A329697 -id:A329697 - cp's OEIS frontend

A334093 Primes p for which A329697(p) == 3.

19, 23, 29, 31, 37, 53, 61, 73, 83, 89, 101, 103, 113, 241, 353, 389, 401, 409, 449, 577, 773, 1097, 1153, 1283, 1361, 1409, 1543, 1553, 1601, 3089, 3329, 5441, 6529, 7681, 13313, 15361, 17477, 18433, 25601, 26113, 49157, 49409, 61441, 82241, 83969, 87041, 98689, 114689, 147457, 295937, 327689, 328961, 417793

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334102(n) + 1, for some n >= 1.

Cf. A329697, A334102, primes in A334103.

Cf. also A019434, A334092, A334094, A334095, A334096.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334093(n) = (isprime(n)&&(3==A329697(n)));

A334094 Primes p for which A329697(p) == 4.

Original entry on oeis.org

43, 47, 59, 67, 71, 79, 107, 109, 131, 149, 151, 157, 167, 179, 181, 227, 233, 239, 251, 281, 293, 307, 313, 337, 433, 443, 521, 593, 601, 613, 673, 809, 821, 823, 881, 929, 953, 971, 977, 1021, 1201, 1217, 1249, 1637, 1697, 1931, 2081, 2113, 2309, 2657, 2689, 2741, 2789, 2819, 3203, 3209, 3299, 3457, 3469, 3593, 3617, 3847, 3881, 4001

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334103(n) + 1, for some n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..330; all terms < 2^31

Primes in A334104.
Cf. A000040, A010051, A052126, A171462, A329697, A334103.
Cf. also A019434, A334092, A334093, A334095, A334096.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334094(n) = (isprime(n)&&(4==A329697(n)));

A334106 Numbers n for which A329697(n) == 6.

Original entry on oeis.org

283, 301, 329, 343, 347, 361, 379, 381, 383, 387, 399, 413, 417, 419, 423, 431, 437, 441, 463, 469, 473, 483, 487, 489, 491, 497, 509, 513, 517, 519, 523, 529, 531, 539, 547, 551, 553, 557, 559, 566, 567, 571, 573, 589, 591, 597, 599, 602, 603, 609, 611, 621, 627, 631, 633, 635, 637, 639, 643, 645, 649, 651, 653, 658, 665

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12096; all terms < 2^17

Row 6 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334103, A334104, A334105.
Cf. A334096 (primes present).

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334106(n) = (6==A329697(n));

A334095 Primes p for which A329697(p) == 5.

Original entry on oeis.org

127, 139, 163, 173, 191, 197, 199, 211, 223, 229, 263, 269, 271, 277, 311, 317, 331, 349, 359, 367, 373, 397, 421, 439, 457, 461, 467, 479, 499, 503, 541, 563, 569, 587, 607, 617, 619, 647, 661, 677, 701, 733, 739, 751, 761, 857, 877, 887, 919, 937, 997, 1009, 1031, 1049, 1061, 1069, 1123, 1187, 1193, 1213, 1229, 1231

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334104(n) + 1, for some n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..1454; all terms < 2^31

Primes in A334105.
Cf. A000040, A010051, A052126, A171462, A329697, A334104.
Cf. also A019434, A334092, A334093, A334094, A334096.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334095(n) = (isprime(n)&&(5==A329697(n)));

A334096 Primes p for which A329697(p) == 6.

Original entry on oeis.org

283, 347, 379, 383, 419, 431, 463, 487, 491, 509, 523, 547, 557, 571, 599, 631, 643, 653, 683, 691, 709, 719, 727, 743, 757, 787, 797, 811, 829, 853, 859, 907, 911, 941, 991, 1013, 1033, 1051, 1087, 1091, 1093, 1109, 1117, 1129, 1151, 1163, 1171, 1181, 1277, 1289, 1381, 1399, 1451, 1453, 1493, 1511, 1523, 1559, 1571, 1583, 1607

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Primes p of the form of the form A334105(n) + 1, for some n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..4827; all terms < 2^31

Primes in A334106.
Cf. A000040, A010051, A052126, A171462, A329697, A334105.
Cf. also A019434, A334092, A334093, A334094, A334095.

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334096(n) = (isprime(n)&&(6==A329697(n)));

A334105 Numbers m for which A329697(m) = 5.

Original entry on oeis.org

127, 129, 133, 139, 141, 147, 161, 163, 171, 173, 177, 189, 191, 197, 199, 201, 203, 207, 209, 211, 213, 215, 217, 223, 229, 231, 235, 237, 243, 245, 247, 253, 254, 258, 259, 261, 263, 266, 269, 271, 273, 277, 278, 279, 282, 285, 294, 295, 297, 299, 311, 315, 317, 319, 321, 322, 326, 327, 331, 333, 335, 341, 342, 345, 346, 349, 351

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

			127 = 63*2 + 1 is a term, as 127 is a prime and 63 is in A334104 as A329697(63) = 4.
2^32 -1 = 4294967295 = 3*5*17*257*65537 is a term as it is a product of five Fermat primes, thus in five steps all odd primes can be eliminated with p -> (p-1) map.
Likewise for 1442840405 = 5 * 17 * 257^3. (The first term with binary weight = 24).

