cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A348931 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) < k.

Original entry on oeis.org

5, 11, 17, 23, 29, 35, 41, 47, 49, 53, 55, 59, 65, 71, 77, 83, 85, 89, 95, 101, 107, 113, 115, 119, 125, 131, 137, 143, 145, 149, 155, 161, 167, 169, 173, 179, 185, 187, 191, 197, 203, 205, 209, 215, 221, 227, 233, 235, 239, 245, 251, 253, 257, 263, 265, 269, 275, 281, 287, 293, 295, 299, 305, 311, 317, 319, 323
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

See comments in A348930.

Links

Crossrefs

Cf. A007310, A348930, A348932, A348933.
Cf. also A348741, A348753.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[350], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348931(n) = ((n%2)&&(n%3)&&(A348930(n)

A348932 Numbers k congruent to 1 or 5 mod 6, for which A348930(k) > k.

Original entry on oeis.org

7, 13, 19, 25, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 175, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 289, 301, 307, 313, 325, 331, 337, 343, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 475, 481, 487, 499, 511, 523, 529, 541
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

See comments in A348930.

Links

Crossrefs

Cf. A007310, A348930, A348931.
Cf. also A348742, A348754.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[550], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #]] > # &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348932(n) = ((n%2)&&(n%3)&&(A348930(n)>n));

A348933 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) < k^2.

Original entry on oeis.org

7, 13, 19, 31, 35, 37, 43, 61, 65, 67, 73, 77, 79, 91, 95, 97, 103, 109, 119, 127, 133, 139, 143, 151, 155, 157, 161, 163, 175, 181, 185, 193, 199, 203, 209, 211, 215, 217, 221, 223, 229, 241, 247, 259, 271, 277, 283, 287, 299, 301, 305, 307, 313, 323, 325, 329, 331, 335, 337, 341, 349, 365, 367, 371, 373, 377, 379
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Any hypothetical odd term y of A005820 must by necessity be a square. If y is also a nonmultiple of 3, then the square root x = A000196(y) of such a number y must satisfy the condition that for all nontrivial unitary divisor pairs d and x/d [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348934. The explanation is similar to the one given in A348738. See also comments in A348935.

Links

Crossrefs

Cf. A000196, A005820, A348738, A348930, A348931, A348934, A348935.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[400], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348933(n) = ((n%2)&&(n%3)&&(A348930(n^2)<(n^2)));

A348934 Numbers k congruent to 1 or 5 mod 6, for which A348930(k^2) > k^2.

Original entry on oeis.org

5, 11, 17, 23, 25, 29, 41, 47, 49, 53, 55, 59, 71, 83, 85, 89, 101, 107, 113, 115, 121, 125, 131, 137, 145, 149, 167, 169, 173, 179, 187, 191, 197, 205, 227, 233, 235, 239, 245, 251, 253, 257, 263, 265, 269, 275, 281, 289, 293, 295, 311, 317, 319, 343, 347, 353, 355, 359, 361, 383, 389, 391, 401, 415, 419, 425, 431
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

See comments in A348933.

Links

Crossrefs

Cf. A348930, A348932, A348933, A348936.

Programs

  • Mathematica
    s[n_] := n / 3^IntegerExponent[n, 3]; Select[Range[450], MemberQ[{1, 5}, Mod[#, 6]] && s[DivisorSigma[1, #^2]] > #^2 &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
isA348934(n) = ((n%2)&&(n%3)&&(A348930(n^2)>(n^2)));

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97
Offset: 1

Views

Author

Antti Karttunen, Nov 02 2021

Keywords

Comments

Claim: If there is an odd term y of A336702 larger than one, and it is the least one of such terms, then it should satisfy condition that for all nontrivial unitary divisor pairs d and x/d of x = A064989(y) [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348739. Proof: Applying A064989 to the odd terms of A336702 gives the fixed points of A326042. Suppose there are other odd terms in A336702 in addition to its initial 1, and let y be the least of these odd terms > 1 and x = A064989(y). Because A326042 (from here on indicated with f) is multiplicative, it follows that if we take any two nontrivial unitary divisors a and b of x, with x = a*b, gcd(a,b) = 1, 1 < a,b < x, then f(a)*f(b) = f(x) = x. Because f(x)/x = 1, we must have f(a)/a * f(b)/b = 1, as also the ratio f(n)/n is multiplicative. But f(a)/a and f(b)/b cannot be equal to 1, because then a and b would also be fixed by f, which contradicts our assumption that x were the least such fixed point larger than one. Therefore f(a) < a and f(b) > b, or vice versa. See also the comments in A348930, A348933.
Moreover, all odd perfect numbers (a subsequence of A336702), if such numbers exist, should also satisfy the same condition, regardless of whether they are the least of such numbers or not, because having a non-deficient proper divisor will push the abundancy index (ratio sigma(n)/n) of any number over 2. That is, for any such pair of nontrivial unitary divisors d and x/d, both A003961(d) and A003961(x/d) should be deficient, i.e., neither one should be in A337386. See also the condition given in A347383.
Terms that occur also in A337386 are: 120, 240, 360, 420, 480, 504, 540, 600, 630, ...

Links

Crossrefs

Positions of positive terms in A348736, positions of 1's in A348737 (characteristic function).
Almost complement of A348739.
Cf. A000203, A003961, A003973, A006039, A027687, A064989, A326042, A336702, A337386, A347383, A348741, A348748 (corresponding odd numbers).
Subsequences: A000040, A374464 (after its initial 1).
Cf. also A348930, A348933.

Programs

  • Mathematica
    f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[100], s1[DivisorSigma[1, s2[#]]] < # &] (* Amiram Eldar, Nov 04 2021 *)
  • PARI
    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); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
isA348738(n) = (A326042(n)
Showing 1-5 of 5 results.

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