A336353 -id:A336353 - cp's OEIS frontend

A336352 Number of prime divisors of sigma(n) that are larger than the largest prime factor of n; a(1) = 0.

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

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A006530, A058062, A336353 (positions of zeros).

A336352(n) = if(1==n,0,my(gp=vecmax(factor(n)[, 1])); #select(p -> (p>gp), factor(sigma(n))[, 1]));

For all n >= 1, a(n) <= A058062(n).

A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

Original entry on oeis.org

9, 18, 25, 27, 36, 45, 49, 50, 54, 55, 63, 72, 75, 81, 90, 98, 99, 100, 108, 110, 117, 121, 125, 126, 135, 144, 147, 150, 162, 165, 169, 175, 180, 196, 198, 200, 216, 220, 225, 234, 242, 243, 245, 250, 252, 270, 275, 288, 289, 294, 300, 315, 324, 325, 330, 338, 343, 350, 360, 361, 363, 375, 385, 392, 396, 400

Offset: 1

Antti Karttunen, Aug 28 2021

nonn

Provided there do not exist any odd perfect numbers, these are numbers k for which A347240(k) >= A006530(k), as for any odd perfect number x, A347240(x) = -1 by its escape clause.
If k is included as a term, then 2*k is also present.
Not all odd squares of primes are present. For example, 67^2 and 79^2 are not included. See also A091490, which seems to be a subsequence of those exceptions.
Conjecture: There are no primes in this sequence. Checked up to the 2^20-th prime, 16290047.

			For n = 55 = 5*11, on the first iteration we get A000593(55) = 72 = 2^3 * 3^2, but both 2 and 3 are less than 11; therefore we iterate a second time to get A000593(72) = 13, which is the first value whose largest prime factor is larger than that of 55 (13 > 11), thus 55 is included in the sequence.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000593, A006530, A091490, A161942, A336353, A336361, A347240, A347241, A347243 (complement), A347244 (characteristic function).

Positions of nonzero terms in A347245.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A000265(n) = (n >> valuation(n, 2));
A000593(n) = sigma(A000265(n));
A347244(n) = { my(gpf=A006530(n)); while(n>1, n = A000593(n); if(A006530(n)>=gpf,return(1))); (0); };
isA347242(n) = A347244(n);

A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

Original entry on oeis.org

1, 6, 40, 126, 154, 204, 1716, 1914, 2772, 8580, 11264, 12090, 12540, 50960, 62790, 64350, 77748, 83200, 104720, 152320, 186116, 193440, 331890, 382720, 432768, 518364, 648788, 684684, 753480, 817344, 895356, 1083852, 1113840, 1619352, 1675044, 1743588, 1759680, 1991340, 2060322, 2360484, 2492028, 2621080, 2932800

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Numbers k for which A336314(k) = A323173(k).
Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.
Note that all terms after 1 seem to be present in A102750. This observation is equal to Ore's conjecture that there are no odd Harmonic numbers larger than one.
Also, all terms after 1 seem to be even, which would imply that apart from its initial 1, A001599 were a subsequence of A102750. However, this is false, as there are terms of A001599 not in A102750, for example 8011798098793361832960 found by David A. Corneth. Note that A122111(8011798098793361832960) = 96922193555635754403846044921625, which is thus an odd term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..61
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.

\\ Standalone program:
isA336317(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); !((x*d)%s));

A336354 Numbers k such that p^2 divides k, where p = A006530(k), the largest prime factor of k, and sigma(k) does not have any prime factor larger than p.

Original entry on oeis.org

343, 686, 1029, 1372, 1715, 2058, 2744, 3430, 4116, 4489, 5145, 6241, 6860, 8232, 8978, 9261, 10290, 10976, 12482, 13467, 13720, 17956, 18522, 18723, 18769, 20580, 22201, 22445, 24964, 26569, 26934, 31205, 31423, 32761, 32928, 35912, 36481, 37044, 37446, 37538, 40401

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

			343 = 7^3 is present, as A000203(343) = 400 = 2^4 * 5^2, with none of the prime factors > 7.
1715 = 5 * 7^3 is present, as sigma(1715) = 2400 = 2^5 * 3 * 5^2.

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A070003 and A336353.

Cf. A000203, A006530, A319988, A336352.

isA336354(n) = ((0==A336352(n))&&(1==A319988(n)));

is(n) = {if(n == 1, return(0));
	my(f = factor(n), s, fs);
	if(f[#f~, 2] < 2, return(0));
	s = sigma(f);
	fs = factor(s, f[#f~, 1]);
	fs[#fs~, 1] <= f[#f~, 1]
} \\ David A. Corneth, Jun 27 2024

A351540 Numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k).

Original entry on oeis.org

30, 51, 66, 96, 102, 120, 138, 159, 165, 174, 204, 210, 213, 246, 255, 264, 267, 282, 294, 306, 318, 321, 330, 345, 354, 357, 364, 390, 408, 426, 435, 462, 477, 480, 498, 510, 534, 537, 552, 561, 570, 591, 606, 615, 636, 642, 660, 663, 672, 678, 679, 690, 696, 699, 705, 714, 735, 745, 750, 753, 759, 760, 765, 786

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

			30 = 2 * 3 * 5 is present as sigma(30) = 72 = 2^3 * 3^2, and thus there is at least one odd prime factor (in this case 3) such that a higher power of the same prime divides the sum of divisors of the same number.

Index entries for sequences related to sigma(n)

Positions of nonzero terms in A351539.
Cf. A000203, A351541 (subsequence).
Probably subsequence: A007691 \ (A323653 U A336702).
Cf. also A336353.

Select[Range[2, 800], Function[{k, s}, AnyTrue[DeleteCases[FactorInteger[k][[All, 1]], 2], Mod[s, #^(1 + IntegerExponent[k, #])] == 0 &]] @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Feb 16 2022 *)

A351539(n) = { my(f=factor(n),s=sigma(n)); sum(k=1,#f~,(f[k,1]%2)&&(0==(s%(f[k,1]^(1+f[k,2]))))); };
isA351540(n) = (A351539(n)>0);

cp's OEIS Frontend

A336352 Number of prime divisors of sigma(n) that are larger than the largest prime factor of n; a(1) = 0.

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A347242 Numbers k such that when iterating the map x -> A000593(x), at some point before 1 is reached (after starting from x=k), a term is encountered whose largest prime factor is at least as large as A006530(k).

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A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

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A336354 Numbers k such that p^2 divides k, where p = A006530(k), the largest prime factor of k, and sigma(k) does not have any prime factor larger than p.

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A351540 Numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k).

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