A051703 -id:A051703 - cp's OEIS frontend

A038701 Prime powers q for which f(g(m(q))) = m(q), where f = A051703, g = A008475 and m = A003418.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 103, 107, 109, 113

Offset: 1

Vladeta Jovovic, May 01 2000

These functions are defined for all natural numbers > 1 by: g(x) = Sum (p_j^k_j) where x = Product (p_j^k_j) is prime factorization of x (A008475); f(n) = max{x:g(x)=n} (A051703); m(n) = lcm(1,2,3,...,n) (A003418).
There are no more prime powers in the list <= 199. Conjecture: The sequence is finite, i.e., f(g(m(q))) > m(q) for sufficiently great prime powers q.
No other terms below 409. - Max Alekseyev, Sep 05 2023

			27 is not in the list because m(27) = 2^4*3^3*5^2*7*11*13*17*19*23, g(m(27))=158, f(158) = 3*5*7*11*13*17*19*23*29*31 > m(27).

D. W. Wilson, Answers to sci.math questions

Cf. A000961, A003418, A008475, A051703.

Offset changed to 1 by Jinyuan Wang, Mar 16 2020

A080743 Array read by rows in which n-th row lists orders of elements of Symm(n) that are not orders of elements of Symm(n-1) (6th row is empty, written as 0 by convention).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 0, 7, 10, 12, 8, 15, 9, 14, 20, 21, 30, 11, 18, 24, 28, 35, 42, 60, 13, 22, 36, 40, 33, 45, 70, 84, 26, 44, 56, 105, 16, 39, 55, 63, 66, 90, 120, 140, 17, 52, 72, 210, 65, 77, 78, 110, 126, 132, 168, 180, 19, 34, 48, 88, 165, 420, 51, 91, 99, 130, 154, 156, 220

Offset: 1

N. J. A. Sloane, Mar 08 2003

A051613 gives number of elements in n-th row.

			1;
2;
3;
4;
5, 6;
0;
7, 10, 12;
8, 15;
...

Alois P. Heinz, Rows n = 1..100, flattened
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A008475, A051613, A080744, A051703.

b:= proc(n) option remember; `if`(n<3, {n},
      {n, seq(map(x-> ilcm(x, i), b(n-i))[], i=2..n-1)}
       minus {seq(b(i)[], i=1..n-1)})
    end:
T:= proc(n) local l; l:= [b(n, n)[]];
      `if`(nops(l)=0, 0, sort(l)[])
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 15 2013

b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<1 || n<0, 0, Max[Join[{b[n, i-1]}, Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n]}]]]]] ]; T[1] = {1}; T[6] = {0}; T[n_] := Reap[For[m = n, m <= b[n, PrimePi[n]], m++,  If[n == Total[Power @@@ FactorInteger[m]], Sow[m]]]][[2, 1]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

n-th row = set of m such that A008475(m) = n, or 0 if no such m exists.

More terms from Vladeta Jovovic, Mar 12 2003

A039952 Maximum cardinality of finite D0L sequence over an alphabet with n symbols.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 15, 20, 30, 31, 60, 61, 84, 105, 140, 210, 211, 420, 421, 422, 423, 840, 841, 1260, 1261, 1540, 2310, 2520, 4620, 4621, 5460, 5461, 9240, 9241, 13860, 13861, 16380, 16381, 27720, 30030, 32760, 60060, 60061, 60062, 60063, 120120, 120121

Offset: 0

Jeffrey Shallit

nonn

			a(11) = 31 because we can write 11 = 1 + 2 + 3 + 5 and 31 = 1+2*3*5.

P. M. B. Vitanyi, Lindenmayer Systems: Structure, Languages and Growth Functions, Mathematisch Centrum, Math. Centre Tracts #96, 1980, p. 25.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
O. Osterby, Prime decompositions with minimum sum, Matematisk Institut, Aarhus Universitet, Technical Report DAIMI PB-19, November 1973; see the third column of Table 5 on page 18.
O. Osterby, Prime decompositions with minimum sum, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), 451-458.

Cf. A051703.

\\ here s is A051703 as a vector
s(n)={my(v=vector(n+1)); v[1]=1; forprime(p=2, n, forstep(i=#v, 1, -1, my(q=1); while(q*pAndrew Howroyd, Jan 05 2018

Max { Prod p^a + d : Sum p^a + d = n }, p prime.

a(n) = max(a(n-1)+1, A051703(n)). - Andrew Howroyd, Jan 05 2018

First 4 values appear incorrectly in cited references; corrected by JOS

a(0)=1 and terms a(35) and beyond from Andrew Howroyd, Jan 05 2018

A051704 Maximal value of products of partitions of n into powers of distinct primes (powers of 1 and 2 excluded).

Original entry on oeis.org

1, 0, 0, 3, 0, 5, 0, 7, 15, 9, 21, 11, 35, 13, 45, 105, 63, 17, 77, 165, 99, 315, 117, 385, 143, 495, 1155, 693, 1365, 819, 221, 1001, 3465, 1287, 4095, 1309, 5005, 1683, 6435, 15015, 9009, 2431, 8415, 19635, 11781, 45045, 13923, 25935, 17017, 58905, 21879

Offset: 0

Vladeta Jovovic

nonn

			a(14)=45 because max{3*11,9*5}=45.

Cf. A051703.

a[n_] := (pp = Reap[ Do[ pk = p^k; If[pk <= n, Sow[pk]], {p, Prime[ Range[2, PrimePi[n]]]}, {k, 1, Ceiling[Log[3, n]]}]][[2, 1]]; sel = Select[ IntegerPartitions[n, All, pp], Length[#] == Length[ Union[#] && !MatchQ[#, {_, x_, _, y_, _} /; GCD[x, y] != 1]] &]; Max[Times @@@ sel]); a[0] = 1; a[1] = a[2] = a[4] = a[6] = 0; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 31 2012 *)

a(43) (typo?) corrected by Jean-François Alcover, Jul 31 2012

A080744 Smallest element of n-th row of A080743.

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 7, 8, 9, 21, 11, 35, 13, 33, 26, 16, 17, 65, 19, 51, 38, 57, 23, 95, 25, 69, 27, 75, 29, 150, 31, 32, 62, 93, 96, 155, 37, 217, 74, 111, 41, 185, 43, 123, 86, 129, 47, 215, 49, 141, 98, 147, 53, 245, 106, 159, 212, 265, 59, 371, 61, 177, 122, 64, 244, 305

Offset: 1

N. J. A. Sloane, Mar 08 2003

J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Cf. A008475, A051613, A080743, A051703.

More terms from Vladeta Jovovic, Mar 09 2003

cp's OEIS Frontend

A038701 Prime powers q for which f(g(m(q))) = m(q), where f = A051703, g = A008475 and m = A003418.

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A080743 Array read by rows in which n-th row lists orders of elements of Symm(n) that are not orders of elements of Symm(n-1) (6th row is empty, written as 0 by convention).

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Maple

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A039952 Maximum cardinality of finite D0L sequence over an alphabet with n symbols.

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PARI

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A051704 Maximal value of products of partitions of n into powers of distinct primes (powers of 1 and 2 excluded).

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Mathematica

Extensions

A080744 Smallest element of n-th row of A080743.

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