Antti Karttunen, Table of n, a(n) for n = 1..50071; all terms < 2^31

Row 5 of A334100.
Cf. A052126, A171462, A209229, A329697, A334101, A334102, A334103, A334104, A334106.
Cf. A334095 (primes present).

Position[Array[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 360], 5][[All, 1]] (* Michael De Vlieger, Apr 30 2020 *)

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
isA334105(n) = (5==A329697(n));

A334107 a(n) = A329697(A122111(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001248, A008578 (positions of zeros), A064989, A105560, A122111, A322865, A329697, A334108, A334109, A334201.

Map[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, Array[Times @@ Table[Prime[LengthWhile[#1, # >= j &] /. 0 -> 1], {j, #2}] & @@ {#, Max[#]} &@ PrimePi@ Flatten[ConstantArray[#1, {#2}] & @@@ FactorInteger@ #] &, 105] ] (* Michael De Vlieger, May 14 2020, after Robert G. Wilson v at A329697 *)

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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334107(n) = A329697(A122111(n));

a(n) = A329697(A122111(n)) = A329697(A322865(n)).
a(n) = A329697(A105560(n)) + a(A064989(n)).
For n >= 1, a(A001248(n)) = n, and these seem to be also the first occurrences of each n.

A334861 a(n) = A329697(n) + A331410(n).

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 3, 0, 4, 3, 4, 2, 4, 3, 5, 0, 4, 4, 6, 3, 5, 4, 5, 2, 6, 4, 6, 3, 7, 5, 4, 0, 6, 4, 6, 4, 7, 6, 6, 3, 5, 5, 7, 4, 7, 5, 6, 2, 6, 6, 6, 4, 7, 6, 7, 3, 8, 7, 8, 5, 5, 4, 7, 0, 7, 6, 8, 4, 7, 6, 7, 4, 8, 7, 8, 6, 7, 6, 7, 3, 8, 5, 6, 5, 7, 7, 9, 4, 8, 7, 7, 5, 6, 6, 9, 2, 5, 6, 8, 6, 8, 6, 6, 4, 8

Offset: 1

Antti Karttunen, May 14 2020

nonn

Completely additive because A329697 and A331410 are. No 1's occur as terms.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079 (positions of zeros), A329697, A331410, A334862.

A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A334861(n) = (A329697(n)+A331410(n));
\\ Or alternatively as:
A334861(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(2+A329697(f[k,1]-1)+A331410(f[k,1]+1)))); };

a(n) = A329697(n) + A331410(n).

a(2) = 0, a(p) = 2+A329697(p-1)+A331410(p+1) for odd primes p, a(m*n) = a(m)+a(n), if m,n > 1.

A335875 a(n) = min(A329697(n), A331410(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

			A329697(67) = A331410(67) = 4, thus a(67) = min(4,4) = 4.
A329697(15749) = 9 and A331410(15749) = 10, thus a(15749) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A329697, A331410, A334861, A335877, A335881, A335884, A335885, A335904.

A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A335875(n) = min(A329697(n),A331410(n));

a(n) = min(A329697(n), A331410(n)).

For all n >= 1, A335904(n) >= A335884(n) >= A335881(n) >= a(n) >= A335885(n).

A336471 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 3, 3, 5, 1, 2, 4, 6, 2, 7, 3, 6, 2, 4, 3, 8, 3, 6, 5, 6, 1, 7, 2, 7, 4, 6, 6, 7, 2, 3, 7, 9, 3, 10, 6, 9, 2, 11, 4, 5, 3, 6, 8, 7, 3, 12, 6, 9, 5, 6, 6, 13, 1, 7, 7, 9, 2, 12, 7, 9, 4, 6, 6, 10, 6, 12, 7, 9, 2, 14, 3, 6, 7, 5, 9, 12, 3, 6, 10, 12, 6, 12, 9, 12, 2, 3, 11, 13, 4, 6, 5, 6, 3, 15

Offset: 1

Antti Karttunen, Jul 22 2020

nonn

Restricted growth sequence transform of the ordered pair [A329697(n), A336158(n)].
For all i, j:
  A336470(i) = A336470(j) => a(i) = a(j)
  a(i) = a(j) => A336396(i) = A336396(j),
  a(i) = a(j) => A336469(i) = A336469(j) => A336477(i) = A336477(j).
This sequence has an ability to see where the terms of A003401 are, as they are the indices of zeros in A336469. Specifically, they are numbers k that satisfy the condition A329697(k) = A001221(A336158(k)), i.e., numbers for which A329697(k) is equal to the number of distinct prime divisors of the odd part of k. See also comments in array A334100.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A003401, A046523, A329697, A334100, A336158.

Cf. also A003602, A324400, A336159, A336160, A336396, A336469, A336470, A336472, A336473, A336477.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
Aux336471(n) = [A329697(n), A336158(n)];
v336471 = rgs_transform(vector(up_to, n, Aux336471(n)));
A336471(n) = v336471[n];

cp's OEIS Frontend

A334093 Primes p for which A329697(p) == 3.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A334094 Primes p for which A329697(p) == 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A334106 Numbers n for which A329697(n) == 6.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A334095 Primes p for which A329697(p) == 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A334096 Primes p for which A329697(p) == 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A334105 Numbers m for which A329697(m) = 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A334107 a(n) = A329697(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334861 a(n) = A329697(n) + A331410(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A335875 a(n) = min(A329697(n), A331410(n)).

Views

Author

Keywords

Examples

Links