A337479 -id:A337479 - cp's OEIS frontend

A337539 Number of primitive non-deficient numbers (A006039) dividing A337479(n).

2, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 6, 2

Offset: 1

Antti Karttunen and Peter Munn, Sep 20 2020

nonn

The numbers in A337479 are those that become a primitive nondeficient number (term of A006039) when each of their prime factors is replaced by the next larger prime number.

			Table of n, A337479(n), a(n) and the relevant divisors starts:
   n   A337479(n)  a(n)  divisors in A006039
   1      120       2     6, 20;
   2      180       2     6, 20;
   3      300       2     6, 20;
   4      420       4     6, 20, 28, 70;
   5      504       2     6, 28;
   6      630       2     6, 70;
   7      660       2     6, 20;
   8      780       2     6, 20;
   9      924       2     6, 28;
  10      990       1     6;
  11     1020       2     6, 20;
  12     1050       2     6, 70;

Antti Karttunen, Table of n, a(n) for n = 1..24814

A006039, A337479 are used to define this sequence.
See A000203 and A023196 for definitions of deficient and nondeficient.
Subsequence of A337690.
Cf. A337386.

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
isA006039(n) = ((sigma(n)==(2*n))||isA071395(n));
A337690(n) = sumdiv(n,d,isA006039(d));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
isA337479(n) = (isA337386(n)&&(1==sumdiv(n,d,isA337386(d))));
k=0; for(n=1,2^15,if(isA337479(n),k++; print1(A337690(n), ", ")));

a(n) = A337690(A337479(n)).

A006039 Primitive nondeficient numbers.

Original entry on oeis.org

6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030

Offset: 1

N. J. A. Sloane, R. K. Guy

nonn

A number n is nondeficient (A023196) iff it is abundant or perfect, that is iff A001065(n) is >= n. Since any multiple of a nondeficient number is itself nondeficient, we call a nondeficient number primitive iff all its proper divisors are deficient. - Jeppe Stig Nielsen, Nov 23 2003
Numbers whose proper multiples are all abundant, and whose proper divisors are all deficient. - Peter Munn, Sep 08 2020
As a set, shares with the sets of k-almost primes this property: no member divides another member and each positive integer not in the set is either a divisor of 1 or more members of the set or a multiple of 1 or more members of the set, but not both. See A337814 for proof etc. - Peter Munn, Apr 13 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..8671
L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jared Duker Lichtman, The reciprocal sum of primitive nondeficient numbers, Journal of Number Theory, Vol. 191 (2018), pp. 104-118.
Joshua Zelinsky, The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-deficient Number, Integers (2025) Vol. 25, Art. No. A59. See p. 2.
Index entries for sequences where any odd perfect numbers must occur

Cf. A001065 (aliquot function), A023196 (nondeficient), A005101 (abundant), A091191.
Subsequences: A000396 (perfect), A071395 (primitive abundant), A006038 (odd primitive abundant), A333967, A352739.
Positions of 1's in A341620 and in A337690.
Cf. A180332, A337479, A337688, A337689, A337691, A337814, A338133 (sorted by largest prime factor), A338427 (largest prime(n)-smooth), A341619 (characteristic function), A342669.

k = 1; lst = {}; While[k < 4050, If[DivisorSigma[1, k] >= 2 k && Min@Mod[k, lst] > 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 09 2017 *)

Union of A000396 (perfect numbers) and A071395 (primitive abundant numbers). - M. F. Hasler, Jul 30 2016

Sum_{n>=1} 1/a(n) is in the interval (0.34842, 0.37937) (Lichtman, 2018). - Amiram Eldar, Jul 15 2020

A337386 Numbers k for which A003973(k) >= 2*A003961(k).

Original entry on oeis.org

120, 180, 240, 300, 360, 420, 480, 504, 540, 600, 630, 660, 720, 780, 840, 900, 924, 960, 990, 1008, 1020, 1050, 1080, 1092, 1140, 1170, 1200, 1260, 1320, 1380, 1440, 1470, 1500, 1512, 1560, 1620, 1650, 1680, 1740, 1800, 1848, 1860, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2220, 2280, 2310, 2340, 2400, 2460

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Provided that there are no odd perfect numbers, then these are equal to numbers k for which A003961(k) is in A005231, i.e., numbers that become odd abundant numbers when prime-shifted once.

