A246068 -id:A246068 - cp's OEIS frontend

A246066 Least k such that Lppf(k) > Gpf(k) + n, where Lppf(k) is the largest prime power factor of k and Gpf(k) is the greatest prime factor of k.

4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 27, 27, 27, 27, 32, 32, 32, 32, 32, 32, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81

Offset: 0

Arkadiusz Wesolowski, Aug 23 2014

nonn

All terms belong to A025475. See comment in A246067 for missing terms of A025475. - Jens Kruse Andersen, Aug 26 2014

Jens Kruse Andersen, Table of n, a(n) for n = 0..10000

Cf. A006530 (Gpf), A034699 (Lppf), A246067, A246068.

Cf. A025475.

k=2; for(n=0, 66, k=k-1; until(vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))>vecmax(f[, 1])+n, k++; f=factor(k)); print1(k, ", "));

A246067 Record values in A246066.

Original entry on oeis.org

4, 8, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 7921, 8192, 9409, 10201, 10609, 11449

Offset: 1

Arkadiusz Wesolowski, Aug 23 2014

nonn

Subsequence of A025475. The missing terms of A025475 are 1 and prime powers q^n for which there is a smaller prime power p^m with q^n-q <= p^m-p, i.e., q^n = p^m+d for some d <= q-p. The first cases: 9 = 3^2 = 2^3+1, 529 = 23^2 = 2^9+17, 2197 = 13^3 = 3^7+10, 2209 = 47^2 = 3^7+22, 6889 = 83^2 = 19^3+30. - Jens Kruse Andersen, Aug 26 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A025475, A246066, A246068.

k=2; for(n=0, 10506, k=k-1; r=k+1; until(vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))>vecmax(f[, 1])+n, k++; f=factor(k)); if(k>r, print1(k, ", ")));

A308324 Numbers which can be written in the form m^k - m with m an odd prime and k a positive integer.

Original entry on oeis.org

0, 6, 20, 24, 42, 78, 110, 120, 156, 240, 272, 336, 342, 506, 620, 726, 812, 930, 1320, 1332, 1640, 1806, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 9312

Offset: 1

Craig J. Beisel, May 20 2019

Besides the trivial example a(1)=0, the only known term which has two representations is a(24) = 2184 = 3^7 - 3 = 13^3 - 13. It is conjectured by Bennett to be the only term with this property.

			a(3) = 5^2 - 5 = 20.

Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

Cf. A057895, A246068.

x=List([]);lim=10000;forprime(m=3,lim,for(k=1,100,y=(m^k-m);if(y>lim,break,i=setsearch(x,y,1);if(i>0,listinsert(x, y, i)))));for(i=1, #x,print(x[i]));

A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

Original entry on oeis.org

0, 2, 6, 14, 20, 24, 30, 42, 62, 78, 110, 120, 126, 156, 240, 254, 272, 336, 342, 506, 510, 620, 726, 812, 930, 1022, 1320, 1332, 1640, 1806, 2046, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4094, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 8190, 9312

Offset: 1

Craig J. Beisel, May 24 2019

The only known terms which have two representations where m is prime are 6 and 2184. It is conjectured by Bennett these are the only terms with this property.

			a(9) = 2^6 - 2 = 62.
For the two terms known to have two representations we have a(3) = 6 = 2^3 - 2 = 3^2 - 3 and a(33)= 2184 = 3^7 - 3 = 13^3 - 13.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

Cf. A057895, A057896, A246068, A308324.

Subsequences: A000918 (2^n - 2), A036689 (p^2 - p), A058809 (3^n - 3), A178671 (5^n - 5).

N:= 10^6; # to get all terms <= N
P:= select(isprime,[2,seq(i,i=3..floor((1+sqrt(1+4*N))/2),2)]):
S:= {0,seq(seq(m^k-m,k=2..floor(log[m](N+m))),m=P)}:
sort(convert(S,list)); # Robert Israel, Aug 11 2019

x=List([]); lim=10000; forprime(m=2, lim, for(k=1, 100, y=(m^k-m); if(y>lim, break, i=setsearch(x, y, 1); if(i>0, listinsert(x, y, i))))); for(i=1, #x, print(x[i]));

isok(n) = {forprime(p=2, oo, my(keepk = 0); for (k=1, oo, if ((x=p^k - p) == n, return(1)); if (x > n, keepk = k; break);); if (keepk == 2, break););} \\ Michel Marcus, Aug 06 2019

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A246066 Least k such that Lppf(k) > Gpf(k) + n, where Lppf(k) is the largest prime power factor of k and Gpf(k) is the greatest prime factor of k.

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A246067 Record values in A246066.

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A308324 Numbers which can be written in the form m^k - m with m an odd prime and k a positive integer.

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A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

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