A064549 -id:A064549 - cp's OEIS frontend

A361265 Multiplicative with a(p^e) = e * p^(e + 1), e > 0.

1, 4, 9, 16, 25, 36, 49, 48, 54, 100, 121, 144, 169, 196, 225, 128, 289, 216, 361, 400, 441, 484, 529, 432, 250, 676, 243, 784, 841, 900, 961, 320, 1089, 1156, 1225, 864, 1369, 1444, 1521, 1200, 1681, 1764, 1849, 1936, 1350, 2116, 2209, 1152, 686, 1000, 2601, 2704

Offset: 1

Vaclav Kotesovec, Mar 06 2023

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

Cf. A005361, A064549, A203639, A361268.

g[p_, e_] := e*p^(e+1); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, 1 + p^2 * X / (1 - p*X)^2)[n], ", "))

Dirichlet g.f.: Product_{primes p} (1 + p^(s + 2)/(p^s - p)^2).
Dirichlet g.f.: zeta(s-2) * zeta(s-1)^2 * Product_{primes p} (1 - p^(4 - 3*s) + p^(2 - 2*s) + 2*p^(3 - 2*s) - p^(4 - 2*s) - 2*p^(1 - s)).
Sum_{k=1..n} a(k) ~ c * Pi^4 * n^3 / 108, where c = Product_{primes p} (1 - 3/p^2 + 2/p^3 + 1/p^4 - 1/p^5) = 0.3086489554825164955853322259998244718829914385...
a(n) = A005361(n) * A064549(n).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - log(1-1/p))/p = 1.6843597117... . - Amiram Eldar, Sep 01 2023

A361266 Multiplicative with a(p^e) = p^(e + 3), e > 0.

Original entry on oeis.org

1, 16, 81, 32, 625, 1296, 2401, 64, 243, 10000, 14641, 2592, 28561, 38416, 50625, 128, 83521, 3888, 130321, 20000, 194481, 234256, 279841, 5184, 3125, 456976, 729, 76832, 707281, 810000, 923521, 256, 1185921, 1336336, 1500625, 7776, 1874161, 2085136, 2313441, 40000

Offset: 1

Vaclav Kotesovec, Mar 06 2023

In general, if the function is multiplicative with a(p^e) = p^(e + m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(m+1)/(p^s - p)).
Equivalently, Dirichlet g.f.: zeta(s-m-1) * zeta(s-1) * Product_{primes p} (1 + p^(2 + m - 2*s) - p^(2 + 2*m - 2*s) - p^(1 - s)).
Sum_{k=1..n} a(k) ~ c(m) * zeta(m+1) * n^(m+2) / (m+2), where c(m) = Product_{primes p} (1 - 1/p^2 - 1/p^(m+1) + 1/p^(m+2)).
Limit_{m->oo} c(m) = 6/Pi^2 = A059956.

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

Cf. A003557, A007947, A064549, A361264.

f:= proc(n) local t;
  mul(t[1]^(t[2]+3), t = ifactors(n)[2])
end proc:
map(f, [$1..50]); # Robert Israel, Mar 07 2023

g[p_, e_] := p^(e+3); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

for(n=1, 100, print1(direuler(p=2, n, 1 + p^4*X/(1 - p*X))[n], ", "))

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,2] +=3); factorback(f); \\ Michel Marcus, Mar 07 2023

Dirichlet g.f.: Product_{primes p} (1 + p^4/(p^s - p)).
Dirichlet g.f.: zeta(s-4) * zeta(s-1) * Product_{primes p} (1 + p^(5 - 2*s) - p^(8 - 2*s) - p^(1 - s)).
Sum_{k=1..n} a(k) ~ c * Pi^4 * n^5 / 450, where c = Product_{primes p} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.5761527353856670595206110782641172754062471168028961885...
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = n * A007947(n)^3 = A064549(n) * A007947(n)^2 = A361264(n) * A007947(n) = A064549(A064549(A064549(n))).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.148846213921... . (End)

A363172 Primitive terms of A363171: terms of A363171 with no proper divisor in A363171.

Original entry on oeis.org

6, 10, 14, 44, 52, 105, 136, 152, 184, 232, 248, 286, 374, 418, 442, 495, 506, 592, 656, 688, 752, 848, 944, 976, 1292, 1564, 1748, 1755, 1972, 2108, 2144, 2145, 2204, 2272, 2336, 2356, 2516, 2528, 2656, 2668, 2788, 2805, 2812, 2848, 2852, 2924, 2925, 3104, 3116

Offset: 1

Amiram Eldar, May 19 2023

nonn

If k is a term then m*k is a term of A363171 for all m >= 1.

The least odd term is a(6) = 105, and the least term that is coprime to 6 is a(34832) = 37182145.

