A003593 -id:A003593 - cp's OEIS frontend

A264998 Number of partitions of n into distinct parts of the form 3^a*5^b or 2.

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

Offset: 0

Joseph Myers, Nov 29 2015

			15 = 15 = 9 + 5 + 1 = 9 + 3 + 2 + 1, so a(15) = 3.

Joseph Myers and Reinhard Zumkeller, Table of n, a(n) for n = 0..20000 (first 1000 terms from Joseph Myers)
British Mathematical Olympiad 2015/16, Olympiad Round 1, Problem 6, Friday, 27 November 2015.

Cf. A003593, A264997.

import Data.MemoCombinators (memo2, list, integral)
a264998 n = a264998_list !! (n-1)
a264998_list = f 0 [] (1 : 2 : tail a003593_list) where
   f u vs ws'@(w:ws) | u < w = (p' vs u) : f (u + 1) vs ws'
                     | otherwise = f u (vs ++ [w]) ws
   p' = memo2 (list integral) integral p
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p' ks (m - k) + p' ks m
-- Reinhard Zumkeller, Dec 18 2015

nmax = 100; A003593 = Select[Range[nmax], PowerMod[15, #, #] == 0 &]; CoefficientList[Series[(1 + x^2) * Product[(1 + x^(A003593[[k]])), {k, 1, Length[A003593]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 18 2015 *)

G.f.: (1+x)(1+x^2)(1+x^3)(1+x^5)(1+x^9)(1+x^15)....

A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

Original entry on oeis.org

6, 10, 12, 18, 20, 21, 24, 30, 33, 35, 36, 39, 40, 42, 48, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 78, 80, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 120, 123, 126, 129, 130, 132, 138, 140, 141, 144, 145, 147, 150

Offset: 1

Michael De Vlieger, Aug 02 2018

Complement of A000027 and union of A003593 and A229829.
Analogous to A081062 and A105115 that apply to A120944(1)=6 and A120944(2)=10, respectively.
This sequence applies to term A120944(4)=15.

			6 is in the sequence since gcd(6, 15) = 3 and 6 does not divide 15^e with integer e >= 0.
2 and 4 are not in the sequence since they are coprime to 15.
3 and 5 are not in the sequence since they are divisors of 15.
9 is not in the sequence since 9 | 15^2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A003593, A081062, A105115, A120944, A229829, A316991.

With[{nn = 150, k = 15}, Select[Range@ nn, And[1 < GCD[#, k] < #, PowerMod[k, Floor@ Log2@ nn, #] != 0] &]]

A354180 Numbers k such that d(k) = 3^i5j with i,j >= 0, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 625, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 3025, 3249

Offset: 1

Amiram Eldar, May 18 2022

nonn

All the terms are squares since their number of divisors is odd.

			4 is a term since A000005(4) = 3 = 3^1*5^0;
16 is a term since A000005(16) = 5 = 3^0*5^1;
144 is a term since A000005(144) = 15 = 3^1*5^1;

Amiram Eldar, Table of n, a(n) for n = 1..10000
Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000290, A003593.

p35Q[n_] := n == 3^IntegerExponent[n, 3] * 5^IntegerExponent[n, 5]; Select[Range[60]^2, p35Q[DivisorSigma[0, #]] &]

is(n) = n==3^valuation(n, 3)*5^valuation(n, 5); \\ A003593
isok(m) = is(numdiv(m)); \\ Michel Marcus, May 19 2022

The number of terms <= x is c*sqrt(x) + O(x^(1/6)), where c = Product_{p prime} (1 - 1/p)*(Sum_{k in A003593} 1/p^((k-1)/2)) = 0.8747347138... (Hilberdink, 2022).

A022338 Index of 5^n within sequence of numbers of form 3^i*5^j.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 45, 57, 71, 86, 103, 121, 141, 162, 184, 208, 233, 260, 288, 318, 349, 382, 416, 452, 489, 528, 568, 610, 653, 697, 743, 790, 839, 889, 941, 994, 1049, 1105, 1163, 1222, 1283, 1345, 1408, 1473, 1539, 1607, 1676, 1747, 1819, 1893, 1968

Offset: 1

Clark Kimberling

nonn

Write down the numbers 3^i * 5^j in an ordered list and then record where the powers of 5 appear.

			The first twenty odd 5-smooth numbers are 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243, 375, 405, 625, 675, 729, 1125.
In that subset, the powers of 5 occur at positions 1 (corresponding to 1), 3 (corresponding to 5), 6 (corresponding to 25), 11 (corresponding to 125) and 17 (corresponding to 625).

