A056892 -id:A056892 - cp's OEIS frontend

A176035 Difference between product of two distinct primes and previous perfect square.

2, 1, 5, 6, 5, 6, 1, 8, 9, 10, 2, 3, 10, 2, 6, 8, 9, 13, 1, 5, 10, 13, 1, 4, 5, 6, 10, 12, 13, 14, 6, 11, 15, 18, 19, 1, 2, 8, 12, 13, 20, 21, 22, 1, 2, 11, 14, 15, 17, 22, 8, 9, 14, 16, 18, 25, 5, 6, 7, 9, 10, 13, 17, 18, 19, 21, 22, 23, 25, 1, 10, 12, 22, 24, 28, 29, 3, 6, 9, 11, 18, 22, 31

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

6-4=2, 10-9=1, 14-9=5, 15-9=6, 21-16=5,..

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A006881, A056892, A106044, A176032, A176033, A176034

A006881:= [n: n in [1..1000] | EulerPhi(n) + DivisorSigma(1, n) eq 2*(n+1)];
[A006881[n] - Floor(Sqrt(A006881[n]))^2: n in [1..100]]; // G. C. Greubel, Oct 26 2022

A006881=Sort@Flatten@Table[Prime[m]*Prime[n], {n, 2, 150}, {m, n-1}];
Table[A006881[[n]]-Floor[Sqrt[A006881[[n]]]]^2, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 02 2011 *)

A006881=[n for n in (1..750) if euler_phi(n) + sigma(n,1) == 2*n+2]
[A006881[n] - isqrt(A006881[n])^2 for n in range(101)] # G. C. Greubel, Oct 26 2022

a(n) = A053186(A006881(n)). - R. J. Mathar, Aug 25 2025

A158038 Difference between n-th prime and next cube.

Original entry on oeis.org

6, 5, 3, 1, 16, 14, 10, 8, 4, 35, 33, 27, 23, 21, 17, 11, 5, 3, 58, 54, 52, 46, 42, 36, 28, 24, 22, 18, 16, 12, 89, 85, 79, 77, 67, 65, 59, 53, 49, 43, 37, 35, 25, 23, 19, 17, 5, 120, 116, 114, 110, 104, 102, 92, 86, 80, 74, 72, 66, 62, 60, 50, 36, 32, 30, 26, 12, 6, 165, 163

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

Could be read as a table, since there are always several primes between two cubes. Whenever a(n+1) > a(n), the n-th prime is the largest one below a given cube and prime(n+1) is the smallest prime larger than that cube. For n > 1, these are also the indices where the parity of the terms changes. - M. F. Hasler, Oct 19 2018

			The first terms are: 8 - 2 = 6, 8 - 3 = 5, 8 - 5 = 3, 8 - 7 = 1, 27 - 11 = 16, ...
From _M. F. Hasler_, Oct 19 2018: (Start)
Starting a new row when going to the next cube, the sequence reads:
  6, 5, 3, 1,                      // = 8 - {primes between 1^3 = 1 and 2^3 = 8}
  16, 14, 10, 8, 4,                // = 27 - {primes between 2^3 = 8 and 3^3 = 27}
  35, 33, 27, 23, 21, 17, 11, 5, 3, // = 64 - {primes between 27 and 4^3 = 64}
  58, 54, 52, ..., 18, 16, 12,     // = 125 - {primes between 64 and 5^3 = 125}
  89, 85, 79, ..., 19, 17, 5,      // = 216 - {primes between 125 and 6^3 = 216}
  120, 116, 114, ..., 26, 12, 6,   // = 343 - {primes between 216 and 7^3 = 343}
  etc. (End)

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A106044, A104492, A056892, A158037

lst={};Do[p=Prime[n];s=p^(1/3);f=Floor[s];a=(f+1)^3;d=a-p;AppendTo[lst,d],{n,6!}];lst
nc[n_]:=(Floor[Surd[n,3]]+1)^3-n; Table[nc[n],{n,Prime[Range[70]]}] (* Harvey P. Dale, Jun 19 2014 *)

A158038(n)=(sqrtnint(n=prime(n),3)+1)^3-n \\ M. F. Hasler, Oct 19 2018

first(n) = my(res = vector(n), t = 0, c = 2, c3 = 8); forprime(p = 2, oo, t++; if(p > c3, c++; c3 = c^3); res[t] = c3 - p; if(t==n, return(res))) \\ David A. Corneth, Oct 19 2018

a(n) > 0. - David A. Corneth, Oct 19 2018

Edited by M. F. Hasler, Oct 19 2018

A176034 Difference between product of two distinct primes and next perfect square.

