A147516 -id:A147516 - cp's OEIS frontend

A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.

1, 3, 9, 15, 45, 105, 225, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815

Offset: 1

Stefano Lanfranco (lastefano(AT)yahoo.it), Mar 21 2000

Also numbers k such that the number of partitions of k into consecutive integers is a record. For example, 45 = 22+23 = 14+15+16 = 7+8+9+10+11 = 5+6+7+8+9+10 = 1+2+3+4+5+6+7+8+9, six such partitions, but all smaller terms have fewer such partitions (15 has four). See A000005 comments and A038547 formula. - Rick L. Shepherd, Apr 20 2008
From Hartmut F. W. Hoft, Mar 29 2022: (Start)
Also the odd parts of the numbers in A340506, see also comments in A250071.
A140864 is a subsequence. (End)
Positions of records in A001227, i.e., integers whose number of odd divisors sets a new record. - Bernard Schott, Jul 18 2022
Conjecture: all terms after the first three terms are congruent to 5 mod 10. - Harvey P. Dale, Jul 05 2023
From Keith F. Lynch, Jan 12 2024: (Start)
Dale's conjecture is correct.  a(n) can't be even, since then a(n)/2 would be a smaller number with the same number of odd divisors.  The respective powers of the successive odd primes can't increase, since if they did, swapping them would give a smaller number with the same number of divisors, e.g., 3^2 * 5^4 has the same number of divisors as 3^4 * 5^2, and the latter is smaller.  As such, every a(n) must be an odd multiple of 5, hence congruent to 5 mod 10, unless it's simply a power of 3.  But multiplying a power of 3 by 3 gives just one more divisor while multiplying a power of 3 by 5 doubles the number of divisors, so after a(n) = 9 all a(n) must be congruent to 5 mod 10, i.e., have a rightmost decimal digit of 5.
This has three equivalent definitions:
* Odd numbers with more divisors than any smaller odd number.
* Numbers with more odd divisors than any smaller number, i.e., record high values of A001227.
* Numbers with a greater excess of odd divisors over even divisors than any smaller number, i.e., record high values of A048272. (End)

			9 is in the sequence because 9 has 3 divisors {1, 3, 9}, which is more than any previous odd number.

Amiram Eldar, Table of n, a(n) for n = 1..363 (terms 1..170 from Ray Chandler)
David Ryan, Mathematical Harmony Analysis, arXiv preprint arXiv:1603.08904 [cs.SD], 2016.

Cf. A002182, A053640, A000005. Subsequence of A147516.
Cf. A140864, A237593, A250071, A279387, A340506.
Cf. A005408, A001227, A038547.

nn = 10^6; maxd = 0;
Reap[For[n = 1, n <= nn, n += 2, If[(nd = DivisorSigma[0, n]) > maxd, Print[n]; Sow[n]; maxd = nd]]][[2, 1]] (* Jean-François Alcover, Sep 20 2018, from PARI *)
next[n_] := Module[{k=n, r=DivisorSigma[0, n]}, While[DivisorSigma[0, k]<=r, k+=2]; k]
a053624[n_] := NestList[next, 1, n-1]/; n>=1 (* returns n numbers *)
a053624[31] (* Hartmut F. W. Hoft, Mar 29 2022 *)
DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,1,131*10^6,2}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 05 2023 *)

lista(nn) = {maxd = 0; forstep (n=1, nn, 2, if ((nd = numdiv(n)) > maxd, print1(n, ", "); maxd = nd;););} \\ Michel Marcus, Apr 21 2014

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 5, 3, 8, 3, 5, 9, 10, 3, 8, 3, 8, 9, 5, 3, 11, 7, 5, 12, 8, 3, 13, 3, 14, 9, 5, 9, 15, 3, 5, 9, 11, 3, 13, 3, 8, 16, 5, 3, 17, 7, 8, 9, 8, 3, 11, 9, 11, 9, 5, 3, 18, 3, 5, 16, 19, 9, 13, 3, 8, 9, 13, 3, 20, 3, 5, 16, 8, 9, 13, 3, 17, 21, 5, 3, 18, 9, 5, 9, 11, 3, 18, 9, 8, 9, 5, 9, 22, 3, 8, 16, 15, 3, 13, 3, 11, 23, 5, 3, 20, 3, 13

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1, and for n > 1, b(n) = A046523(n) + A000035(n), which starts as 1, 2, 3, 4, 3, 6, 3, 8, 5, 6, 3, 12, 3, 6, 7, 16, 3, 12, 3, 12, ...

