A336700 -id:A336700 - cp's OEIS frontend

A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, 3, 4, 7, 10, 15, 18, 23, 27, 31, 47, 57, 63, 95, 119, 127, 255, 348, 383, 415, 447, 511, 575, 695, 767, 959, 1023, 1054, 1071, 1535, 1919, 2047, 2626, 3471, 3839, 4095, 4415, 6815, 8191, 8703, 13823, 16383, 31743, 32767, 39895, 42367, 48127, 64607, 65535, 68727, 74495, 81919, 92159, 98303, 113535, 124671, 131071

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to select for numbers with a long tail of trailing 1-bits. Terms that are not in A004767 are: 1, 4, 10, 18, 57, 348, 1054, 2626, 675348, 1869741, 12371554, 14070141, 1158654378, 1673018314, etc.

Antti Karttunen, Table of n, a(n) for n = 1..159
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003961, A004767.
Subsequences: A000225, A348514 (which is also a subsequence of A387414).
For similar sequences, see A336700, A387410, A387415, A387410, A387418, A387419.

a3961[x_] := Apply[Times, Prime[PrimePi[#1] + 1]^#2 & @@@ FactorInteger[x]] - Boole[x == 1];
a265[x_] := x/2^IntegerExponent[x, 2];
Select[Range[2^17], Divisible[1 + a3961[#], a265[# + 1] ] &] (* Michael De Vlieger, Sep 01 2025 *)

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA387411(n) = !((1+A003961(n))%A000265(1+n));

A337196 The 3-adic valuation of 1+A000265(sigma(n)), where A000265 gives the odd part.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 18 2020

nonn

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

Cf. A000203, A000265, A007949, A074941, A161942, A336698, A336700, A337194, A337195.

Cf. A337197 (the first occurrence of each n).

Array[IntegerExponent[1 + #/2^IntegerExponent[#, 2], 3] &@ DivisorSigma[1, #] &, 105] (* Michael De Vlieger, Feb 22 2021 *)

A007949(n) = valuation(n,3);
A000265(n) = (n>>valuation(n,2));
A337196(n) = A007949(1+A000265(sigma(n)));

a(n) = A007949(A337194(n)) = A007949(1+A000265(A000203(n))).

a(n) = A007949(A336698(n)).

A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 639, 1023, 2047, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 126975, 131071, 204799, 229375, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A004767, A048250.

For similar sequences, see A336700, A387411, A387415, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
isA387410(n) = !((1+A000265(A048250(n)))%A000265(1+n));

A387415 Numbers k such that the odd part of (1+k) divides (1 + odd part of A001615(k)), where A001615 is Dedekind's psi-function.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 2431, 4095, 8191, 14335, 16383, 27135, 32767, 44031, 57855, 65535, 75775, 131071, 204799, 262143, 376831, 524287, 667135, 923647, 1048575, 1441791, 1632255, 2056191, 2097151, 2315775, 2744319, 4194303, 6768639, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A001615.

For similar sequences, see A336700, A387410, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
isA387415(n) = !((1+A000265(A001615(n)))%A000265(1+n));

A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 1791, 2047, 2431, 4095, 8191, 14335, 14847, 16383, 27391, 32767, 44031, 57855, 65535, 114687, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 8978431, 12058623, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A002827, A004767, A034448.

For similar sequences, see A336700, A387410, A387415, A387419.

A000265(n) = (n>>valuation(n,2));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
isA387418(n) = !((1+A000265(A034448(n)))%A000265(1+n));

A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 3, 4, 7, 15, 31, 40, 63, 127, 255, 511, 639, 1023, 2047, 2175, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. Terms 1, 4 and 40 are probably the only terms that are not in A004767.

Cf. A000225 (subsequence), A000265, A003959, A004767.

For similar sequences, see A336700, A387410, A387411, A387415, A387418.

