A003973 -id:A003973 - cp's OEIS frontend

A337382 Numbers k for which A003973(k) < 2*sigma(k).

1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 116, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139, 141

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Cf. A000203, A003961, A003973.
Cf. A337381 (complement).
Positions of zeros in A337383.
Subsequence of A337379.
Cf. also A246281.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337382(n) = (A003973(n)<2*sigma(n));

A336931 Difference between the 2-adic valuation of A003973(n) [= the sum of divisors of the prime shifted n] and the 2-adic valuation of the number of divisors of n.

Original entry on oeis.org

0, 1, 0, 0, 2, 1, 1, 1, 0, 3, 0, 0, 0, 2, 2, 0, 1, 1, 2, 2, 1, 1, 0, 1, 0, 1, 0, 1, 4, 3, 0, 1, 0, 2, 3, 0, 0, 3, 0, 3, 1, 2, 3, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 5, 0, 2, 1, 1, 1, 0, 2, 1, 2, 1, 0, 4, 0, 1, 3, 1, 0, 2, 1, 1, 1, 2, 0, 2, 0, 1, 3, 4, 4, 1, 0, 3, 1, 0, 0, 1, 4, 1, 0, 1, 0, 0, 2, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Aug 17 2020

nonn

Note that A295664(n) = A295664(A003961(n)).

Cf. A003961, A003973, A007814, A007913, A295664, A336930 (positions of zeros), A336932, A336937.

A007814(n) = valuation(n, 2);
A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A007814(n) = valuation(n, 2);
A336931(n) = (A007814(A003973(n)) - A007814(numdiv(n)));

from math import prod
from sympy import factorint, nextprime, divisor_count
def A336931(n): return (~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()))& m-1).bit_length()-(~(k:=int(divisor_count(n))) & k-1).bit_length() # Chai Wah Wu, Jul 05 2022

Additive with a(p^e) = 0 when e is even, a(p^e) = A007814(1+A003961(p))-1 when e is odd.
a(n) = A336932(n) - A295664(n).
a(n) = a(A007913(n)).

A337384 Numbers k for which A003973(k) is equal to 2*sigma(k).

Original entry on oeis.org

6, 14, 15, 35, 286, 470, 715, 874, 969, 2001, 2185, 2261, 3021, 4669, 7049, 10509, 24521, 30362, 34694, 34918, 46189, 54610, 58102, 58179, 62698, 65570, 69513, 73628, 75905, 79431, 82510, 86735, 87295, 94658, 95381, 108862, 109810, 120524, 133023, 135751, 144001, 145255, 147572, 156745, 162197, 185339, 192062, 216717

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Cf. A000203, A003961, A003973.

Subsequence of A337381.

Select[Range[250000], If[# == 1, 1, DivisorSigma[1, Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Prime[PrimePi@ p + 1]^e] ]] == 2 DivisorSigma[1, #] &] (* Michael De Vlieger, Aug 27 2020 *)

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337384(n) = (A003973(n)==2*sigma(n));

A353796 Numbers k such that k divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 36, 44, 72, 96, 112, 128, 132, 160, 180, 220, 288, 336, 352, 360, 384, 396, 480, 528, 560, 640, 660, 880, 1044, 1056, 1152, 1232, 1344, 1404, 1440, 1680, 1760, 1920, 1980, 2088, 2352, 2376, 2464, 2496, 2640, 3168, 3600, 3696, 3920, 4032, 4400, 4736, 5220, 5280, 5376, 5760, 5824, 6075, 6144, 6160

Offset: 1

Antti Karttunen, May 12 2022

nonn

Of 5263 initial terms (terms < 2^32), only 67 are odd, and of these, only two, 1 and 1525391261 (= 503^2 * 6029) are in A007310. Of 5263 initial terms, 4653 are multiples of 3, 2331 are multiples of 81, and 3780 are multiples of 5.

Cf. A000010, A000203, A003961, A003973, A353790, A353797 (subsequence).

Cf. also A353795.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };
isA353796(n) = !(A353790(n)%n);

A336919 Numbers k such that A000005(k) does not divide A003973(k); numbers k for which A336839(k) > 1.

