1,2

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 52.

M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.

Don Reble, Table of n, a(n) for n = 1..2000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a(p) = 2^(p-1) for primes p: a(A000040(n)) = A061286(n); a(p^2) = 6^(p-1) for primes p: a(A001248(n)) = A061234(n); a(p*q) = 2^(q-1)*3^(p-1) for primes p<=q: a(A001358(n)) = A096932(n); a(p*m*q) = 2^(q-1) * 3^(m-1) * 5^(p-1) for primes p<m<q: A005179(A007304(n)) = A061299(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 15 2004

a(p^n)=(2*3...*p_n)^(p-1) for p > log p_n / log 2. Unpublished proof from Andrzej Schinzel. - Tomasz Ordowski (ordot(AT)poczta.onet.pl), Jul 22 2005

If p is a prime and n=p^k then a(p^k)=(2*3*...*s_k)^(p-1) where (s_k) is the numbers of the form q^(p^j) for every q and j>=0, according to Grost (1968), Theorem 4. For example, if p=2 then a(2^k) is the product of the first k members of the A050376 sequence: number of the form q^(2^j) for j>=0, according to Ramanujan (1915). - Tomasz Ordowski (ordot(AT)poczta.onet.pl), Aug 30 2005

If n=2^k then a(2^k)=A037992(k). - Tomasz Ordowski (ordot(AT)poczta.onet.pl), Aug 30 2005

a = Table[ 0, {43} ]; Do[ d = Length[ Divisors[ n ]]; If[ d < 44 && a[[ d ]] == 0, a[[ d]] = n], {n, 1, 1099511627776} ]; a

(PARI) prodR(n, maxf)={ local(dfs, a=[], r, tmp ) ; dfs=divisors(n) ; for(i=2, length(dfs), if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)) ; for(j=1, length(r), tmp=concat(dfs[i], r[j]) ; a=concat(a, [tmp]) ; ) ; ) ; ) ; ) ; return(a) ; } A005179(n)={ local(pf=prodR(n, n), a=1, b) ; for(i=1, length(pf), b=prod(j=1, length(pf[i]), prime(j)^(pf[i][j]-1)) ; if(b<a || i==1, a=b ) ; ) ; return(a) ; } { for(n=1, 100, print1(A005179(n)", ") ; ) } /* R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 26 2008 */

Cf. A007416, A099316, A003586, A099311, A099313.

Cf. A050376 and A037992.

Adjacent sequences: A005176 A005177 A005178 this_sequence A005180 A005181 A005182

Sequence in context: A049022 A136033 A099315 this_sequence A037019 A096174 A096173

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com), David Singmaster

More terms from David W. Wilson (davidwwilson(AT)comcast.net)