A056781 -id:A056781 - cp's OEIS frontend

A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2046, 4095, 8190, 16380, 32760, 65520, 131040, 262080, 524160, 1048320, 2097144, 4193280, 8386560, 16773900, 33547800, 67095600, 134191200, 268382400, 536215680, 1073709000, 2144142000, 4288284000, 8527559040, 16908091200, 27935107200

Offset: 1

Labos Elemer, Aug 16 2000

			These maximal terms are usually "near" to 2^n. For n=1..10 they are equal to 2^n. At n=21, a(21)=2097144, 1048576 < a(21) < 2097144 = 8*27*7*19*73 has d=128 divisors, of which the cube is d^3d=2097152. So this maximum is near to but still less than d^3.

Cf. A000005, A029837, A034884, A035033, A035034, A035035, A056757-A056767, A056781.

Table[Last@ Select[Range @@ (2^{n - 1, n}), DivisorSigma[0, #]^3 > # &], {n, 22}] (* Michael De Vlieger, Dec 31 2016 *)

a(n) = {k = 2^n; while(numdiv(k)^3 <= k, k--); k;} \\ Michel Marcus, Dec 11 2013

Largest terms of A056757 between 2^(n-1) and 2^n.

a(32) from Michel Marcus, Dec 11 2013

a(33)-a(35) and keyword "full" added by Amiram Eldar, Feb 23 2025

A056761 Odd numbers less than the cube of their number of divisors.

Original entry on oeis.org

3, 5, 7, 9, 15, 21, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 75, 81, 99, 105, 117, 135, 147, 153, 165, 171, 175, 189, 195, 207, 225, 231, 255, 273, 285, 297, 315, 345, 351, 357, 375, 385, 399, 405, 429, 435, 441, 455, 459, 465, 483, 495, 525, 567, 585, 675, 693

Offset: 1

Labos Elemer, Aug 16 2000

Last term is a(267) = 883575, confirming the author's conjecture. - Charles R Greathouse IV, Apr 27 2011

			14175 = 81*25*7 has 30 divisors, and 30^3 = 27000 > 14175.

Zak Seidov, Table of n, a(n) for n = 1..267 (complete sequence)

Cf. A035033-A035035, A034884, A000005, A000265, A056757-A056767, A056781.

Select[Range[1, 10^6 + 1, 2], DivisorSigma[0, #]^3 > # &] (* Michael De Vlieger, Oct 26 2017 *)

isok(n) = (n % 2) && (numdiv(n)^3 > n); \\ Michel Marcus, Dec 19 2013

A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

Original entry on oeis.org

16, 25, 27, 32, 49, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 15625, 16384, 16807, 17161, 19683, 22801, 32761, 32768, 36481, 59049, 65536, 78125, 97969, 117649, 124609, 131072, 139129, 146689, 161051, 177147, 262144

Offset: 1

Bernard Schott, Dec 10 2020

Equivalently: numbers m with only one prime factor such that the LCM of their palindromic divisors is neither 1 nor m: subsequence of A334392.
G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see Simmons p. 98). According to this conjecture, these perfect powers are terms: {2^k, k>=4}, {3^k, k>=3}, {5^k, k>=2}, {7^k, k=2 and k>=4}, {11^k, k>=5}, {101^k, k>= 5}, {131^k, k>=2}, ...
From a(1) = 16 to a(17) = 2187, the data is the same as A056781(10) until A056781(26), then a(18) = 2401 and A056781(27) = 4096.

			5^2 = 25, 2^6 = 64, 3^4 = 81 are terms.
7^2 = 49 is a term, 7^3 = 343 is not a term, and 7^4 = 2401 is a term.
101^2 = 10201 and 11^4 = 14641 are not terms.

Murray S. Klamkin, Problems in applied mathematics: selections from SIAM review, (1990), p. 520.

Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].
Wikipedia, Palindromic number, Perfect Powers.

Intersection of A025475 and A334392.
Cf. A087990, A087999, A334139, A334391.
Subsequences: A000079 \ {1,2,4,8}, A000244 \ {1,3,9}, A000351 \ {1,5}, A000420 \ {1,7,343}, A001020 \ {1,11,121,1331,14641}, A096884 \ {1,101, 10201, 1030301, 104060401}.

q[n_] := Module[{f = FactorInteger[n]}, Length[f] == 1 && f[[1, 2]] > 1 && PalindromeQ[f[[1, 1]]]]; Select[Range[10^5], !PalindromeQ[#] && q[#] &] (* Amiram Eldar, Dec 10 2020 *)

ispal(n) = my(d=digits(n)); Vecrev(d)==d;
isok(k) = my(p); isprimepower(k, &p) && isprime(p) && ispal(p) &&!ispal(k); \\ Michel Marcus, Dec 10 2020

More terms from Amiram Eldar, Dec 10 2020

A056759 The 17 prime powers k = p^w such that d(p^w)^3 > p^w where d = A000005().

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 16, 25, 27, 32, 64, 81, 128, 256, 512, 1024

Offset: 1

Labos Elemer, Aug 16 2000

For all divisors of the LCM of the terms of this sequence (14515200) the defining relation d(x)^3 > x is also satisfied.

			Differences d(x)^3 - x of the 17 entries of this sequence are 6, 5, 23, 3, 1, 56, 18, 109, 2, 37, 184, 279, 44, 384, 473, 488, 307.

Cf. A035033-A035035, A034884, A000005, A056757-A056757, A056781.

A056764 Number of integers k not exceeding 2^n such that the cube of number of divisors [A000005(k)] is larger than k.

Original entry on oeis.org

1, 3, 7, 13, 24, 48, 78, 138, 248, 385, 633, 1032, 1523, 2324, 3470, 4856, 6844, 9369, 12283, 15920, 20001, 24335, 28984, 33563, 37868, 41735, 44946, 47413, 49143, 50262, 50854, 51126, 51230, 51258, 51261, 51261, 51261, 51261, 51261, 51261, 51261, 51261, 51261

Offset: 1

Labos Elemer, Aug 16 2000

a(n) = 51261 for n >= 35 since A056757 is finite with 51261 terms. - Amiram Eldar, Jun 02 2024

			Below 2^29 = 536870912 in A056757 altogether 49143 terms occur, so a(29) = 49143.

Index entries for linear recurrences with constant coefficients, signature (1).

Number of entries in A056757 not exceeding 2^n.

Cf. A000005, A029837, A034884, A035033-A035035, A056757-A056767, A056781.

With[{s = Import["https://oeis.org/A056757/b056757.txt", "Table"][[;; , 2]]}, a[n_] := Count[s, ?(# <= 2^n &)]; Table[a[n], {n, 1, 35}]] (* _Amiram Eldar, Jun 02 2024 *)

More terms from Amiram Eldar, Jun 02 2024

cp's OEIS Frontend

A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.

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A056761 Odd numbers less than the cube of their number of divisors.

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A339624 Perfect powers p^k, k >= 2 of palindromic primes p when p^k is not a palindrome.

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A056759 The 17 prime powers k = p^w such that d(p^w)^3 > p^w where d = A000005().

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A056764 Number of integers k not exceeding 2^n such that the cube of number of divisors [A000005(k)] is larger than k.

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