, 10] (* Robert G. Wilson v, Jul 27 2006 *)

a(n) = p(p[n])#, where p[n] is the n-th prime number and where p(m)# is the m-th primorial number (Cf. A002110).

a(10) from Robert G. Wilson v, Jul 27 2006

cp's OEIS Frontend

User: Walter Carlini

A346150 Alternating runs of primes and composites, with the runs of primes being of composite length and the runs of composites being of prime length.

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A277186 Sum of primes within 2n-wide closed interval centered upon prime(n).

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A277005 Least prime greater than n-th compositorial.

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A255383 Compositorial mod sum-of-composites.

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A255217 Primorial mod sum-of-primes.

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A153284 a(n) = n + Sum_{j=1..n-1} (-1)^j * a(j) for n >= 2, a(1) = 1.

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A153285 a(1)=1; for n > 1, a(n) = n^2 + Sum_{j=1..n-1} (-1)^j*a(j).

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