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A291767, A291768 (bisections), A147516.
Cf. A046523, A101296, A286161, A286251, A286367, A291762 (related or similar filtering sequences).
Cf. A065091 (positions of 3's), A100484 (of 4 and 5's), A001248 (of 4 and 7's), A046388 (of 9's), A030078 (of 6 and 12's).
Cf. A098108 (one of the matching sequences).

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
write_to_bfile(1,rgs_transform(vector(65537,n,((-1)^n)*A046523(n))),"b291761.txt");
\\ Or alternatively:
f(n) = if(1==n,n,A046523(n)+(n%2));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291761.txt");

A357450 a(n) is the smallest integer having exactly n odd square divisors (A298735).

Original entry on oeis.org

1, 9, 81, 225, 6561, 2025, 531441, 11025, 50625, 164025, 3486784401, 99225, 282429536481, 13286025, 4100625, 893025, 1853020188851841, 2480625, 150094635296999121, 8037225, 332150625, 87169610025, 984770902183611232881, 12006225, 2562890625, 7060738412025, 121550625

Offset: 1

Bernard Schott, Sep 29 2022

nonn

All terms are odd and squares (A016754).

			2025 has 6 divisors that are odd squares: {1, 9, 25, 81, 225, 2025}; also, 2025 is the smallest integer that has 6 odd squares divisors, hence a(6) = 2025.

Cf. A000290, A016754, A038547, A130279, A147516, A298735.

f(n) = factorback(apply(e->e\2+1, factor(n/2^valuation(n, 2))[, 2])); \\ A298735
a(n) = my(k=1); while (f(k)!=n, k++); k; \\ Michel Marcus, Sep 29 2022

a(n) = A038547(n)^2. - Thomas Scheuerle, Sep 30 2022
Proof: Suppose a(n) = Product p_i^(2*e_i), where the p_i are odd primes. Then the n odd square divisors are all of the form d = Product p_i^(2*k_i) with 0 <= k_i <= e_i. As a(n) = Product (p_i^e_i)^2 = (Product (p_i^e_i))^2, we get that sqrt(a(n)) = Product (p_i^e_i). This is the prime decomposition of sqrt(a(n)). As there is a bijection between prime factors p_i^(2*k_i) and (p_i^k_i), there is also bijection between odd square divisors of a(n) and odd divisors of sqrt(a(n)). We conclude that sqrt(a(n)) is the smallest integer that has exactly n odd divisors. - Bernard Schott, Oct 01 2022
a(p) = 3^(2*(p-1)) for primes p. - Bernard Schott, Oct 03 2022

a(7)-a(10) from Michel Marcus, Sep 29 2022

More terms from Amiram Eldar, Sep 29 2022

A097273 Least integer with each "mod 2 prime signature".

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 27, 30, 32, 36, 45, 48, 54, 60, 64, 72, 81, 90, 96, 105, 108, 120, 128, 135, 144, 162, 180, 192, 210, 216, 225, 240, 243, 256, 270, 288, 315, 324, 360, 384, 405, 420, 432, 450, 480, 486, 512, 540, 576, 630, 648, 675, 720

Offset: 1

Ray Chandler, Aug 22 2004

nonn

For n = 2^e_0 * p_1^e_1 * ... * p_n^e_n where p_i is odd prime and e_1 >= e_2 >= ... >= e_n, define "mod 2 prime signature" to be ordered prime exponents (e_0,e_1,...,e_n).
Least integer with a given mod 2 prime signature is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.
A097272 sorted and duplicates removed.
Numbers k such that A097272(k) = k.
Verified up to a(68) = 972, 2*a(n) is also the order of a dihedral group D such that the lattice of normal subgroups of D is not isomorphic to the lattice of normal subgroups of any dihedral group of order less than 2*a(n). - Miles Englezou, May 18 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A097272, A097274, A097275.