Original entry on oeis.org

4, 9, 16, 18, 20, 32, 36, 44, 45, 48, 49, 64, 68, 72, 80, 81, 90, 98, 99, 100, 112, 116, 124, 144, 153, 160, 162, 164, 169, 176, 180, 192, 196, 198, 208, 220, 225, 236, 240, 244, 245, 252, 256, 261, 279, 284, 288, 292, 304, 306, 320, 324, 336, 338, 340, 352, 356, 360, 361, 369, 392, 396, 400, 404, 405, 428, 432, 441

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Numbers k such that A003961(k) is not in A003601, but in A049642.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A003601, A003961, A003973, A049642, A336839.

Cf. A336918 (complement).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA336919(n) = !!(sigma(A003961(n))%numdiv(n));

A336930 Numbers k such that the 2-adic valuation of A003973(k), the sum of divisors of the prime shifted k is equal to the 2-adic valuation of the number of divisors of k.

Original entry on oeis.org

1, 3, 4, 9, 11, 12, 13, 16, 23, 25, 27, 31, 33, 36, 37, 39, 44, 47, 48, 49, 52, 59, 64, 69, 71, 75, 81, 83, 89, 92, 93, 97, 99, 100, 107, 108, 109, 111, 117, 121, 124, 131, 132, 139, 141, 143, 144, 147, 148, 151, 156, 167, 169, 176, 177, 179, 188, 191, 192, 193, 196, 207, 208, 213, 225, 227, 229, 236, 239, 243, 249, 251

Offset: 1

Antti Karttunen, Aug 17 2020

nonn

Numbers k for which A295664(k) is equal to A336932(k). Note that A295664(A003961(n)) = A295664(n).
Numbers k such that A003961(A007913(k)) [or equally, A007913(A003961(k))] is in A004613, i.e., has only prime divisors of the form 4k+1.
Subsequences include squares (A000290), and also primes p which when prime-shifted [as A003961(p)] become primes of the form 4k+1 (A002144), and all their powers as well as the products between these.

Positions of zeros in A336931.

Cf. A000290, A002144, A003961, A003973, A004613, A007814, A007913, A295664, A336918, A336932, A336937.

A007814(n) = valuation(n, 2);
A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };
isA336930(n) = !A336931(n);

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA004613(n) = (1==(n%4) && 1==factorback(factor(n)[, 1]%4)); \\ After code in A004613.
isA336930(n) = isA004613(A003961(core(n)));

from math import prod
from itertools import count, islice
from sympy import factorint, nextprime, divisor_count
def A336930_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()))& m-1).bit_length()==(~(k:=int(divisor_count(n))) & k-1).bit_length(),count(max(startvalue,1)))
A336930_list = list(islice(A336930_gen(),30)) # Chai Wah Wu, Jul 05 2022

A337335 a(n) = gcd(A048673(n), 1+A003973(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 19, 1, 13, 1, 1, 1, 13, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 24 2020

nonn

Cf. A003961, A003973, A048673, A337337, A337342 (positions where equal to A048673).

Cf. also A336850.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337335(n) = { my(s=A003961(n)); gcd((1+s)/2, 1+sigma(s)); };

a(n) = gcd(A048673(n), 1+A003973(n)) = (n) = gcd((1+A003961(n))/2, 1+sigma(A003961(n))).

For all n>= 1, a(A000290(n)) = A337337(n).

A353795 Numbers k such that k divides A353794(k), where A353794(n) = A003958(A003973(n)) * A064989(A003973(n)).

Original entry on oeis.org

1, 4, 12, 24, 36, 44, 72, 96, 112, 132, 180, 220, 360, 384, 396, 400, 480, 560, 660, 784, 832, 864, 1044, 1056, 1188, 1200, 1344, 1920, 1980, 2088, 2352, 2376, 2496, 2800, 3168, 3600, 3920, 4320, 4736, 5220, 5280, 5376, 5824, 5940, 6800, 6912, 7056, 7200, 7488, 8400, 8800, 9504, 9900, 10000, 10440, 10800, 11484

Offset: 1

Antti Karttunen, May 12 2022

nonn

Of 2608 initial terms, only 188 are not in A353796. The first three of these are: 400, 784, 832.

