A090076 -id:A090076 - cp's OEIS frontend

A192133 Difference of base and exponent of prime powers (cf. A000961).

1, 1, 2, 0, 4, 6, -1, 1, 10, 12, -2, 16, 18, 22, 3, 0, 28, 30, -3, 36, 40, 42, 46, 5, 52, 58, 60, -4, 66, 70, 72, 78, -1, 82, 88, 96, 100, 102, 106, 108, 112, 9, 2, 126, -5, 130, 136, 138, 148, 150, 156, 162, 166, 11, 172, 178, 180, 190, 192, 196, 198, 210

Offset: 1

Reinhard Zumkeller, Jun 26 2011

sign

a(1) = 1 by convention, in accordance with A025473(1) = 1 and A025474(1) = 0.

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

A006093 and A090076 are subsequences.

Cf. A000961, A025473, A025474, A192134.

s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 1]] - f[[1, 2]], Nothing]]; s[1] = 1; Array[s, 250] (* Amiram Eldar, May 16 2025 *)

a(n) = A025473(n)-A025474(n) = A192134(n)*A025473(n)/A000961(n).

A216124 Primes which are the nearest integer to the geometric mean of the previous prime and the following prime.

Original entry on oeis.org

3, 5, 7, 23, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393

Offset: 1

César Eliud Lozada, Sep 01 2012

nonn

The geometric mean of two primes p and q is sqrt(pq).

			The prime before 3 is 2 and the prime after 3 is 5. 2 * 5 = 10 and the geometric mean of 2 and 5 is therefore sqrt(10) = 3.16227766..., which rounds to 3. Therefore 3 is in the sequence.
The geometric mean of 7 and 13 is 9.539392... which rounds up to 10, well short of 11, hence 11 is not in the sequence.

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

Cf. A216101, A090076.

A := {}: for n from 2 to 1000 do p1 := ithprime(n-1): p := ithprime(n); p2 := ithprime(n+1): if p = round(sqrt(p1*p2)) then A := `union`(A, {p}) end if end do; A := A;

Prime[Select[Range[2, 700], Prime[#] == Round[Sqrt[Prime[# - 1] Prime[# + 1]]] &]] (* Alonso del Arte, Sep 01 2012 *)
Select[Partition[Prime[Range[750]],3,1],Round[GeometricMean[{#[[1]],#[[3]]}]]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Feb 28 2024 *)

lista(nn) = forprime (p=2, nn, if (round(sqrt(precprime(p-1)*nextprime(p+1))) == p, print1(p, ", "))); \\ Michel Marcus, Apr 08 2015

from math import isqrt
from itertools import islice
from sympy import nextprime, prevprime
def A216124_gen(startvalue=3): # generator of terms >= startvalue
    q = max(3,nextprime(startvalue-1))
    p = prevprime(q)
    r = nextprime(q)
    while True:
        if q == (m:=isqrt(k:=p*r))+(k-m*(m+1)>=1):
            yield q
        p, q, r = q, r, nextprime(r)
A216124_list = list(islice(A216124_gen(),20)) # Chai Wah Wu, Jun 19 2024

More terms from Michel Marcus, Apr 08 2015

A276942 Square array A(row,col): A(row,1) = A276937(row), and for col > 1, A(row,col) = A003961(A(row,col-1)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

2, 3, 6, 5, 15, 9, 7, 35, 25, 10, 11, 77, 49, 21, 14, 13, 143, 121, 55, 33, 18, 17, 221, 169, 91, 65, 75, 22, 19, 323, 289, 187, 119, 245, 39, 26, 23, 437, 361, 247, 209, 847, 85, 51, 30, 29, 667, 529, 391, 299, 1859, 133, 95, 105, 34, 31, 899, 841, 551, 493, 3757, 253, 161, 385, 57, 38, 37, 1147, 961, 713, 589, 6137, 377, 319, 1001, 115, 69, 42

Offset: 2

Antti Karttunen, Sep 25 2016

The starting offset is 2 because 1 is not included in the array proper. With it the terms are a permutation of A276078.

All terms on each row have the same prime signature.

			The top left corner of the array:
   2,  3,  5,   7,  11,  13,  17,  19,  23,   29,   31,   37,   41,   43
   6, 15, 35,  77, 143, 221, 323, 437, 667,  899, 1147, 1517, 1763, 2021
   9, 25, 49, 121, 169, 289, 361, 529, 841,  961, 1369, 1681, 1849, 2209
  10, 21, 55,  91, 187, 247, 391, 551, 713, 1073, 1271, 1591, 1927, 2279
  14, 33, 65, 119, 209, 299, 493, 589, 851, 1189, 1333, 1739, 2173, 2537

Antti Karttunen, Table of n, a(n) for n = 2..631; the first 35 antidiagonals of array

Transpose: A276941.
Leftmost column: A276937, second column: A276938.
Rows from the top: A000040, A006094, A001248 (from 9 onward), A090076, A090090.
Cf. A003961.
Cf. A276078 (sorted into ascending order).
Cf. also A276075, A276955.