Cf. A025487, A070826, A147516.

lpsQ[n_] := n==1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1,1]] == Prime[Length[f] + 1]); Select[Range[1000], lpsQ[# / 2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 23 2024 *)

Sum_{n>=1} 1/a(n) = 2 * Product_{n>=2} 1/(1 - 1/A070826(n)) = 3.2482341898... . - Amiram Eldar, Jul 23 2024

Offset corrected by Amiram Eldar, Jul 23 2024

A233819 Least odd integer of each prime signature ordered by prime signatures occurrence.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 81, 135, 105, 243, 225, 405, 315, 729, 675, 1215, 945, 2187, 2025, 1575, 3645, 1155, 3375, 2835, 6561, 6075, 4725, 10935, 3465, 10125, 8505, 19683, 18225, 14175, 32805, 10395, 30375, 11025, 25515, 59049, 23625, 54675, 17325, 50625, 42525, 98415, 31185, 91125, 33075, 76545, 177147, 70875

Offset: 1

Lior Manor, Dec 16 2013

nonn

a(n) is the least odd integer with the same prime signature as A025487(n).

			a(7) is the first odd integer with prime signature {1,3} = 3^3*5 = 135.
a(8) is the first odd integer with prime signature {1,1,1} = 3*5*7 = 105.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Will Nicholes, Prime Signatures

Cf. A003961, A025487, A147516.

a(n) = A003961(A025487(n)). - Amiram Eldar, Jun 23 2019

A291767 Odd bisection of A291761.

Original entry on oeis.org

1, 3, 3, 3, 7, 3, 3, 9, 3, 3, 9, 3, 7, 12, 3, 3, 9, 9, 3, 9, 3, 3, 16, 3, 7, 9, 3, 9, 9, 3, 3, 16, 9, 3, 9, 3, 3, 16, 9, 3, 21, 3, 9, 9, 3, 9, 9, 9, 3, 16, 3, 3, 23, 3, 3, 9, 3, 9, 16, 9, 7, 9, 12, 3, 9, 3, 9, 26, 3, 3, 9, 9, 9, 16, 3, 3, 16, 9, 3, 9, 9, 3, 23, 3, 7, 16, 3, 16, 9, 3, 3, 9, 9, 9, 26, 3, 3, 23, 3, 3, 9, 9, 9, 16, 9

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Records occur at positions: 1, 2, 5, 8, 14, 23, 41, 53, 68, 113, 122, 158, 203, 338, 365, ... (= (A147516(n)+1)/2) that give also all distinct values in this sequence: 1, 3, 7, 9, 12, 16, 21, 23, 26, 32, 34, 37, 40, 46, 48, 53, 58, 59, 64, 69, 72, 77, 81, ... Note that the terms of A291768 are all from the complementary sequence: 2, 4, 5, 6, 8, 10, 11, 13, 14, 15, 17, ...

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A147516, A278223, A291761, A291768.

(define (A291767 n) (A291761 (+ n n -1)))

a(n) = A291761(2n - 1).

A368041 a(n) is the least number k such that k^2 can be written as the difference of two positive squares in exactly n ways.

Original entry on oeis.org

1, 3, 8, 16, 12, 64, 128, 24, 512, 1024, 48, 4096, 72, 60, 32768, 65536, 192, 144, 524288, 384, 2097152, 4194304, 120, 16777216, 432, 1536, 134217728, 576, 3072, 1073741824, 2147483648, 240, 1152, 17179869184, 12288, 68719476736, 137438953472, 360, 1728, 1099511627776

Offset: 0

Ilya Gutkovskiy, Dec 09 2023

nonn

Index of first occurrence of n in A046079.

All the terms are of the form 2^m * A147516(k), m >= 0, k >= 1. - Amiram Eldar, Nov 08 2024

			a(2) = 8: 8^2 = 10^2 - 6^2 = 17^2 - 15^2.

Amiram Eldar, Table of n, a(n) for n = 0..999

Cf. A025487, A046079, A068314, A071571, A094191, A100073, A122842, A147516.

a(n) = min(A122842(n+1), 2*A071571(n)). - Jon E. Schoenfield, Dec 09 2023

a(26)-a(29) from Michel Marcus, Dec 09 2023

a(30)-a(39) from Jon E. Schoenfield, Dec 09 2023

A340155 Numbers k such that A008480(k) > k.

