A113839 -id:A113839 - cp's OEIS frontend

A360840 3-full numbers (A036966) sandwiched between twin primes.

432, 2592, 139968, 444528, 472392, 995328, 3456000, 5174928, 6561000, 10125000, 15552000, 15804072, 17496000, 25299648, 28449792, 37340352, 54675000, 63700992, 85957848, 88723728, 99574272, 120891312, 144027072, 169869312, 177147000, 197413632, 253125000, 259308000

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			432 = 2^4 * 3^3 is a term since it is 3-full and 431 and 433 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Intersection of A014574 and A036966.

Subsequence of A113839.

Select[6*Range[10^6], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 2 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 2;

A360841 4-full numbers (A036967) sandwiched between twin primes.

Original entry on oeis.org

2592, 139968, 995328, 37340352, 63700992, 99574272, 169869312, 414720000, 1399680000, 4076863488, 10871635968, 17714700000, 22781250000, 25312500000, 35888419872, 51840000000, 82012500000, 98802571392, 135304020000, 136136700000, 170749797552, 174960000000, 196730062848

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			2592 = 2^5 * 3^4 is a term since it is 4-full and 2591 and 2593 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..1520 (terms below 3*10^19)
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Intersection of A014574 and A036967.

Subsequence of A113839 and A360840.

Select[6*Range[2*10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 3 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 3;

A360842 5-full numbers (A069492) sandwiched between twin primes.

Original entry on oeis.org

139968, 995328, 63700992, 4076863488, 17714700000, 82012500000, 98802571392, 174960000000, 445240556352, 641194278912, 889223142528, 1059917571072, 1594323000000, 1663012435968, 2348273369088, 3333709317312, 5717741400000, 16260080320512, 19144761127488, 28697814000000

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			139968 = 2^6 * 3^7 is a term since it is 5-full and 139967 and 139969 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..250 (terms below 3*10^19)
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Intersection of A014574 and A069492.

Subsequence of A113839, A360840 and A360841.

Select[6*Range[2*10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 4 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 4;

A360843 6-full numbers (A069493) sandwiched between twin primes.

Original entry on oeis.org

139968, 98802571392, 174960000000, 889223142528, 1594323000000, 2348273369088, 19144761127488, 28697814000000, 56358560858112, 84537841287168, 150289495621632, 186624000000000, 328341017826432, 369056250000000, 392147405854848, 578415690713088, 597871125000000

Offset: 1

Amiram Eldar, Feb 23 2023

nonn

			139968 = 2^6 * 3^7 is a term since it is 6-full and 139967 and 139969 are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..74 (terms below 3*10^19)
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Intersection of A014574 and A069493.

Subsequence of A113839, A360840, A360841 and A360842.

Select[6*Range[10^5], PrimeQ[# - 1] && PrimeQ[# + 1] && Min[FactorInteger[#][[;; , 2]]] > 5 &]

is(n) = isprime(n-1) && isprime(n+1) && vecmin(factor(n)[,2]) > 5;

A360844 a(n) is the least k-full number that is sandwiched between twin primes.

Original entry on oeis.org

4, 432, 2592, 139968, 139968, 174960000000, 56358560858112, 84537841287168, 578415690713088, 578415690713088, 1141260857376768, 61628086298345472, 61628086298345472, 61628086298345472, 322850407500000000000000000000, 322850407500000000000000000000, 62518864539857068333550694039552

Offset: 2

Amiram Eldar, Feb 23 2023

nonn

k-full number is a number m such that if a prime p divides m then so does p^k. All the exponents in the canonical prime factorization of a k-full number are not smaller than k.
a(2)-a(15) are the terms below 3*10^19. Except for a(7) = 174960000000, they are all 3-smooth numbers (A003586, and thus they are terms of A027856). Are there other terms that are not 3-smooth?
a(168) = 2^176 * 3^173 * 7^168 is the first term that is not 5-smooth. - Bert Dobbelaere, Feb 24 2023

			The first 3 terms, their factorizations and the corresponding twin primes are:
  n |   a(n)  prime factorization  A051904(a(n))  {a(n)-1, a(n)+1}
  ----------------------------------------------------------------
  2 |     4                  2^2              2             {3, 5}
  3 |   432            2^4 * 3^3              3         {431, 433}
  4 |  2592            2^5 * 3^4              4       {2591, 2593}

Bert Dobbelaere, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Cf. A001097, A001694, A014574, A036966, A036967, A069492, A069493.
Cf. A113839, A360840, A360841, A360842, A360843.
Cf. A027856, A003586, A051904.

More terms from Bert Dobbelaere, Feb 24 2023

A370395 Numbers k that are neither squarefree nor prime powers sandwiched between twin primes.