Cf. A000010, A003958, A003961, A003973, A064989, A353794.

Cf. also A353796.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353794(n) = { my(s=sigma(A003961(n))); (A003958(s)*A064989(s)); };
isA353795(n) = !(A353794(n)%n);

A353797 Numbers k such that kA003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) A064989(A003973(n)).

Original entry on oeis.org

1, 2, 4, 44, 132, 220, 396, 660, 1980, 3920, 4400, 8800, 11484, 13200, 13328, 22000, 26400, 30800, 39984, 57420, 66640, 74800, 92400, 119952, 149600, 199920, 224400, 269892, 277200, 448800, 523600, 599760, 673200, 771012, 1063692, 1345792, 1346400, 1570800, 3478608, 4037376, 4712400, 5664400, 6344448, 8038800, 10574080

Offset: 1

Antti Karttunen, May 12 2022

nonn

Note that A003557(A003961(n)) [= A003961(A003557(n))] is a divisor of A003972(n), therefore the set of k such that A353789(k) divides A353790(k) is a subset of this sequence.

Of 101 initial terms (terms < 2^32) all others apart from a(1) = 1 and a(2) = 2 are multiples of 4.

Subsequence of A353796.

Cf. A000010, A000203, A003557, A003961, A003972, A003973, A353789, A353790.

A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };
isA353797(n) = !(A353790(n)%(n*A003557(A003961(n))));

A355933 a(n) = A003973(n) / gcd(sigma(n), A003973(n)), where A003973(n) = sigma(A003961(n)) and A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 4, 3, 13, 4, 2, 3, 8, 31, 16, 7, 39, 9, 2, 2, 121, 10, 124, 6, 52, 9, 14, 5, 4, 57, 12, 39, 39, 16, 8, 19, 52, 7, 40, 2, 31, 21, 8, 27, 32, 22, 3, 12, 13, 124, 5, 9, 363, 7, 76, 5, 117, 10, 26, 14, 4, 9, 64, 31, 26, 34, 19, 93, 1093, 12, 7, 18, 130, 15, 8, 37, 248, 40, 28, 171, 78, 7, 18, 21, 484, 71, 88, 15, 117

Offset: 1

Antti Karttunen, Jul 22 2022

Numerator of ratio A003973(n) / A000203(n). This sequence gives the numerators when presented in its lowest terms, while A355934 gives the denominators. As both A000203 and A003973 are multiplicative sequences, their ratio is also: 1, 4/3, 3/2, 13/7, 4/3, 2/1, 3/2, 8/3, 31/13, 16/9, 7/6, 39/14, 9/7, 2/1, 2/1, 121/31, 10/9, 124/39, 6/5, etc.

Cf. A000203, A003961, A003973, A355932, A355934 (denominators).

Cf. also A341525, A349161.

f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n] / DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355933(n) = { my(u=A003973(n)); (u/gcd(sigma(n), u)); };

a(n) = A003973(n) / A355932(n) = A003973(n) / gcd(A000203(n), A003973(n)).

cp's OEIS Frontend

A337382 Numbers k for which A003973(k) < 2*sigma(k).

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A336931 Difference between the 2-adic valuation of A003973(n) [= the sum of divisors of the prime shifted n] and the 2-adic valuation of the number of divisors of n.

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A337384 Numbers k for which A003973(k) is equal to 2*sigma(k).

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A353796 Numbers k such that k divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

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A336919 Numbers k such that A000005(k) does not divide A003973(k); numbers k for which A336839(k) > 1.

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A336930 Numbers k such that the 2-adic valuation of A003973(k), the sum of divisors of the prime shifted k is equal to the 2-adic valuation of the number of divisors of k.

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A337335 a(n) = gcd(A048673(n), 1+A003973(n)).

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A353795 Numbers k such that k divides A353794(k), where A353794(n) = A003958(A003973(n)) * A064989(A003973(n)).

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A353797 Numbers k such that k*A003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

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A353797 Numbers k such that kA003557(A003961(k)) divides A353790(k), where A353790(n) = phi(A003973(n)) A064989(A003973(n)).