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

Cf. A363171.

q[n_] := DivisorSigma[-1, n * Times @@ FactorInteger[n][[;; , 1]]] > 2; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || ! q[#] &]; Select[Range[3200], primQ]

A064549(n) = { my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]+1)); };
isA363171(n) = sigma(A064549(n), -1) > 2;
is(n) = { if(!isA363171(n), return(0)); fordiv(n, d, if(d < n && isA363171(d), return(0))); return(1) };

A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

3600, 25200, 28224, 39600, 46800, 61200, 68400, 82800, 104400, 111600, 133200, 141120, 147600, 154800, 169200, 190800, 212400, 219600, 241200, 255600, 262800, 277200, 284400, 298800, 310464, 320400, 327600, 349200, 363600, 366912, 370800, 385200, 392400, 406800

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947) (the term "coreful divisor" was used by Hardy and Subbarao, 1983).
If k is a term, then also m*k is, for any squarefree m coprime to k. Thus there are infinitely many coreful triperfect numbers, and all of them can be generated from the sequence of primitive terms, which is the subsequence of powerful terms of this sequence. This sequence is c(n) = A064549(A005820(n)), i.e., the triperfect numbers (A005820), multiplied by their squarefree kernel (A007947): 3600, 28224, 1071645696, 1651818858099200, 532098668445696, 317519577357516800, ...
The asymptotic density of this sequence is Sum_{i>=1} beta(c(i))/c(i), where beta(n) = (6/Pi^2) * Product_{p|n} (p/(p+1)) = 0.0000797856... . If there are only 6 triperfect numbers, then the exact value of this density is 18575679807276818039685539/(23589576231586703755916083200 * Pi^2).

			3600 is in the sequence since its coreful divisors are {30, 60, 90, 120, 150, 180, 240, 300, 360, 450, 600, 720, 900, 1200, 1800, 3600}, whose sum is 10800 = 3 * 3600.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A005820, A007947, A057723, A064549, A307958.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500000], s[#] == 3*# &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) == 3*n;

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467

Offset: 1

Gus Wiseman, Feb 20 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.

All terms are odd.

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]-Times@@Union[prix[n]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A250209 a(n) = least k such that k * n is in A072226, or 0 if no such k exists.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 5, 1, 1, 1, 1, 1, 8, 34, 8, 1, 2, 1, 10, 1, 2, 350, 2, 1, 111, 4, 1, 3, 16, 4, 15, 28, 3, 1, 206, 3, 10, 2, 1, 1, 2, 3, 1, 15, 637, 12, 1, 4, 22, 17, 104, 4, 2, 1012, 1, 1

Offset: 1

Eric Chen, Jan 18 2015

nonn

Conjecture: a(n) > 0 for all n.
a(n) is currently unknown for n = 121, 124, 143, 162, 171, 172, 185, 188, 197, 215, ..., for which we have n * a(n) > 130000.
a(121) = (A117545(2048))/11 and they are both currently unknown.
A117545(2^n) = a(A064549(n)).
a(130) = 917, a(144) = 820, a(164) = 720, a(201) = 606. - Max Alekseyev, Dec 04 2024

Eric Chen, Table of n, a(n) for n = 1..120
Eric Chen, Table of n, a(n) for n = 1..300 status

Table[k=1; While[!PrimeQ[Cyclotomic[n*k, 2]], k++]; k, {n, 43}]

a(n) = {k = 1; while (!isprime(polcyclo(k*n, 2)), k++); k;} \\ Michel Marcus, Jan 18 2015

A334151 Numbers k such that k / rad(k) > m / rad(m) for all m < k.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 128, 243, 256, 512, 1024, 2048, 4096, 6561, 8192, 16384, 32768, 59049, 65536, 131072, 262144, 524288, 1048576, 1594323, 2097152, 4194304, 8388608, 14348907, 16777216, 33554432, 67108864, 129140163, 134217728, 268435456

Offset: 1

Ilya Gutkovskiy, Apr 16 2020

nonn

The terms listed in the Data section are numbers of the form 2^i or 3^ceiling(j*(1 + sqrt(2))), i >= 2, j >= 0 (empirical observation).

Michael De Vlieger, Table of n, a(n) for n = 1..4190
Michael De Vlieger, 256 X 256 pixel bitmap showing black if 3 | a(n) else white for n = 1..2^16, read in rows from left to right, stacked top to bottom.

Cf. A003557, A006899, A007947, A064549.

pp = 4; nn = 2^29; j = 0; c = e[_] = 1; r = Prime@ Range[pp];
Do[(e[#1]++; Set[{k, m}, {#1^#2, #1^(#2 - 1)}]) & @@
  First@ MinimalBy[Array[{#, e[#]} &[r[[#]]] &, pp], Power @@ # &];
 If[m > j, Set[{a[c], j}, {k, m}]; c++];
 If[k > nn/2, Break[]], {n, Infinity}];
{1}~Join~Array[a, c - 2, 2] (* Michael De Vlieger, Mar 11 2023 *)

A343881 Table read by antidiagonals upward: T(n,k) is the least integer m > k such that k^x * m^y = c^n for some positive integers c, x, and y where x < n and y < n; n >= 2, k >= 1.

