A073848 Smallest of n consecutive numbers in A.P. with a common difference of n with a sum that is an n-th power.

1, 1, 6, 58, 615, 7761, 117628, 4, 2151, 999999955, 25937424546, 181398462, 23298085122403, 793714773254053, 29192926025390520, 3976, 48661191875666868345, 5642219814759, 104127350297911241532670

Offset

1,3

Links

Table of n, a(n) for n=1..19.

Formula

For n!=4: a(n)=(p_1*p_2*...p_r)^n/n-n*(n-1)/2 with prime factorization n=p_1^b_1*...*p_r^b_r.

Proof of formula: If a is the first member in A.P. then there must exist an integer m with n*a+n*n*(n-1)/2=m^n. Therefore all prime divisors of n divide m. For n>4 this gives a positive solution for a.

Example

a(4) = 58 and 58 +62 +66 +70 = 256 = 4^4.

Maple

for n from 1 to 50 do a := ifactors(n); b := 1:for j from 1 to nops(a[2]) do b := b*a[2][j][1]; od; c[n] := b^n/n-n*(n-1)/2; od:c[4] := 58:seq(c[j], j=1..50);

Crossrefs

Sequence in context: A223030 A292574 A354228 * A141382 A212426 A259612

Adjacent sequences: A073845 A073846 A073847 * A073849 A073850 A073851

Keyword

nonn

Author

Amarnath Murthy, Aug 14 2002

Extensions

Formula and more terms from Sascha Kurz, Aug 14 2002

Status

approved