A000265(n) = (n>>valuation(n,2));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
isA387419(n) = !((1+A000265(A003959(n)))%A000265(1+n));

A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 01 2025

Positions of 0's in this sequence is given by such numbers n that sigma(n) = 2^k * n + r, for some n >= 1, k >= 0, 0 <= r < 2^k. These would include also quasi-perfect numbers and their generalizations, numbers n such that sigma(n) = 2^k * n + 2^k - 1, for some n > 1, k > 0 (see comments in A332223), if such numbers exist. However, it is conjectured that there are no other zeros than those given by A336702.

Cf. A000203, A000523, A332223, A336700, A336701, A336702 (conjectured positions of 0's), A387422.

Cf. also A347381, A387413.

A387423[n_] := BitLength[n] - LengthWhile[Transpose[IntegerDigits[{n, DivisorSigma[1, n]}, 2][[All, ;; BitLength[n]]]], Equal @@ # &];
Array[A387423, 100] (* Paolo Xausa, Sep 03 2025 *)

A387423(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };

from os.path import commonprefix
from sympy import divisor_sigma
def A387423(n): return n.bit_length()-len(commonprefix([bin(n)[2:],bin(divisor_sigma(n))[2:]])) # Chai Wah Wu, Sep 03 2025

a(n) = (1+A000523(n)) - A387422(n).

A379483 a(n) is the number of trailing 1-bits in the binary representation of sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2024

nonn

Cf. A003961, A003973, A007814, A048673, A379222, A379482, A379483.

Cf. also A336700.

{1}~Join~Array[Length@ Last@ Split[IntegerDigits[#, 2]][[1 ;; -1 ;; 2]] &[
DivisorSigma[1,
  Apply[Times, Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]] ]^2] ] &,
    105, 2] (* Michael De Vlieger, Dec 27 2024 *)

A379483(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1); f[i, 2] *= 2); valuation(1+sigma(factorback(f)),2); };

a(n) = A007814(1+A379482(n)).

a(n) = A379222(A048673(n)).

A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).

Original entry on oeis.org

399736269009, 1013616036225, 1393148751631700625, 2998748839068013955625, 3547850289210724050225

Offset: 1

Antti Karttunen, Jan 13 2025

If there are any quasiperfect numbers, i.e., numbers x for which sigma(x) = 2*x+1, then they should occur also in this sequence.
Square roots of these terms are: 632247, 1006785, 1180317225, 54760833075, 59563833735.
Question: Are there any solutions to similar equations "Odd squares s such that 2*s is equal to bitwise-AND of 2*s and A001065(s)" and "Odd squares s such that 3*s is equal to bitwise-AND of 3*s and sigma(s)"? Such sequences would contain odd triperfect numbers, if they exist (cf. A005820, A347391, A347884). - Antti Karttunen, Aug 19 2025
a(6) > 4*10^21. - Giovanni Resta, Aug 19 2025

Paolo Cattaneo, Sui numeri quasiperfetti, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62.
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.

Odd squares in A324647.
Intersection of A016754 and A324647.
Subsequence of A325311, which is a subsequence of A005231.
Cf. A000203, A001065, A004198, A318468.
Cf. also A336700, A336701, A337339, A337342, A348742, A379474, A379503, A379505, A379949 for other conditions that quasiperfect numbers should satisfy.

k=0; forstep(n=1,oo,2, if(!((n-1)%(2^27)),print1("("n")")); if(!isprime(n) && omega(n)>=3, f = factor(n); sq=n^2; sig=prod(i=1,#f~,((f[i,1]^(1+(2*f[i,2])))-1) / (f[i,1]-1)); if(((2*sq)==bitand(2*sq, sig)), k++; print1(sq,", "))));

a(4) and a(5) from Giovanni Resta, Aug 19 2025

cp's OEIS Frontend

A387411 Numbers k such that the odd part of (1+k) divides (1+A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A337196 The 3-adic valuation of 1+A000265(sigma(n)), where A000265 gives the odd part.

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A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

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A387415 Numbers k such that the odd part of (1+k) divides (1 + odd part of A001615(k)), where A001615 is Dedekind's psi-function.

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A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

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A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

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A379483 a(n) is the number of trailing 1-bits in the binary representation of sigma(A003961(n^2)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A379490 Odd squares s such that 2*s is equal to bitwise-AND of 2*s and sigma(s).

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A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).