A077037 -id:A077037 - cp's OEIS frontend

A077038 Least difference of primes p, q such that p < n^3 < q.

4, 6, 6, 14, 12, 10, 12, 6, 12, 34, 10, 24, 8, 16, 6, 10, 12, 6, 16, 20, 12, 34, 22, 10, 6, 6, 18, 12, 18, 14, 22, 18, 12, 36, 14, 20, 8, 52, 10, 10, 16, 38, 34, 6, 40, 24, 10, 16, 12, 14, 8, 18, 20, 30, 20, 32, 18, 34, 40, 48, 10, 6, 8, 18, 10, 18, 18, 30, 30, 30, 42, 20, 6, 44

Offset: 2

Reinhard Zumkeller, Oct 21 2002

nonn

Least m such that a(m)=2*n for n=1,2,3,... are: {2,3,14,7,6,5,15,28,21,24,13,...}. - Zak Seidov, May 10 2016

There are numbers k other than 2 such that a(k) = 4. The first few (up to 1000) are 129 189 369 435 549 555 561 819. Conjecture: every even integer greater than 2 occurs infinitely often in this sequence. - Franklin T. Adams-Watters, May 13 2016

Zak Seidov, Table of n, a(n) for n = 2..10000

Cf. A058043, A000578.

Table[c=n^3;NextPrime[c]-NextPrime[c,-1],{n,2,80}] (* Harvey P. Dale, Sep 14 2012 *)

a(n) = A014220(n) - A077037(n).

A242382 Lesser of consecutive primes whose average is a perfect cube.

Original entry on oeis.org

61, 1723, 4093, 17573, 21943, 46649, 110587, 195103, 287491, 314423, 405221, 474547, 1061189, 1191013, 1404919, 1601609, 1906621, 2000371, 2146687, 2196979, 3241783, 3511799, 4912991, 5268017, 6229501, 6751267, 6858997, 7077883, 11239421, 20346407, 21951997, 26198063

Offset: 1

Antonio Roldán, May 12 2014

nonn

			1723 is in the sequence because it is prime, nextprime(1723) = 1733, and average(1723,1733) = 1728 = 12^3.

Amiram Eldar, Table of n, a(n) for n = 1..128

Subsequence of A077037 and A242380.

Cf. A225195, A091624.

Select[Partition[Prime[Range[2, 10^5]], 2, 1], IntegerQ[Surd[(First[#] + Last[#])/2, 3]] &][[;; , 1]] (* Amiram Eldar, Jul 04 2022 *)

{for(i=3,3*10^7,if(isprime(i),k=(i+nextprime(i+1))/2;if(ispower(k,3),print1(i,", "))))}

A243893 a(n) = prime(k-1) with k = n^2 + prime(n)^2.

Original entry on oeis.org

7, 37, 137, 311, 829, 1249, 2269, 2939, 4483, 7411, 8681, 12653, 15877, 17827, 21673, 28087, 35393, 38317, 46957, 53327, 56897, 67493, 75269, 87523, 105143, 115057, 120427, 130811, 136547, 147863, 189067, 202481, 222991, 230393, 267401, 275677

Offset: 1

Freimut Marschner, Jun 14 2014

nonn

prime(k-1) is also the largest prime number < (n^2 + prime(n)^2). Remark : Largest prime number < n^2 is A053001. Largest prime number < n^3 is A077037.

			n=1, 1^2=1, prime(1)^2 = 4, 1 + 4 = 5, 5 - 1= 4, prime(4) = 7 ;
n=2, 2^2=4, prime(2)^2 = 9, 4 + 9= 13, 13 - 1= 12, prime(12) = 37.

Freimut Marschner, Table of n, a(n) for n = 1..63

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A106587 (n^2 + prime(n)^2).

a[n_]:=Prime[(n^2 + Prime[n]^2) - 1]; Array[a,36] (* Stefano Spezia, Mar 12 2025 *)

a(n) = prime((n^2 + prime(n)^2) - 1) = prime(A106587(n) - 1).

A173831 Largest prime < n^4.