Original entry on oeis.org

12, 18, 60, 72, 108, 150, 180, 192, 198, 228, 240, 270, 312, 348, 420, 432, 522, 600, 660, 810, 828, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1278, 1320, 1428, 1452, 1488, 1608, 1620, 1668, 1788, 1872, 1932, 1950, 1998, 2028, 2088, 2112, 2142, 2268, 2340, 2550

Offset: 1

Michael De Vlieger, Mar 27 2024

Contains A113839 \ {4}.

This sequence contains 1062, 1278, 1608 while A258838 does not; A258838 includes 4, 6, 30, 42, 462, 570, etc.

			The number 12 is neither squarefree nor a prime power but comes between primes 11 and 13.
The number 30 is squarefree, though it comes between primes 29 and 31, it is not in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A014574, A113839, A126706, A258838.

Select[1 + Select[Prime@ Range[100], PrimeQ[# + 2] &], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]

Intersection of A014574 and A126706.

A173518 Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.

Original entry on oeis.org

21, 960540, 16418498901144294337512360, 436066841882071117095002459324085167366543342937477344818646196279385305441506861017701946929489111120

Offset: 1

Michel Lagneau, Feb 20 2010

nonn

A. Nitaj proved Erdős's conjecture (1975) and claimed that there exist infinitely many triples of 3-powerful numbers a,b,c with (a,b) = 1, such that a+b=c, because the equation x^3 + y^3 = 6z^3 admits an infinite number of solutions, and given by the recurrence equations (see formula). It is proved that a=x(k)^3, b=y(k)^3, and c=6c^3, and are 3-powerful numbers for each k >= 1.

			37^3 + 17^3 = 6*21^3.

J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 348.
Mordell, L. J. (1969). Diophantine equations. Academic Press. ISBN 0-12-506250-8

P. Erdős, C. Pomerance, and A. Sárközy, On locally repeated values of certain arithmetic functions, II, Acta Math. Hungarica 49 (1987), pp. 251-259. [alternate link]
A. Nitaj, On a conjecture of Erdős on 3-powerful numbers, Bull. London Math. Soc. 27 (1995), no. 4, 317-318.
Wikipedia, Diophantine equation

Cf. A050240, A050241, A057521, A060859, A113839, A115645, A115651, A115676, A115686, A115687, A115689, A115691, A115693, A115695, A115697, A116064, A140172.

x0:=37:y0:=17:z0:=21: for p from 1 to 5 do: x1:=x0*(x0^3+ 2*y0^3):y1:=-y0*(2*x0^3+ y0^3):z1:=z0*(x0^3- y0^3): print(z1) : x0 :=x1 :y0 :=y1 :z0 :=z1 :od :

We generate the solutions (x(k),y(k),z(k)) from the initial solution x(0) = 37, y(0)=17, z(0)=21 x(k+1) = x(k)*(x(k)^3 + 2*y(k)^3) y(k+1) = -y(k)*(2*x(k)^3 + y(k)^3) z(k+1) = z(k)*(x(k)^3 - y(k)^3).

A370355 Highly touchable numbers sandwiched between untouchable twin pairs.

Original entry on oeis.org

1681, 5251, 7771, 36961, 39271, 170941, 196351, 360361, 510511, 1009471, 9699691

Offset: 1

Amiram Eldar, Feb 16 2024

Highly touchable numbers k have a record number of solutions x to A001065(x) = k, while untouchable numbers k have no solution to this equation.

Intersection of A238895 and {A231964(n) + 1};
Cf. A001065, A005114.
Similar sequences: A068507, A113839.

seq[nmax_] := Module[{v = Table[0, {nmax}], i, s = {}, vmax = -1}, Do[i = DivisorSigma[1, n] - n; If[0 < i <= nmax, v[[i]]++], {n, 1, nmax^2}]; Do[If[v[[n]] > vmax, vmax = v[[n]]; If[v[[n - 1]] == 0 && v[[n + 1]] == 0, AppendTo[s, n]]], {n, 2, nmax - 1}]; s]; seq[8000]

cp's OEIS Frontend

A360840 3-full numbers (A036966) sandwiched between twin primes.

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A360841 4-full numbers (A036967) sandwiched between twin primes.

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A360842 5-full numbers (A069492) sandwiched between twin primes.

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A360843 6-full numbers (A069493) sandwiched between twin primes.

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A360844 a(n) is the least k-full number that is sandwiched between twin primes.

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A370395 Numbers k that are neither squarefree nor prime powers sandwiched between twin primes.

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A173518 Solutions z of the Diophantine equation x^3 + y^3 = 6z^3.

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A370355 Highly touchable numbers sandwiched between untouchable twin pairs.

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