A174090 -id:A174090 - cp's OEIS frontend

A383962 Irregular triangle read by rows: T(n,k) is the index of the k-th odd divisor in the list of divisors of n, with n >= 1, k >= 1.

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

Offset: 1

Omar E. Pol, May 26 2025

Row n lists the indices of the odd divisors in the list of divisors of n.
If n is odd then row n lists the first A000005(n) positive integers (A000027).
Row n is [1] if and only if n is a power of 2 (A000079).
Row n is [1, 2] if and only if n is an odd prime (A065091).

			Triangle begins (n = 1..21):
  1;
  1;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1;
  1, 2, 3;
  1, 3;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2, 3, 4;
  1;
  1, 2;
  1, 3, 5;
  1, 2;
  1, 4;
  1, 2, 3, 4;
  ...
For n = 20 the divisors of 20 are [1, 2, 4, 5, 10, 20]. The odd divisors are [1, 5] and their indices in the list of divisors are [1, 4] respectively, so the 20th row of the triangle is [1, 4].

Column 1 gives A000012.
Row lengths gives A001227.
Right border gives A383401.
Cf. A000005, A000027, A000079, A027750, A065091, A174090, A182469.

row[n_] := Position[Divisors[n], ?OddQ] // Flatten; Array[row, 45] // Flatten (* _Amiram Eldar, May 26 2025 *)

A166159 Numbers k such that phi(k) + number of perfect partitions of (k-1) = k.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 11, 12, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 60, 61, 64, 67, 71, 73, 79, 80, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Juri-Stepan Gerasimov, Oct 08 2009

nonn

Numbers k such that A000010(k) + A002033(k-1) = k.

Also numbers k such that A000010(k) + A074206(k) = k. Union of the primes (A000040), the powers of 2 (A000079) above 1, and the terms 12, 60, 80, 448, 528, 560, 3648, 4560, 11264, 22272, 24320, 53248, 125952, 146432, 1114112, 3489792, 3850240, 4145152, 4980736, 12931072, 17498112, 19333120, 20905984, 21168128, 85721088, 96468992, ... - Amiram Eldar, Feb 29 2020

Cf. A000010, A002033, A074206, A174090.

f[1] = 1; f[n_] := f[n] = Plus @@ (f /@ Most @ Divisors[n]); Select[Range[1000], f[#] + EulerPhi[#] == # &] (* Amiram Eldar, Feb 29 2020 *)

Index in the definition corrected, and extended by R. J. Mathar, Oct 10 2009

A376235 Prime numbers with Hamming weight neither prime nor a power of 2.

Original entry on oeis.org

311, 317, 347, 349, 359, 373, 461, 467, 571, 599, 619, 683, 691, 739, 797, 811, 821, 839, 853, 857, 881, 907, 937, 977, 991, 1019, 1021, 1103, 1117, 1181, 1223, 1229, 1237, 1279, 1303, 1307, 1319, 1381, 1427, 1429, 1433, 1471, 1481, 1489, 1531, 1559, 1579, 1607, 1613, 1619, 1621, 1637, 1663

Offset: 1

M. F. Hasler, Oct 24 2024

nonn

Surprisingly, all primes less than 311 have a Hamming weight (sum of their binary digits: A000120) equal to either a prime or a power of two (cf. A174090). But for larger primes, exceptions become more and more frequent.

Cf. A000120 (binary or Hamming weight), A174090 (union of primes and powers of two).

select( {is_A376235(n)=!is_A174090(hammingweight(n))&&isprime(n)}, [0..2222])

cp's OEIS Frontend

A383962 Irregular triangle read by rows: T(n,k) is the index of the k-th odd divisor in the list of divisors of n, with n >= 1, k >= 1.

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A166159 Numbers k such that phi(k) + number of perfect partitions of (k-1) = k.

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A376235 Prime numbers with Hamming weight neither prime nor a power of 2.

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