Original entry on oeis.org

326918592000, 435891456000, 653837184000, 871782912000, 980755776000, 1089728640000, 1307674368000, 1401079680000, 1508855040000, 1525620096000, 1569209241600, 1587890304000, 1634592960000, 1710035712000, 1852538688000, 1868106240000, 1961511552000, 2070484416000

Offset: 1

Amiram Eldar, Dec 29 2020

nonn

a(1) is given in the paper by Montgomery and Tenenbaum (2017).
If k is a term then all the numbers below k with the same prime signature as k are also terms. In particular A046523(k) is a term.
1710035712000 is the least term which is not in A025487.
Are there any odd terms in this sequence? If there are, the least of them is a term of A147516 and has more than 90 prime divisors (counted with multiplicity).

			326918592000 is a term since A008480(326918592000) = 358500542400 > 326918592000.

Hugh L. Montgomery and Gérald Tenenbaum, On Multiplicative Compositions Of Integers, Mathematika, Vol. 63, No. 3 (2017), pp. 1081-1090; altermative link. See p. 1088.

Cf. A008480, A025487, A046523, A147516, A260987, A270308, A318170.

A352797 Let S(k) be the subsequence of multiples of k from k*positive integers where each element of S(k) sets a new record of divisors in that sequence. Then f(k) is the first element from S(k)/k that is not a power of 2. Sequence lists records for f(k).

Original entry on oeis.org

1, 3, 9, 45, 135, 945, 2835, 14175, 155925, 467775, 6081075

Offset: 1

J. Lowell, Apr 03 2022

Conjecture: Subsequence of A147516.

			For k=1, the sequence of multiples of 1 that set records for numbers of divisors (divided by 1) starts 1,2,4,6. (A002182)
For k=3, the sequence starts 1,2,4,8,12. (A351623)
For k=9, the sequence starts 1,2,4,8,16,20.
For k=45, the sequence starts 1,2,4,8,16,24.

Cf. A002182, A351623, A147516.

isp2(n) = if (n<=2, return(1)); my(m); ispower(n,,&m) && (m==2);
f(n) = {my(m=1, nm, k=1, ok=0); until(ok, nm = numdiv(k*n); if (nm > m, m = nm; if (!isp2(k), ok = 1);); if (!ok, k++);); k;}
lista(nn) = {my(m=1, fm); for (n=1, nn, fm = f(n); if (fm > m, m = fm; print1(n, ", ");););} \\ Michel Marcus, May 05 2022

A385511 Numbers that are less than the number of their ordered factorizations into squarefree numbers greater than 1.

Original entry on oeis.org

2520, 5040, 7560, 10080, 10800, 12600, 15120, 20160, 21600, 22680, 23760, 25200, 27720, 30240, 32400, 35280, 37800, 43200, 45360, 47520, 50400, 52920, 55440, 60480, 64800, 65520, 70560, 71280, 75600, 79200, 83160, 86400, 88200, 90720, 95040, 98280, 100800, 105840

Offset: 1

Amiram Eldar, Jul 01 2025

nonn

Numbers k such that A050328(k) > k.
If k is a term then all the smaller numbers with the same prime signature (A118914) as k are also terms.
The least term that is not divisible by 5 is a(112) = 399168.
The least term that is not divisible by 3 is 144848704000.
The least odd term is A147516(43302) = 16639855392913235373515625.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A050328, A118914, A147516, A270308, A340155.

f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n && SquareFreeQ[n/#] &]; Select[Range[110000], f[#] > # &]

f(n) = if(n == 1, 1, sumdiv(n, d, if((d k;

cp's OEIS Frontend

A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.

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A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

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A357450 a(n) is the smallest integer having exactly n odd square divisors (A298735).

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A097273 Least integer with each "mod 2 prime signature".

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A233819 Least odd integer of each prime signature ordered by prime signatures occurrence.

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A291767 Odd bisection of A291761.

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A368041 a(n) is the least number k such that k^2 can be written as the difference of two positive squares in exactly n ways.

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A340155 Numbers k such that A008480(k) > k.

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