Not all terms are even. The first odd term is a(8313165) = 334639305 = A064989(A115414(1)). (See A337385). For any odd term x present, A064989(x) is also present, for example, A064989(334639305) = 19399380 = a(482324).

Cf. A000203, A003961, A003973, A005231, A115414, A336389, A337468.
Subsequence of A005101, of A337381, and of A246282.
Subsequences: A337385 (odd terms), A337479 (primitive elements).

Select[Range[2500], If[# == 1, 1, DivisorSigma[1, # ]] >= 2# &@ Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Prime[PrimePi@ p + 1]^e] &] (* Michael De Vlieger, Aug 27 2020 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = (sigma(A003961(n))>=2*A003961(n));

A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Sep 15 2020

nonn

As a simple consequence of the definition of a primitive nondeficient number, a(n) is nonzero if and only if n is nondeficient.

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = a(2) = a(3) = a(4) = a(5) = 0 as all primitive nondeficient numbers are larger, and therefore not divisors; and a(6) = 1, as only 1 primitive nondeficient number divides 6, namely 6 itself.
60 has the following 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Of these, only 6 and 20 are in A006039, thus a(60) = 2.

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

A006039 (or equivalently, its characteristic function, A341619) is used to define this sequence.
See A000203 and A023196 for definitions of deficient and nondeficient.
Sequences with similar definitions: A080224, A294927, A337539, A341620.
Cf. A302110, A341604.
Positions of 0's: A005100.
Positions of numbers >= k: A023196 (k=1), A337688 (k=2), A337689 (k=3).
Positions of first appearances are given in A337691.
Differs from its derived sequence A341618 for the first time at n=120, where a(120)=2, while A341618(120)=1.

A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d= 2*d), return(0))); (1)); \\ After code in A071395
A337690(n) = sumdiv(n, d, A341619(d));

a(n) = Sum_{d|n} A341619(d) = Sum_{d|n} [1==A341620(d)]. - Corrected by Antti Karttunen, Feb 21 2021
a(A005100(n)) = 0.
a(A006039(n)) = 1.
a(A023196(n)) >= 1.
a(A337479(n)) = A337539(n).
a(n) <= A341620(n). - Antti Karttunen, Feb 22 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A006039(n) = 0.3... (see A006039 for a better estimate of this constant). - Amiram Eldar, Jan 01 2024

Data section extended to 120 terms by Antti Karttunen, Feb 21 2021

A337538 a(n) is the least k such that A003961(k*A071395(n)) is abundant.

Original entry on oeis.org

6, 6, 15, 15, 15, 15, 15, 15, 3, 6, 6, 6, 9, 2, 15, 15, 15, 3, 15, 2, 15, 3, 15, 15, 6, 3, 2, 3, 3, 3, 9, 3, 9, 3, 3, 3, 2, 15, 6, 15, 6, 3, 2, 15, 15, 15, 3, 15, 15, 15, 3, 15, 15, 3, 15, 15, 2, 15, 15, 15, 15, 2, 3, 15, 2, 15, 15, 15, 2, 15, 15, 15, 15, 2, 15, 15, 15, 15, 15, 15, 2, 2, 15, 15, 15, 15, 2

Offset: 1

Antti Karttunen and Peter Munn, Sep 07 2020

nonn

A071395(n) is the n-th primitive abundant number. A003961(k) replaces each prime factor of k with the next larger prime.

See also the table in the example section of A337469.

Antti Karttunen, Table of n, a(n) for n = 1..13037

Cf. A000203, A003961, A003973, A071395, A337386, A337469, A337479.

Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; k] &, Select[Range[10^4], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k,", "); break))));

a(n) = A337469(n) / A071395(n).

A337469 a(n) is the least k that is a multiple of A071395(n) (the n-th primitive abundant number) for which A003961(k) is abundant.

