A003973 -id:A003973 - cp's OEIS frontend

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

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));

A337381 Numbers k for which A003973(k) >= 2*sigma(k).

Original entry on oeis.org

6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Note that A003973(n) >= sigma(n) for all n. See A336852.
Like the abundancy index (ratio A000203(n)/n), and ratio A003961(n)/n, the ratio A003973(n)/sigma(n) is also multiplicative and > 1 for all n > 1. Thus if the number has a proper divisor that is in this sequence, then the number itself is also. See A337543 for those terms included here, but which have no proper divisor in this sequence. - Antti Karttunen, Aug 31 2020
All terms are in A246282 because A341528(n) < A341529(n) for all n > 1. - Antti Karttunen, Feb 22 2021

Cf. A000203, A003961, A003973, A336852, A337541, A337542, A341528, A341529.
Cf. A337382 (complement), A337383 (characteristic function).
Subsequences: A337378, A337384, A337386, A337543 (primitive terms).
Subsequence of A246282.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337381(n) = (A003973(n)>=2*sigma(n));

A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

Original entry on oeis.org

0, 2, 4, 14, 6, 36, 10, 68, 44, 52, 12, 192, 16, 84, 92, 284, 18, 326, 22, 274, 148, 100, 28, 840, 90, 132, 344, 438, 30, 648, 36, 1094, 176, 148, 212, 1622, 40, 180, 232, 1192, 42, 1032, 46, 520, 802, 228, 52, 3324, 230, 654, 260, 684, 58, 2376, 252, 1896, 316, 244, 60, 3156, 66, 292, 1278, 4010, 332, 1224, 70, 766

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

All terms are even because A003973 and A336845 match parity-wise. Also in the sum formulas, only even terms are summed (only one of which is zero).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A003961, A048673, A094471, A336837, A336839, A336840, A336845, A336856.
Cf. A336846 [= gcd(a(n), A003973(n))].
Twice the terms of A336854.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));

A336841(n) = sumdiv(n,d,(A003961(n)-A003961(d)));

A336841(n) = sumdiv(A003961(n),d,(A003961(n)-d));

a(n) = A336845(n) - A003973(n) = (A000005(n)*A003961(n)) - A000203(A003961(n)).
a(n) = A094471(A003961(n)).
a(n) = Sum_{d|n} (A003961(n)-A003961(d)) = Sum_{d|A003961(n)} (A003961(n)-d).
a(n) = 2*A336854(n) = 2*Sum_{d|n} (A048673(n)-A048673(d)).
a(n) = ((A003961(n)+1)*A000005(n)) - 2*A336840(n).
a(n) = 2 * ((A000005(n)*A048673(n)) - A336840(n)).
a(n) = A000005(n) * (A336837(n)/A336839(n)) = A336837(n) * A336856(n).

A336848 a(n) = A003973(n) / A336846(n).

Original entry on oeis.org

1, 2, 3, 13, 4, 2, 6, 10, 31, 8, 7, 13, 9, 4, 12, 121, 10, 62, 12, 52, 18, 14, 15, 2, 19, 6, 39, 26, 16, 8, 19, 182, 21, 20, 24, 403, 21, 8, 27, 40, 22, 12, 24, 7, 124, 10, 27, 121, 133, 38, 6, 13, 30, 26, 4, 20, 36, 32, 31, 52, 34, 38, 62, 1093, 36, 14, 36, 130, 9, 16, 37, 62, 40, 14, 57, 52, 42, 18, 42, 484, 781

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

If there are no more 1's in this sequence after the initial one, then there are no odd terms of A001599 (Ore's Harmonic Numbers) larger than one.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001599, A003961, A003973, A336845, A336846, A336847, A336849.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336848(n) = { my(u=A003961(n),s=sigma(u)); (s/gcd(s, numdiv(n)*u)); };

a(n) = A003973(n) / A336846(n).

A341525 Numerator of A003973(n) / A003961(n).

Original entry on oeis.org

1, 4, 6, 13, 8, 8, 12, 40, 31, 32, 14, 26, 18, 16, 48, 121, 20, 124, 24, 104, 72, 56, 30, 16, 57, 24, 156, 52, 32, 64, 38, 364, 84, 80, 96, 403, 42, 32, 108, 320, 44, 96, 48, 14, 248, 40, 54, 242, 133, 76, 24, 26, 60, 208, 16, 160, 144, 128, 62, 208, 68, 152, 372, 1093, 144, 112, 72, 260, 36, 128, 74, 248, 80, 56

Offset: 1

Antti Karttunen, Feb 16 2021

Also numerator of the ratio (A341528(n)/A341529(n)) / (n/sigma(n)).

Cf. A000203, A000290 (positions of odd terms), A003961, A003973, A017665, A151800, A336850, A341526, A341527, A341528, A341529, A341530.

Cf. A336849 (denominators).

f[p_, e_] := (p^(e + 1) - 1)/((p - 1)*p^e); g[p_, e_] := f[NextPrime[p], e]; a[1] = 1; a[n_] := Numerator[Times @@ g @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341525(n) = { my(u=A003961(n), s=sigma(u)); (s/gcd(u, s)); };

a(n) = A017665(A003961(n)).
a(n) = A003973(n) / A336850(n) = A003973(n) / gcd(A003961(n), A003973(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A341525(k)/A336849(k) = 1 / Product_{p prime} (1 - 1/(p*nextprime(p))) = 1.3766054560..., where nextprime(p) = A151800(p). - Amiram Eldar, Dec 28 2024

A336856 Prime-shifted analog of gcd(d(n), sigma(n)): a(n) = gcd(A000005(n), A003973(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 4, 1, 4, 2, 6, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 8, 2, 8, 2, 2, 2, 4, 2, 2, 1, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 1, 4, 8, 2, 2, 4, 8, 2, 4, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4, 8, 2, 4, 4, 6, 4, 4, 4, 12, 2, 2, 2, 3, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Cf. A000005, A003961, A003973, A009205, A336837, A336838, A336839, A336840, A336841.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A336856(n) = gcd(numdiv(n), A003973(n));

a(n) = A009205(A003961(n)).
a(n) = gcd(A000005(n), A003973(n)) = gcd(A000005(n), A336841(n)).
a(n) = gcd(A000005(n), 2*A336840(n)).
a(n) = A003973(n) / A336838(n) = A000005(n) / A336839(n).
For n > 1, a(n) = A336841(n) / A336837(n).
For all primes p, and n >= 0, a(p^((2^n)-1)) = 2^n.