Original entry on oeis.org

3, 6, 2, 1, 4, 3, 10, 3, 2, 1, 11, 10, 3, 13, 9, 7, 6, 2, 16, 12, 7, 4, 18, 15, 14, 13, 9, 7, 6, 5, 15, 10, 6, 3, 2, 22, 21, 15, 11, 10, 3, 2, 1, 24, 23, 14, 11, 10, 8, 3, 19, 18, 13, 11, 9, 2, 24, 23, 22, 20, 19, 16, 12, 11, 10, 8, 7, 6, 4, 30, 21, 19, 9, 7, 3, 2, 30, 27, 24, 22, 15, 11, 2

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

nonn

9-6=3, 16-10=6, 16-14=2, 16-15=1, 25-21=4,...

Cf. A006881, A056892, A106044, A176032, A176033

f1[n_]:=Floor[Sqrt[n]];f2[n_]:=Last/@FactorInteger[n]=={1,1};lst={};Do[If[f2[n],AppendTo[lst,(f1[n]+1)^2-n]],{n,0,6!}];lst

A067614 a(n) is the second partial quotient in the simple continued fraction for sqrt(prime(n)).

Original entry on oeis.org

2, 1, 4, 1, 3, 1, 8, 2, 1, 2, 1, 12, 2, 1, 1, 3, 1, 1, 5, 2, 1, 1, 9, 2, 1, 20, 6, 2, 2, 1, 3, 2, 1, 1, 4, 3, 1, 1, 1, 6, 2, 2, 1, 1, 28, 9, 1, 1, 15, 7, 3, 2, 1, 1, 32, 4, 2, 2, 1, 1, 1, 8, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 40, 4, 2, 1, 1, 1, 1, 21, 5, 2, 2, 1, 1, 1, 14, 6, 2, 2, 1, 1, 1

Offset: 1

Roger L. Bagula, Feb 01 2002

			For n=8, prime(n)=19, floor(sqrt(19))=4 and 1/(sqrt(19)-4) = 2.786..., so a(8)=2.

Cf. A000006, A000040, A056892.

a[n_] := Floor[1/(Sqrt[Prime[n]]-Floor[Sqrt[Prime[n]]])]

a(n) = my(r); sqrtint(prime(n),&r)<<1 \ r; \\ Kevin Ryde, May 06 2022

a(n) = floor(1/(sqrt(prime(n))-floor(sqrt(prime(n))))), where prime(n) is the n-th prime.

a(n) = floor(2*s/r) where s = floor(sqrt(p)) = A000006(n), r = p - s^2 = A056892(n), and p = prime(n). - Kevin Ryde, May 06 2022

Edited by Dean Hickerson, Feb 14 2002

A158037 A106044 sorted and duplicates removed.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

All natural numbers except squares, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49, 8^2=64, ... are missing from the current sequence.

Cf. A106044, A104492, A056892

lst={};Do[p=Prime[n];s=p^(1/2);f=Floor[s];a=(f+1)^2;d=a-p;AppendTo[lst,d],{n,7!}];Take[Union[lst],5! ]
(Floor[Sqrt[#]]+1)^2-#&/@Prime[Range[500]]//Union (* Harvey P. Dale, Aug 02 2021 *)

A104493 Numbers n for which the cube excess of the n-th prime is prime.

Original entry on oeis.org

2, 5, 6, 8, 10, 19, 20, 23, 26, 28, 31, 48, 49, 50, 51, 52, 55, 56, 57, 59, 61, 65, 66, 99, 100, 105, 110, 112, 114, 117, 121, 125, 127, 170, 171, 173, 178, 184, 185, 186, 190, 192, 194, 196, 200, 201, 206, 208, 214, 270, 271, 272, 274, 277, 278, 279, 280, 282

Offset: 1

Jonathan Vos Post, Mar 19 2005

			99 is an element of this sequence because the 99th prime is 523, 523 - 8^3 = 523-512 = 11 and 11 is prime. 100 is in this sequence because the 100th prime is 541 and 541-8^3 = 29, which is prime.