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A000351 (powers of 5), A003593 (odd 5-smooth numbers), A025715.

Position[Sort@ Flatten@ Table[3^i * 5^j, {i, 0, Log[3, #]}, {j, 0, Log[5, #/(3^i)]}] &[15^31], ?(IntegerQ@ Log[5, #] &)][[All, 1]] (* _Michael De Vlieger, May 22 2018 *)

From David A. Corneth, May 14 2018: (Start)
Numbers between 5^n and 5^(n + 1) are of the form 5^m * 3^j where j > 0 and so m < n.
Thus 5^n < 5^m * 3^j < 5^(n + 1) if and only if 5^(n - m) < 3^j < 5^(n - m + 1).
Taking logs give (n - m) * log(5) < j * log(3) < (n - m + 1) * log(5).
Dividing by log(3) > 0 gives (n - m) * log(5) / log(3) < j < (n - m + 1) * log(5) / log(3).
(End)

A036315 Composite numbers whose prime factors contain no digits other than 3 and 5.

Original entry on oeis.org

9, 15, 25, 27, 45, 75, 81, 125, 135, 159, 225, 243, 265, 375, 405, 477, 625, 675, 729, 795, 1059, 1125, 1215, 1325, 1431, 1765, 1875, 2025, 2187, 2385, 2809, 3125, 3177, 3375, 3645, 3975, 4293, 5295, 5625, 6075, 6561, 6625, 7155, 8427, 8825, 9375, 9531

Offset: 1

Patrick De Geest, Dec 15 1998

Products of at least two terms of A020462. - David A. Corneth, Oct 09 2020

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to prime factors.

Cf. A003593, A020462, A036302-A036325.

Select[Range[10000],SubsetQ[{3,5},Union[Flatten[IntegerDigits/@ FactorInteger[ #][[All,1]]]]]&&CompositeQ[#]&] (* Harvey P. Dale, May 30 2021 *)

Sum_{n>=1} 1/a(n) = Product_{p in A020462} (p/(p - 1)) - Sum_{p in A020462} 1/p - 1 = 0.3620363317... . - Amiram Eldar, May 22 2022

A112752 Greatest common divisors of consecutive terms of numbers of the form 3^i*5^j.

Original entry on oeis.org

1, 1, 1, 3, 5, 1, 9, 15, 3, 1, 5, 45, 9, 3, 15, 5, 25, 27, 9, 45, 15, 75, 81, 1, 125, 135, 45, 225, 243, 3, 375, 405, 5, 625, 675, 729, 9, 1125, 1215, 15, 1875, 2025, 2187, 1, 3125, 3375, 3645, 45, 5625, 6075, 6561, 3, 9375, 10125, 10935, 5, 15625, 16875, 18225

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A003593, A112757.

s = {}; m = 12; Do[n = 5^k; While[n <= 5^m, AppendTo[s, n]; n *= 3], {k, 0, m}]; Map[GCD @@ # &, Partition[Union[s], 2, 1]] (* Amiram Eldar, Feb 06 2020 *)

a(n) = gcd(A003593(n), A003593(n+1)).

A356241 a(n) is the number of distinct Fermat numbers dividing n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1

Offset: 1

Amiram Eldar, Jul 30 2022

nonn

A051179(n) is the least number k such that a(k) = n.
The asymptotic density of occurrences of 0 is 1/2.
The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/2^(2^k) = (1/2) * A007404 = 0.4082107545... .

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Number.
Wikipedia, Fermat number.

Cf. A000215, A007404, A051158, A051179, A356242.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Count[f, _?(Divisible[n, #] &)]; Array[a, 100]

a(A000215(n)) = 1.
a(A051179(n)) = n.
a(A003593(n)) = A112753(n).
a(n) <= A356242(n).
a(A080307(n)) > 0 and a(A080308(n)) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)+1) = 0.5960631721... (A051158).

A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 1, 2, 0, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 1, 4, 0, 0, 1, 2, 0, 1

Offset: 1

Amiram Eldar, Jul 30 2022

nonn

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).
A000244(n) is the least number k such that a(k) = n.
The asymptotic density of occurrences of 0 is 1/2.
The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/(2^(2^k)+1) = (1/2) * A051158 = 0.2980315860... .

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Number.
Wikipedia, Fermat number.

Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]

a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.
a(A000215(n)) = 1.
a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.
a(A003593(n)) = A112754(n).
a(n) >= A356241(n).
a(A051179(n)) = n.
a(A080307(n)) > 0 and a(A080308(n)) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)) = 0.8164215090... (A007404).