A336932 The 2-adic valuation of A003973(n), the sum of divisors of prime shifted n.

Original entry on oeis.org

0, 2, 1, 0, 3, 3, 2, 3, 0, 5, 1, 1, 1, 4, 4, 0, 2, 2, 3, 3, 3, 3, 1, 4, 0, 3, 2, 2, 5, 6, 1, 2, 2, 4, 5, 0, 1, 5, 2, 6, 2, 5, 4, 1, 3, 3, 1, 1, 0, 2, 3, 1, 2, 4, 4, 5, 4, 7, 1, 4, 2, 3, 2, 0, 4, 4, 3, 2, 2, 7, 1, 3, 4, 3, 1, 3, 3, 4, 2, 3, 0, 4, 1, 3, 5, 6, 6, 4, 1, 5, 3, 1, 2, 3, 6, 3, 1, 2, 1, 0, 3, 5, 2, 4, 6

Offset: 1

Antti Karttunen, Aug 16 2020

nonn

Cf. A000203, A000290 (positions of zeros), A003961, A003973, A007814, A295664, A336937.

A336932(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); valuation(sigma(factorback(f)),2); };

A007814(n) = valuation(n, 2);
A336932(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))+A007814(1+f[i, 2])-1)); };

from math import prod
from sympy import factorint, nextprime
def A336932(n): return (~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()))& m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Additive with a(p^2e) = 0, a(p^(2e-1)) = A007814(1+A003961(p)) + A007814(e).
a(n) = A007814(A003973(n)).
a(n) = A336937(A003961(n)).
For all n >= 1, a(n) >= A295664(n).

A336847 a(n) = A003973(n) - A336846(n).

Original entry on oeis.org

0, 2, 4, 12, 6, 12, 10, 36, 30, 28, 12, 72, 16, 36, 44, 120, 18, 122, 22, 102, 68, 52, 28, 120, 54, 60, 152, 150, 30, 168, 36, 362, 80, 76, 92, 402, 40, 84, 104, 312, 42, 264, 46, 156, 246, 108, 52, 720, 132, 222, 100, 216, 58, 600, 84, 456, 140, 124, 60, 612, 66, 148, 366, 1092, 140, 312, 70, 258, 160, 360, 72

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

Cf. A000005, A000203, A003961, A003973, A336845, A336846, A336848.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336847(n) = { my(u=A003961(n),s=sigma(u)); (s-gcd(s, numdiv(n)*u)); };

a(n) = A003973(n) - A336846(n).

A336918 Numbers k such that A000005(k) divides A003973(k); numbers k for which A336839(k) = 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 101

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Numbers k such that A003961(k) is in A003601. Numbers which become (or stay as) arithmetic numbers when all primes in their prime factorization are replaced by the next larger primes.

Numbers k for which A003973(k) is equal to A000005(k)*A336838(k).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Wikipedia, Arithmetic number
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A003601, A003961, A003973, A336838, A336930.
Positions of ones in A336839.
Cf. A336919 (complement).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA336918(n) = !(sigma(A003961(n))%numdiv(n));

A337342 Numbers k such that A048673(k) divides 1+A003973(k).

Original entry on oeis.org

1, 10, 584, 3824, 23008, 5033216

Offset: 1

Antti Karttunen, Aug 24 2020

Numbers k such that A048673(k) = A337335(k). Equivalently, numbers k such that (A003961(k)+1)/2 divides 1+A003973(k).
No squares larger than one in this sequence => No quasiperfect numbers. See also A337339. For any x corresponding to a quasiperfect number qp = A003961(x), the quotient (1+A003973(x)) / A048673(x) should be 4. Thus that A003961(x) should also be a member of A325311.
At least for the terms x = a(2) .. a(6) here, the quotient (1+A003973(x)) / A048673(x) = 3. The terms for which the quotient is 3 are precisely those which by prime shifting become the terms of A007593 (that are all odd), thus the terms y = A064989(A007593(n)), for n >= 1, form a subsequence of this sequence.
a(7) > 2^28.
Terms 65810851904356352, 30943274395471606363637940224, 40102483616531202199118491418624 are also in the sequence, but their positions are unknown. (Adapted from Jud McCranie's Dec 16 1999 comment in A007593).

Cf. A003961, A003973, A007593, A048673, A064989, A209922, A324899, A325311, A336700, A337335, A337339.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337342(n) = { my(s=A003961(n)); !((1+sigma(s))%((1+s)/2)); };

cp's OEIS Frontend

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

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A337381 Numbers k for which A003973(k) >= 2*sigma(k).

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A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

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A336848 a(n) = A003973(n) / A336846(n).

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A341525 Numerator of A003973(n) / A003961(n).

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A336856 Prime-shifted analog of gcd(d(n), sigma(n)): a(n) = gcd(A000005(n), A003973(n)).

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A336932 The 2-adic valuation of A003973(n), the sum of divisors of prime shifted n.

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A336847 a(n) = A003973(n) - A336846(n).

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