A037992 -id:A037992 - cp's OEIS frontend

A375269 Partial products of A115975.

1, 2, 6, 24, 120, 840, 6720, 60480, 665280, 8648640, 147026880, 2793510720, 64250746560, 1606268664000, 43369253928000, 1257708363912000, 38988959281272000, 1247646697000704000, 46162927789026048000, 1892680039350067968000, 81385241692052922624000, 3825106359526487363328000

Offset: 1

Amiram Eldar, Aug 09 2024

nonn

First differs from A334395 at n = 42.
Numbers with a record number of dual-Zeckendorf-infinitary divisors (A331109). Also, indices of records in A375272.
a(n) is the least number k such that A375272(k) = n-1 and A331109(k) = 2^(n-1).

			A115975 begins with 1, 2, 3, 4, 5, 7, ..., so, a(1) = 1, a(2) = 1 * 2 = 2, a(3) = 1 * 2 * 3 = 6, ..., a(6) = 1 * 2 * 3 * 4 * 5 * 7 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..358

Cf. A037992 (analogous with "Fermi-Dirac primes", A050376), A115975, A331109, A334395, A375271, A375272.

Subsequence of A025487.

fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k++; f = Fibonacci[k]]; s];
seq[max_] := Module[{s = {}, p = 2, e = 1, f = {}}, While[e > 0, e = Floor[Log[p, max]]; If[f == {}, f = fib[e], f = Select[f, # <= e &]]; s = Join[s, p^f]; p = NextPrime[p]]; FoldList[Times, 1, Sort[s]]]; seq[250]

fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k++; f = fibonacci(k)); Vec(s);}
lista(pmax) = {my(s = [1], p = 2, e = 1, f = [], r = 1); while(e > 0, e = logint(pmax, p); if(#f == 0, f = fib(e), f = select(x -> x <= e, f)); s = concat(s, apply(x -> p^x, f)); p = nextprime(p+1)); s = vecsort(s); for(i = 1, #s, r *= s[i]; print1(r, ", "));}

a(n) = Product_{k=1..n} A115975(k).

A348630 Nonexponential superabundant numbers: numbers m such that nesigma(m)/m > nesigma(k)/k for all k < m, where nesigma(m) is the sum of nonexponential divisors of m (A160135).

Original entry on oeis.org

1, 24, 30, 96, 120, 480, 840, 3360, 13440, 30240, 36960, 120960, 147840, 272160, 332640, 1330560, 2993760, 4324320, 17297280, 38918880, 73513440, 220540320, 294053760, 661620960, 1396755360, 2646483840, 5587021440, 12570798240

Offset: 1

Amiram Eldar, Oct 26 2021

The least term k with nesigma(k)/k > m for m = 2, 3, 4, ... is 480, 332640, 1396755360, ...

Cf. A160135, A348604.
Subsequence of A348629.
The nonexponential version of A004394.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1 ;s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; rm = -1; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]],{n, 1, 10^6}]; seq

A350641 Numbers k such that the product of k and all terms < k in A050376 has more divisors than the product of all terms < k in A050376 and the smallest term > k in A050376.

Original entry on oeis.org

42, 66, 72, 78, 88, 104, 110, 130, 136, 152, 156, 160, 170, 184, 190, 200, 204, 224, 228, 230, 232, 238, 240, 248, 255, 285, 345, 435, 460, 465, 483, 525, 555, 580, 600, 609, 615, 620, 651, 696, 744, 777, 783, 812, 837, 861, 868, 888, 903, 930, 984, 987, 999

Offset: 1

J. Lowell, Jan 09 2022

nonn

Multiplying a number in this sequence by all numbers in A050376 less than it will give a number less than, but with more divisors than, a number in A037992 with comparable magnitude.

			The product of 42 and all terms < 42 in A050376 has 276480 divisors. The product of all terms < 42 in A050376 and the smallest term > 42 (i.e., 43) in A050376 has only 262144 divisors. Thus, 42 is a term of this sequence.

Cf. A050376, A037992, A002183.

list(lim) = my(v=primes(primepi(lim)), t); forprime(p=2, sqrt(lim), t=p; while((t=t^2)<=lim, v=concat(v, t))); vecsort(v); \\ A050376
lista(nn) = my(vfd=list(nn), res=List()); for (n=1, nn, my(vless = select(x->(x(x>n), vfd)); if (#vmore, my(p = vecprod(vless)); if (numdiv(p*n) > numdiv(p*vmore[1]), listput(res, n));););); res; \\ Michel Marcus, Jan 10 2022

More terms from Jinyuan Wang, Jan 09 2022

A380836 a(n) is the smallest k such that tau(2*k) is equal to 2^n, where tau = A000005.

Original entry on oeis.org

1, 3, 12, 60, 420, 3780, 41580, 540540, 8648640, 147026880, 2793510720, 64250746560, 1606268664000, 46581791256000, 1444035528936000, 53429314570632000, 2190601897395912000, 94195881588024216000, 4427206434637138152000, 216933115297219769448000, 11497455110752647780744000

Offset: 1

Juri-Stepan Gerasimov, Feb 06 2025

nonn

a(n) = A000028(n+1) | a(n+1) ? [I interpret this as saying "Is it true that if a(n) = A000028(n+1) then a(n) divides a(n+1)?" - N. J. A. Sloane, Feb 15 2025]

For n > 2, a(n) is a Zumkeller number (A083207). Proof: Let p_r be the maximum prime in the prime factorization of a(n). First, in the prime factorization p_r must be to the power of one (otherwise we could build a smaller term with the same number of divisors, which would contradict the definition). Second, p_1 must be 2 and floor(log_2(p_r)) <= e_1, where e_1 is the exponent of p_1 (same reason as above). Therefore, there exists a power e <= e_1 such that p_1^e*p_r is a primitive Zumkeller number (see A180332). Then p_1^e1*p is a Zumkeller number (see Theorem 4.13 in Mahanta et al. JNT paper at A083207). Then a(n) = p_1*e_1*p_2^e_2*...*p_r is a Zumkeller number (see Corollary 5 in Rao/Peng JNT paper at A083207). - Ivan N. Ianakiev, Feb 15 2025

			1 is in the sequence because tau(1*2) = tau(2) = 2 = 2^1;
3 is in the sequence because tau(3*2) = tau(6) = 4 = 2^2;
12 is in the sequence because tau(12*2) = tau(24) = 8 = 2^3;

Cf. A000005, A000028, A005843, A037992, A099777.

a(n) = A037992(n)/2. - Amiram Eldar, Feb 06 2025

cp's OEIS Frontend

A375269 Partial products of A115975.

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A348630 Nonexponential superabundant numbers: numbers m such that nesigma(m)/m > nesigma(k)/k for all k < m, where nesigma(m) is the sum of nonexponential divisors of m (A160135).

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A350641 Numbers k such that the product of k and all terms < k in A050376 has more divisors than the product of all terms < k in A050376 and the smallest term > k in A050376.

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A380836 a(n) is the smallest k such that tau(2*k) is equal to 2^n, where tau = A000005.

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