Cf. A000040, A053186, A055400, A056892, A102821, A104492.

f[n_] := Block[{k = 1, p = Prime[n]}, While[k^3 < p, k++ ]; p - (k - 1)^3]; Select[ Range[ 284], PrimeQ[ f[ # ]] &] (* Robert G. Wilson v, Mar 19 2005 *)

n such that A055400(A000040(n)) is an element of A000040. n such that A104492(n) is prime.

More terms from Robert G. Wilson v, Mar 19 2005

A134969 List of pairs of primes that are separated by the equivalent of 2 quadratic intervals. Both primes are greater than their preceding perfect squares by the same amount, or offset. The respective perfect squares can be both odd, in which case the offset is even, or both even, in which case the offset is odd.

Original entry on oeis.org

3, 11, 5, 17, 7, 19, 13, 29, 17, 37, 23, 43, 29, 53, 43, 71, 67, 103, 71, 107, 73, 109, 97, 137

Offset: 1

Michael M. Ross, Feb 04 2008

			Prime pair 71 and 107 have an odd offset of 7 from 64 and 100:
  Interval 1: 8*8 = 64 + 7 = 71.
  Interval 2: 9*9 = 81 + 7 = 88.
  Interval 3: 10*10 = 100 + 7 = 107.
Prime pair 97 and 137 have an even offset of 16 from 81 and 121:
  Interval 1: 9*9 = 81 + 16 = 97.
  Interval 2: 10*10 = 100 + 16 = 116.
  Interval 3: 11*11 = 121 + 16 = 137.
In all cases there is a complete intermediate quadratic interval (#2).
The pair (11,83), 11-9 = 83-81 = 2, does not work because there are 6 squares in between: 16,25,36,49,64,81.

Michael M. Ross, What Are Odd Squared, or Biquadratic, Paired Primes?
Michael M. Ross, Diagram of paired primes showing how two perfect pairs partially overlap: 23 and 43 span perfect squares 25 and 36; 29 and 53 span perfect squares 36 and 49.
Michael M. Ross, VBA source code providing an Excel visualization of the odd-offset prime pairs and colocation with twin primes.

Cf. A056892.

PS1 + OfN = P1, PS3 + OfN = P2, where
  PS1 = first perfect square,
  PS3 = 3rd perfect square,
  OfN = an equal positive offset from preceding perfect square, and
  P1 and P2 = the prime pair.

A176036 Absolute values of A176035(n)-A176034(n).

Original entry on oeis.org

1, 5, 3, 5, 1, 3, 9, 5, 7, 9, 9, 7, 7, 11, 3, 1, 3, 11, 15, 7, 3, 9, 17, 11, 9, 7, 1, 5, 7, 9, 9, 1, 9, 15, 17, 21, 19, 7, 1, 3, 17, 19, 21, 23, 21, 3, 3, 5, 9, 19, 11, 9, 1, 5, 9, 23, 19, 17, 15, 11, 9, 3, 5, 7, 9, 13, 15, 17, 21, 29, 11, 7, 13, 17, 25, 27, 27, 21, 15, 11, 3, 11, 29, 31, 23, 17

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

3-2=1, 6-1=5, 5-2=3, 6-1=5, 5-4=1,..

Cf. A006881, A056892, A106044, A176032, A176033, A176034, A176035.

f1[n_]:=Floor[Sqrt[n]];f2[n_]:=Last/@FactorInteger[n]=={1,1};lst={};Do[If[f2[n],AppendTo[lst,Abs[(n-f1[n]^2)-((f1[n]+1)^2-n)]]],{n,0,5!}];lst

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A176035 Difference between product of two distinct primes and previous perfect square.

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A158038 Difference between n-th prime and next cube.

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A176034 Difference between product of two distinct primes and next perfect square.

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A067614 a(n) is the second partial quotient in the simple continued fraction for sqrt(prime(n)).

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A158037 A106044 sorted and duplicates removed.

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A104493 Numbers n for which the cube excess of the n-th prime is prime.

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A176036 Absolute values of A176035(n)-A176034(n).

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