(define (A276942 n) (A276941bi (A004736 (- n 1)) (A002260 (- n 1)))) ;; Code for A276941bi given in A276941.

A(row,1) = A276937(row); for col > 1, A(row,col) = A003961(A(row,col-1)).

A302047 a(n) = 1 if n = prime(k)*prime(2+k) for some k, otherwise 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Apr 24 2018

nonn

Characteristic function for prime(n)*prime(n+2) (A090076).

Cf. A090076, A246277, A280710, A302048, A302049.

Array[Boole@ And[Length@ # == 2, Max@ #[[All, -1]] == 1, Subtract @@ PrimePi[#[[All, 1]]] == -2 ] &@ FactorInteger@ # &, 120] (* or *)
With[{s = Array[Prime[#] Prime[# + 2] &, 5]}, ReplacePart[ConstantArray[0, Max@ s], Map[# -> 1 &, s] ] ] (* Michael De Vlieger, Apr 27 2018 *)
Module[{pp2=Table[Prime[n]Prime[n+2],{n,5}],nn},nn=Max[pp2];Table[ If[ MemberQ[ pp2,k],1,0],{k,nn}]] (* Harvey P. Dale, Dec 13 2021 *)

A302047(n) = if((2!=bigomega(n))||(2!=omega(n)),0,my(f=factor(n)); (f[2,1] == nextprime(1+nextprime(1+f[1,1]))));

a(n) = A185015(A246277(n)).

A226502 Let P(k) denote the k-th prime (P(1)=2, P(2)=3 ...); a(n) = P(n+1)P(n+3) - P(n)P(n+2).

Original entry on oeis.org

11, 34, 36, 96, 60, 144, 160, 162, 360, 198, 320, 336, 352, 494, 460, 720, 378, 560, 718, 450, 972, 1020, 938, 1002, 816, 420, 864, 1752, 960, 2596, 810, 2204, 576, 2404, 1220, 1606, 1980, 1694, 1420, 2876, 744, 2694, 780, 3160, 2810, 3520, 3170, 1824, 1840, 1422, 3836

Offset: 1

Ed Smiley, Jun 09 2013

Differences of the products of alternate primes.

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

First differences of A090076.

#[[2]]#[[4]]-#[[1]]#[[3]]&/@Partition[Prime[Range[60]],4,1] (* Harvey P. Dale, Jul 14 2025 *)

p=2;q=3;r=5;forprime(s=7,1e2,print1(q*s-p*r", ");p=q;q=r;r=s) \\ Charles R Greathouse IV, Jun 10 2013

a(n) >> n log n and this is probably sharp: on Dickson's conjecture there are infinitely many a(n) < kn log n for any k > 4. The constant 4 comes from 8 + 2 - 6 - 0 n the prime quadruplet (p+0, p+2, p+6, p+8). On Cramér's conjecture a(n) = O(n log^3 n). Unconditionally a(n) << n^1.525 log n. - Charles R Greathouse IV, Jun 10 2013

A333747 Numbers that are either the product of two consecutive primes or two primes with a prime in between.

Original entry on oeis.org

6, 10, 15, 21, 35, 55, 77, 91, 143, 187, 221, 247, 323, 391, 437, 551, 667, 713, 899, 1073, 1147, 1271, 1517, 1591, 1763, 1927, 2021, 2279, 2491, 2773, 3127, 3233, 3599, 3953, 4087, 4331, 4757, 4891, 5183, 5609, 5767, 6059, 6557, 7031, 7387, 8051, 8633, 8989, 9797, 9991

Offset: 1

Bobby Jacobs, Apr 03 2020

In other words, these are numbers that are the product of two distinct primes whose prime indices differ by at most two.

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

Subsequence of A001358.

Cf. A000040, A006094, A090076, A110654, A332765, A332877.

R:= NULL;
p:= 2; q:= 3;
for n from 1 to 100 by 2 do
  r:= nextprime(q);
  R:= R, p*q, p*r;
  p:= q; q:= r;
od:
R; # Robert Israel, Apr 22 2020

a[n_] := Prime[Ceiling[n/2]] * Prime[Ceiling[(n + 3)/2]]; Array[a, 50] (* Amiram Eldar, Apr 04 2020 *)

Union of A006094 and A090076.
a(n) = prime(ceiling(n/2))*prime(ceiling((n+3)/2)).
a(2*n-1) = prime(n)*prime(n+1).
a(2*n) = prime(n)*prime(n+2).

A096968 a(n) = (prime(n)*prime(n+2))^4.