Original entry on oeis.org

4, 8, 8, 4, 4, 12, 32, 4, 9, 9, 4, 4, 9, 16, 20, 128, 4, 9, 8, 25, 24, 4, 4, 9, 8, 20, 36, 28, 8, 4, 9, 8, 25, 24, 49, 18, 4, 4, 9, 8, 20, 36, 28, 27, 16, 2048, 4, 9, 8, 25, 24, 49, 18, 24, 40, 4, 4, 9, 8, 20, 36, 28, 16, 12, 80, 44, 8192, 4, 9, 8, 25, 24, 49

Offset: 2

Peter Kagey, May 02 2021

For prime p, the p-th row consists of distinct integers.

Conjecture: T(p,k) = A064549(k) for fixed k > 1 and sufficiently large p.

			Table begins:
  n\k|    1  2   3   4   5   6   7   8   9   10
-----+-----------------------------------------
   2 |    4, 8, 12,  9, 20, 24, 28, 18, 16,  40
   3 |    8, 4,  9, 16, 25, 36, 49, 27, 24,  80
   4 |    4, 4,  9,  8, 20, 24, 28, 18, 12,  40
   5 |   32, 4,  9,  8, 25, 36, 49, 16, 27, 100
   6 |    4, 4,  9,  8, 20, 24, 28,  9, 16,  40
   7 |  128, 4,  9,  8, 25, 36, 49, 16, 27, 100
   8 |    4, 4,  9,  8, 20, 24, 28, 16, 12,  40
   9 |    8, 4,  9,  8, 25, 36, 49, 16, 24,  80
  10 |    4, 4,  9,  8, 20, 24, 28, 16, 16,  40
  11 | 2048, 4,  9,  8, 25, 36, 49, 16, 27, 100
T(2, 3) = 12 with  3   * 12   =  6^2.
T(3,10) = 80 with 10^2 * 80   = 20^3.
T(4, 5) = 20 with  5^2 * 20^2 = 10^4.
T(5, 1) = 32 with  1   * 32   =  2^5.
T(6, 8) =  9 with  8^2 *  9^3 =  6^6.

Rows: A072905 (n=2), A277781 (n=3).

Cf. A064549, A343825.

T(n,1) = 2^A020639(n).

A349574 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344696(i) = A344696(j) and A344697(i) = A344697(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1, 5, 1, 4, 1, 2, 1, 1, 1, 3, 6, 1, 7, 2, 1, 1, 1, 8, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 5, 10, 6, 1, 2, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 4, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 6, 2, 1, 1, 1, 5, 13, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 8, 1, 10, 4, 14, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Nov 22 2021

nonn

Restricted growth sequence transform of the ordered pair [A344696(n), A344697(n)].
For all i, j, A003557(i) = A003557(j) => a(i) = a(j); in other words, this sequence is a function of A003557. This follows because A344696(n) = A344696(A057521(n)), A344697(n) = A344696(A057521(n)), and A057521(n) = A064549(A003557(n)).
Apparently, A081770 gives the positions of 2's, which occur on those n where A344696(n) = 7 and A344697(n) = 6.

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

Cf. A000203, A001615, A003557, A005117 (positions of 1's), A057521, A064549, A081770, A280292, A344695, A344696, A344697.

up_to = 65537;
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; };
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
Aux349574(n) = { my(s=sigma(n),u=A001615(n),g=gcd(u,s)); [s/g, u/g]; };
v349574 = rgs_transform(vector(up_to, n, Aux349574(n)));
A349574(n) = v349574[n];

For all n >= 1, a(n) = a(A057521(n)). [See comments]

A351434 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j + 1)).

Original entry on oeis.org

1, 1, 4, 1, 16, 4, 36, 1, 8, 16, 100, 4, 144, 36, 64, 1, 256, 8, 324, 16, 144, 100, 484, 4, 64, 144, 16, 36, 784, 64, 900, 1, 400, 256, 576, 8, 1296, 324, 576, 16, 1600, 144, 1764, 100, 128, 484, 2116, 4, 216, 64, 1024, 144, 2704, 16, 1600, 36, 1296, 784, 3364, 64, 3600, 900, 288, 1, 2304

Offset: 1

Ilya Gutkovskiy, Feb 11 2022

Cf. A003557, A003958, A003959, A023900, A064549, A326297, A327564, A351419, A351435.

a:= n-> mul((i[1]-1)^(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..65);  # Alois P. Heinz, Feb 11 2022

f[p_, e_] := (p - 1)^(e + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]--; f[k,2]++); factorback(f); \\ Michel Marcus, Feb 11 2022

a(n) = A003958(n) * |A023900(n)|.

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - (3*p^2 - 4*p + 2)/(p*(p^3 - p + 1))) = 0.1161464566... . - Amiram Eldar, Nov 19 2022

cp's OEIS Frontend

A361265 Multiplicative with a(p^e) = e * p^(e + 1), e > 0.

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A361266 Multiplicative with a(p^e) = p^(e + 3), e > 0.

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A363172 Primitive terms of A363171: terms of A363171 with no proper divisor in A363171.

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A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

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A250209 a(n) = least k such that k * n is in A072226, or 0 if no such k exists.

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A334151 Numbers k such that k / rad(k) > m / rad(m) for all m < k.

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A343881 Table read by antidiagonals upward: T(n,k) is the least integer m > k such that k^x * m^y = c^n for some positive integers c, x, and y where x < n and y < n; n >= 2, k >= 1.

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