Original entry on oeis.org

13, 79, 251, 619, 1291, 2399, 4093, 6553, 9973, 14639, 20731, 28559, 38393, 50599, 65521, 83497, 104971, 130307, 159979, 194479, 234239, 279823, 331769, 390581, 456959, 531383, 614639, 707279, 809993, 923513, 1048573, 1185907, 1336333

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 25 2010

nonn

Cf. A014220, A053001, A077037

PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];f[n_]:=n^4;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst
NextPrime[Range[2,40]^4,-1] (* Harvey P. Dale, May 05 2018 *)

A121844 Excess of n^3 over previous prime.

Original entry on oeis.org

1, 4, 3, 12, 5, 6, 3, 2, 3, 4, 5, 18, 3, 2, 3, 4, 5, 2, 7, 4, 9, 4, 17, 6, 3, 2, 9, 10, 7, 2, 19, 4, 3, 12, 7, 2, 3, 38, 3, 4, 11, 14, 25, 4, 9, 10, 5, 6, 9, 4, 5, 4, 7, 12, 15, 4, 9, 22, 17, 38, 7, 4, 5, 16, 5, 2, 9, 28, 11, 28, 17, 14, 3, 28, 9, 4, 5, 12, 3, 58, 5, 4, 11, 12, 13, 16, 21, 22, 7, 2, 7

Offset: 2

Zak Seidov, Aug 29 2006

nonn

			a(5)=12 because 5^3=125 and previous prime is 113 hence a(5)=125-113=12

Cf. A077037.

Table[n^3-Prime[PrimePi[n^3]],{n,2,100}]
#-NextPrime[#,-1]&/@((Range[2,100])^3) (* Harvey P. Dale, Apr 06 2013 *)

a(n) = n^3 - precprime(n^3 - 1) \\ Michel Marcus, Jun 03 2013

A173832 Largest prime < n^5.

Original entry on oeis.org

31, 241, 1021, 3121, 7759, 16787, 32749, 59029, 99991, 161047, 248827, 371291, 537811, 759371, 1048573, 1419839, 1889561, 2476081, 3199997, 4084081, 5153623, 6436327, 7962607, 9765619, 11881357, 14348891, 17210353, 20511143, 24299981

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 25 2010

nonn

Cf. A014220, A053001, A077037, A173831

PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];f[n_]:=n^5;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst
NextPrime[Range[2, 30]^5, -1]  (* Ray Chandler, Dec 08 2018 *)

A329931 Reversal of base-n digits of largest prime < n^3.

Original entry on oeis.org

7, 23, 31, 89, 71, 97, 383, 647, 799, 967, 1151, 1507, 2351, 3149, 3583, 4045, 4535, 6497, 5599, 7937, 6775, 10579, 4607, 12499, 16223, 18953, 15679, 16819, 21599, 28829, 14335, 32669, 36991, 29399, 38879, 49283, 51983, 3041, 60799, 63877, 56447, 55469, 38719

Offset: 2

Robert Dougherty-Bliss, Nov 24 2019

Terms are listed in decimal.

Conjecture: a(n) < prevprime(n^3) for n >= 4. In other words, the most-significant base-n digit is larger than the least-significant base-n digit. This conjecture seems to hold for the analogous sequence with n^2, but fails for powers higher than 3.

			For n = 3, prevprime(3^3) = 23 = 212_3, and reversal gives a(3) = 212_3 = 23. For n = 5, prevprime(5^3) = 113 = 423_5, and reversal gives a(5) = 324_5 = 89.

Base-n reversal of A077037(n).

a[n_] := FromDigits[ Reverse@ IntegerDigits[ NextPrime[n^3, -1], n], n]; Array[a, 43, 2] (* Giovanni Resta, Nov 24 2019 *)

a(n) = fromdigits(Vecrev(digits(precprime(n^3-1), n)), n); \\ Michel Marcus, Nov 25 2019

cp's OEIS Frontend

A077038 Least difference of primes p, q such that p < n^3 < q.

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A242382 Lesser of consecutive primes whose average is a perfect cube.

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A243893 a(n) = prime(k-1) with k = n^2 + prime(n)^2.

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A173831 Largest prime < n^4.

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A121844 Excess of n^3 over previous prime.

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A173832 Largest prime < n^5.

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A329931 Reversal of base-n digits of largest prime < n^3.

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