Original entry on oeis.org

10000, 194481, 9150625, 68574961, 1222830961, 3722098081, 23372600161, 92173567201, 258439040161, 1325558466241, 2609649624481, 6407383500961, 13788812262241, 26975984333281, 59128856241841, 109250345339521

Offset: 1

Pierre CAMI, Aug 19 2004

			a(1) = (prime(1)*prime(3))^4 = 10000.

Equals A090076(n)^4.

a[n_] := (Prime[n]*Prime[n + 2])^4; Table[ a[n], {n, 15}] (* Robert G. Wilson v, Aug 22 2004 *)

More terms from Robert G. Wilson v, Aug 22 2004

A141078 a(n) = abs(prime(n)prime(n+4) - prime(n+1)prime(n+3)).

Original entry on oeis.org

1, 16, 6, 54, 6, 14, 24, 10, 130, 24, 134, 34, 140, 150, 84, 190, 24, 72, 48, 260, 50, 72, 440, 468, 234, 186, 990, 174, 130, 1692, 714, 632, 520, 736, 270, 96, 96, 390, 584, 800, 330, 1608, 1894, 1472, 342, 2326, 1904, 406, 24, 1326, 108, 920, 1184, 1112, 84, 610

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2008

			a(4) = abs(prime(4)*prime(4+4) - prime(4+1)*prime(4+3)) = abs(7*19 - 11*17) = abs(133 - 187) = abs(-54) = 54.

Aaron Toponce, Table of n, a(n) for n = 1..100000

Cf. A090076.

P:=Filtered([1..300],IsPrime);;
List([1..56], n->AbsInt(P[n]*P[n+4]-P[n+1]*P[n+3])); # Muniru A Asiru, Feb 09 2018

p:=ithprime: seq(abs(p(n)*p(n+4)-p(n+1)*p(n+3)),n=1..60); # Emeric Deutsch, Aug 14 2008

a[n_]:=Abs[Prime[n]*Prime[n+4]-Prime[n+1]*Prime[n+3]];Array[a,56] (* James C. McMahon, Jul 23 2025 *)

Corrected and extended by Emeric Deutsch, Aug 14 2008

A176098 a(n) = prime(n) times the n-th nonnegative noncomposite.

Original entry on oeis.org

0, 3, 10, 21, 55, 91, 187, 247, 391, 551, 713, 1073, 1271, 1591, 1927, 2279, 2773, 3233, 3953, 4331, 4891, 5609, 6059, 7031, 8051, 8989, 9991, 10807, 11227, 12091, 13843, 14803, 17399, 18209, 20413, 20989, 23393, 24613, 26219, 28199, 29893, 31313

Offset: 1

Juri-Stepan Gerasimov, Apr 08 2010

			a(5) = prime(5)*(nonnegative noncomposite(5)) = 11*5 = 55.

Cf. A000040, A158611.

Essentially the same as A090076.

a(n) = A000040(n)*A158611(n).

Entries checked by R. J. Mathar, Apr 16 2010

A348569 a(n) = (prime(n) + prime(n+2))^2 + prime(n+1)^2.

Original entry on oeis.org

58, 125, 305, 521, 953, 1313, 1961, 2833, 3757, 5317, 6553, 8081, 9593, 11425, 14045, 16477, 19597, 21913, 24641, 27829, 30577, 35113, 40321, 45509, 50201, 53873, 56393, 60281, 68465, 75665, 86857, 91669, 101117, 106301

Offset: 1

Burak Muslu, Oct 23 2021

The square of the shortest distance between the start and end points of the path followed by a person moving 90 degrees left and right (or left and right) at the end of each path on a path of three consecutive prime numbers.

			For n = 1 the a(1) = (prime(1) + prime(3))^2 + prime(2)^2 = (2 + 5)^2 + 3^2 = 49 + 9 = 58.

Cf. A000040, A090076, A133529.

Table[(Prime[n] + Prime[n + 2])^2 + Prime[n + 1]^2,{n,34}]

a(n) = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + 2*prime(n)*prime(n+2).

cp's OEIS Frontend

A192133 Difference of base and exponent of prime powers (cf. A000961).

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A216124 Primes which are the nearest integer to the geometric mean of the previous prime and the following prime.

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A276942 Square array A(row,col): A(row,1) = A276937(row), and for col > 1, A(row,col) = A003961(A(row,col-1)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A302047 a(n) = 1 if n = prime(k)*prime(2+k) for some k, otherwise 0.

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A226502 Let P(k) denote the k-th prime (P(1)=2, P(2)=3 ...); a(n) = P(n+1)P(n+3) - P(n)P(n+2).

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A333747 Numbers that are either the product of two consecutive primes or two primes with a prime in between.

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Maple

Mathematica

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A096968 a(n) = (prime(n)*prime(n+2))^4.

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A141078 a(n) = abs(prime(n)*prime(n+4) - prime(n+1)*prime(n+3)).

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A141078 a(n) = abs(prime(n)prime(n+4) - prime(n+1)prime(n+3)).