A363814 Intersection of A126706 and A055932.

Original entry on oeis.org

12, 18, 24, 36, 48, 54, 60, 72, 90, 96, 108, 120, 144, 150, 162, 180, 192, 216, 240, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458

Offset: 1

Michael De Vlieger, Dec 18 2023

Products m*P(i) of primorials P(i) = A002110(i) such that rad(m) | P(i), i > 1, m > 1, where rad(m) = A007947(m).

			Sequence contains terms k > 1 in {6 * A003586} since all are divisible by P(2) = 6 and by no prime q that does not divide 6. Therefore 12, 18, 24, etc. are in the sequence.
Sequence does not contain k > 1 in {10 * A003592} since such k are divisible by 5 but not 3. Hence, 20, 40, etc. are not in this sequence.
Sequence does not contain k > 1 in {15 * A003593} since such k are odd. Hence, 45, 135, etc. are not in this sequence, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002110, A007947, A013929, A024619, A055932, A056808, A126706, A364710.

Cf. A003586, A003592, A003593.

Select[Range[12, 1080, 2], And[AnyTrue[#2, # > 1 &], Length[#1] > 1, Union@ Differences@ PrimePi[#1] == {1}] & @@ Transpose@ FactorInteger[#] &]

Union of A056808 and A364710. - Michael De Vlieger, Jan 31 2024

A366807 a(n) = A020639(A120944(n))*A120944(n).

Original entry on oeis.org

12, 20, 28, 45, 63, 44, 52, 60, 99, 68, 175, 76, 117, 84, 92, 153, 275, 171, 116, 124, 325, 132, 207, 140, 148, 539, 156, 164, 425, 172, 261, 637, 279, 188, 475, 204, 315, 212, 220, 333, 228, 575, 236, 833, 244, 369, 387, 260, 931, 268, 276, 423, 284, 1573, 725

Offset: 1

Michael De Vlieger, Dec 16 2023

Define f(x) to be lpf(k)*k, where lpf(k) = A020639(k). This sequence contains the mappings of f(x) across composite squarefree numbers A120944.
Define sequence R_k = { m : rad(m) | k }, where rad(n) = A007947(n) and k is squarefree. Then the sequence k*R_k contains all numbers divisible by squarefree k that are also not divisible by any prime q coprime to k. It is plain to see that k is the first term in the sequence k*R_k. This sequence gives the second term in k*R_k since lpf(k) is the second term in R_k.
Permutation of A366825. Contains numbers whose prime signature has at least 2 terms, of which is 2, the rest of which are 1s.
Proper subset of A364996, which itself is contained in A126706.

			Let b(n) = A120944(n).
a(1) = 12 = 2^2*3^1 = b(1)*lpf(b(1)) = 6*lpf(6) = 6*2. In {6*A003586}, 12 is the second term.
a(2) = 20 = 2^2*5^1 = b(2)*lpf(b(2)) = 10*lpf(10) = 10*2. In {10*A003592}, 20 is the second term.
a(4) = 45 = 3^2*5^1 = b(4)*lpf(b(4)) = 15*lpf(15) = 15*3. In {15*A003593}, 45 is the second term, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A007947, A020639, A065642, A120944, A126706, A285109, A364996, A366825.

nn = 150; s = Select[Range[nn], And[SquareFreeQ[#], CompositeQ[#]] &];
Array[#*FactorInteger[#][[1, 1]] &[s[[#]]] &, Length[s]]

from math import isqrt
from sympy import primepi, mobius, primefactors
def A366807(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m*min(primefactors(m)) # Chai Wah Wu, Aug 02 2024

a(n) = A065642(A120944(n)), n > 1.

a(n) = A285109(A120944(n)).

cp's OEIS Frontend

A264998 Number of partitions of n into distinct parts of the form 3^a*5^b or 2.

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A316992 Numbers m such that 1 < gcd(m, 15) < m and m does not divide 15^e for e >= 0.

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A354180 Numbers k such that d(k) = 3^i*5*j with i,j >= 0, where d(k) is the number of divisors of k (A000005).

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A022338 Index of 5^n within sequence of numbers of form 3^i*5^j.

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A036315 Composite numbers whose prime factors contain no digits other than 3 and 5.

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A112752 Greatest common divisors of consecutive terms of numbers of the form 3^i*5^j.

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A356241 a(n) is the number of distinct Fermat numbers dividing n.

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A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

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A354180 Numbers k such that d(k) = 3^i5j with i,j >= 0, where d(k) is the number of divisors of k (A000005).