Original entry on oeis.org

120, 420, 1320, 1560, 4080, 4560, 5520, 6960, 1650, 3432, 3900, 4488, 7524, 1890, 17760, 19680, 20640, 4290, 22560, 3150, 25440, 5610, 28320, 29280, 12012, 6270, 4410, 6630, 7410, 7590, 23256, 8970, 28152, 9570, 9690, 10230, 6930, 52440, 22620, 59160, 24180, 12210, 8190, 63240, 64320

Offset: 1

Antti Karttunen and Peter Munn, Sep 07 2020

nonn

A003961(k) replaces each prime factor of k with the next larger prime. Thus for all terms a(n), A003961(a(n)) is an odd abundant number (some of which are also primitive abundant numbers, starting with n = 1, 2, 9, 10, 12, ...).

			The table below shows a(n), for n less than 16, alongside A071395(n) and its prime factors, and the additional prime factors that are needed to produce a(n).
   n   a(n)               A071395(n)
   1    120 / (2 * 3)  =    20  =  2^2 * 5,
   2    420 / (2 * 3)  =    70  =  2 * 5 * 7,
   3   1320 / (3 * 5)  =    88  =  2^3 * 11,
   4   1560 / (3 * 5)  =   104  =  2^3 * 13,
   5   4080 / (3 * 5)  =   272  =  2^4 * 17,
   6   4560 / (3 * 5)  =   304  =  2^4 * 19,
   7   5520 / (3 * 5)  =   368  =  2^4 * 23,
   8   6960 / (3 * 5)  =   464  =  2^4 * 29,
   9   1650 / (3)      =   550  =  2 * 5^2 * 11,
  10   3432 / (2 * 3)  =   572  =  2^2 * 11 * 13,
  11   3900 / (2 * 3)  =   650  =  2 * 5^2 * 13,
  12   4488 / (2 * 3)  =   748  =  2^2 * 11 * 17,
  13   7524 / (3 * 3)  =   836  =  2^2 * 11 * 19,
  14   1890 / (2)      =   945  =  3^3 * 5 * 7,
  15  17760 / (3 * 5)  =  1184  =  2^5 * 37, ...

See A000203 and A005101 for the definition of abundant.
A003961 and A071395 are used to define the sequence.
Sequences with related definitions: A337386, A337479, A337538.
Cf. A003973.

Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; # k] &, Select[Range[5*10^3], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k*n,", "); break))));

a(n) = A071395(n) * A337538(n).

A341604 Those primitive elements of A337386 that have exactly one primitive nondeficient divisor (A006039).

Original entry on oeis.org

990, 1170, 4590, 7650, 8550, 19470, 23562, 23868, 26334, 27324, 27846, 31050, 31878, 34452, 35190, 39330, 40194, 44370, 47430, 49590, 53010, 56610, 60030, 62730, 63270, 64170, 65790, 70110, 71910, 73530, 76590, 80370, 80910, 81090, 84870, 90270, 90630, 93330, 93366, 100890, 102510, 104310, 108630, 111690, 117450

Offset: 1

Antti Karttunen, Feb 20 2021

nonn

Terms k of A337479 for which A337690(k) = 1.

Cf. A003961, A006039, A337386, A337479, A337539, A337690.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
isA337479(n) = (isA337386(n)&&(1==sumdiv(n,d,isA337386(d))));
isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code  in A071395
isA006039(n) = ((sigma(n)==(2*n))||isA071395(n));
A337690(n) = sumdiv(n,d,isA006039(d));
isA341604(n) = (isA337479(n)&&(1==A337690(n)));

cp's OEIS Frontend

A337539 Number of primitive non-deficient numbers (A006039) dividing A337479(n).

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A006039 Primitive nondeficient numbers.

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A337386 Numbers k for which A003973(k) >= 2*A003961(k).

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A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

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A337538 a(n) is the least k such that A003961(k*A071395(n)) is abundant.

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A337469 a(n) is the least k that is a multiple of A071395(n) (the n-th primitive abundant number) for which A003961(k) is abundant.

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A341604 Those primitive elements of A337386 that have exactly one primitive nondeficient divisor (A006039).

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