A106044 -id:A106044 - cp's OEIS frontend

A176032 Absolute values of A106044-A056892.

1, 1, 3, 1, 3, 1, 7, 3, 5, 3, 1, 11, 3, 1, 9, 7, 5, 9, 11, 3, 1, 13, 15, 3, 13, 19, 15, 7, 3, 5, 11, 3, 9, 13, 15, 11, 1, 13, 21, 19, 7, 3, 17, 21, 27, 23, 1, 25, 27, 23, 15, 3, 1, 21, 31, 19, 7, 3, 9, 17, 21, 27, 1, 9, 13, 21, 23, 11, 9, 13, 21, 33, 27, 15, 3, 5, 17, 33, 39, 23, 3, 1, 21, 25

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

nonn

A106044 2,1,4,2,5,3,8,6,2,7,5,12,8,6,2,11,.. A056892 1,2,1,3,2,4,1,3,7,4,6,1,5,7,11,4,10,.. 2-1=1,2-1=1,4-1=3,3-2=1,5-2=3,...

Cf. A056892, A106044

f[n_]:=Floor[Sqrt[n]];lst={};Do[p=Prime[n];AppendTo[lst,Abs[((f[p]+1)^2-p)-(p-f[p]^2)]],{n,3*5!}];lst

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 *)

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

Original entry on oeis.org

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

A320688 Sum of the square excess A056892 of the primes between two squares.

Original entry on oeis.org

3, 4, 6, 11, 10, 24, 26, 34, 26, 33, 50, 67, 72, 46, 70, 109, 96, 132, 122, 153, 132, 145, 174, 229, 208, 175, 194, 287, 232, 244, 338, 267, 276, 345, 374, 239, 392, 396, 424, 390, 484, 373, 514, 563, 618, 424, 654, 821, 442, 557, 890, 814, 668, 741, 580, 642, 990, 811, 982, 968, 772

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Consider the primes p1,...,pK between two squares n^2 and (n+1)^2, and take the sum of the differences: (p1 - n^2) + ... + (pK - n^2). Obviously this equals (sum of these primes) - (number of these primes) * n^2.

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

Cf. A014085, A000290, A108314, A106044.

Row sums of A056892, read as a table.

R:= NULL: p:= 2: n:= 1: t:= 0:
while n <= 100 do
    t:= t + p-n^2;
    p:= nextprime(p);
    if p > (n+1)^2 then
     R:= R, t; t:= 0; n:= n+1;
    fi:
od:
R; # Robert Israel, Dec 17 2024

a(n,s=0)={forprime(p=n^2,(n+1)^2,s+=p-n^2);s}

a(n) = A108314(n) - A014085(n)*A000290(n), where A000290(n) = n^2.

A229067 Sum of n-th prime and next perfect square.

Original entry on oeis.org

6, 7, 14, 16, 27, 29, 42, 44, 48, 65, 67, 86, 90, 92, 96, 117, 123, 125, 148, 152, 154, 160, 183, 189, 197, 222, 224, 228, 230, 234, 271, 275, 281, 283, 318, 320, 326, 332, 336, 369, 375, 377, 387, 389, 422, 424, 436, 448, 483, 485, 489, 495, 497, 507, 546, 552

Offset: 1

Vincenzo Librandi, Sep 25 2013

Primes in the sequence: 7, 29, 67, 197, 271, 281, 283, 389, 617, 631, 641, ...

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

Cf. A000040, A145445, A106044.

(Floor[Sqrt[#]] + 1)^2 + # &/@Prime[Range[80]]

a(n) = A000040(n) + A145445(n).

A229497 Product between n-th prime and next perfect square.

Original entry on oeis.org

8, 12, 45, 63, 176, 208, 425, 475, 575, 1044, 1116, 1813, 2009, 2107, 2303, 3392, 3776, 3904, 5427, 5751, 5913, 6399, 8300, 8900, 9700, 12221, 12463, 12947, 13189, 13673, 18288, 18864, 19728, 20016, 25181, 25519, 26533, 27547, 28223, 33908, 35084, 35476, 37436

Offset: 1

Vincenzo Librandi, Sep 25 2013

			63 is in the sequence because 7*9=63.

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

Cf. A000040, A145445, A106044.

(Floor[Sqrt[#]] + 1)^2 # &/@Prime[Range[80]]

a(n) = A000040(n) * A145445(n).

A320687 Sum of differences of the larger square and primes between two squares.

Original entry on oeis.org

3, 6, 8, 16, 12, 28, 19, 34, 31, 72, 42, 58, 63, 70, 116, 122, 79, 90, 112, 134, 169, 170, 108, 212, 200, 196, 246, 226, 240, 244, 292, 318, 394, 276, 336, 418, 283, 528, 445, 582, 429, 392, 530, 416, 565, 506, 581, 634, 548, 554, 655, 866, 616, 676, 641, 714, 965, 710, 922, 968, 827

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Consider the primes p1,...,pK between two squares n^2 and (n+1)^2, and take the sum of the differences (listed as A106044): ((n+1)^2 - p1) + ... + ((n+1)^2 - pK).

			a(1) = 3 = 2 + 1, where {2, 1} = 4 - {2, 3: primes between 1^2 = 1 and 2^2 = 4}.
a(2) = 6 = 4 + 2, with {4, 2} = 9 - {5, 7: primes between 2^2 = 4 and 3^2 = 9}.
a(3) = 8 = sum of {5, 3} = 16 - {11, 13: primes between 3^2 = 9 and 4^2 = 16}.
a(4) = 16 = sum of {8, 6, 2} = 25 - {17, 19, 23: primes between 4^2 and 5^2 = 25}.
a(5) = 12 = sum of {7, 5} = 36 - {29, 31: primes between 5^2 = 25 and 6^2 = 36}.

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

Equals A014085 * A000290(.+1) - A108314.

Row sums of A106044 read as a table.

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
p:= 1;
do
   p:= nextprime(p);
   n:= floor(sqrt(p));
   if n > N then break fi;
   V[n]:= V[n]+(n+1)^2-p;
od:
convert(V,list); # Robert Israel, Jun 17 2019

a(n,s=0)={forprime(p=n^2,(n+=1)^2,s+=n^2-p);s}

a(n) = A014085(n)*A000290(n+1) - A108314(n), where A000290(n) = n^2.

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

nonn

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

cp's OEIS Frontend

A176032 Absolute values of A106044-A056892.

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A158037 A106044 sorted and duplicates removed.

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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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PARI

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

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A320688 Sum of the square excess A056892 of the primes between two squares.

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Maple

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A229067 Sum of n-th prime and next perfect square.

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Mathematica

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A229497 Product between n-th prime and next perfect square.

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