A052126 -id:A052126 - cp's OEIS frontend

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

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));

A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 2, 2, 4, 1, 4, 2, 3, 0, 3, 2, 5, 2, 3, 4, 6, 1, 4, 4, 3, 2, 6, 3, 4, 0, 5, 3, 4, 2, 8, 5, 5, 2, 6, 3, 8, 4, 4, 6, 8, 1, 4, 4, 4, 4, 8, 3, 6, 2, 6, 6, 10, 3, 8, 4, 4, 0, 6, 5, 9, 3, 7, 4, 7, 2, 11, 8, 5, 5, 6, 5, 8, 2, 4, 6, 10, 3, 5, 8, 7, 4, 9, 4, 6, 6, 5, 8, 7, 1, 6, 4, 6, 4, 9, 4, 9, 4, 5

Offset: 1

Antti Karttunen, Jun 29 2020

nonn

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

Cf. A000244, A052126, A087436, A171462, A335875, A335876, A335881, A335884, A335885, A335905, A335906, A335915, A336118.

A335904(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A335904(f[k,1]-1)+A335904(f[k,1]+1)))); };

Totally additive with a(2) = 0, and for odd primes p, a(p) = 1 + a(p-1) + a(p+1).
a(n) = A336118(n) +  A087436(n).
For all n >= 1, a(A335915(n)) = A336118(n).
For all n >= 1, a(n) >= A335884(n) >= A335881(n) >= A335875(n) >= A335885(n).
For all n >= 0, a(3^n) = n.

A323077 Number of iterations of map x -> (x - (largest divisor d < x)) needed to reach 1 or a prime, when starting at x = n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

When iteration is started from n, the first noncomposite reached is A006530(n), from which follows the new formula a(n) = A064097(A052126(n)) = A064097(n/A006530(n)), as A064097 is completely additive sequence. - Antti Karttunen, May 15 2020

Antti Karttunen, Table of n, a(n) for n = 1..12005
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A006530, A032742, A052126, A060681, A064097, A323076, A334202, A334203.
Cf. A334198 (positions of the records, also the first occurrence of each n).
Differs from A334201 for the first time at n=169, where a(169) = 5, while A334201(169) = 6.

Nest[Append[#1, If[PrimeOmega[#2] <= 1, 0, 1 + #1[[Max@ Differences@ Divisors[#2] ]] ]] & @@ {#, Length@ # + 1} &, {}, 105] (* Michael De Vlieger, May 26 2020 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323077(n) = if(1>=bigomega(n),0,1+A323077(A060681(n)));

If A001222(n) <= 1 [when n is 1 or a prime], a(n) = 0, otherwise a(n) = 1 + a(A060681(n)).
a(n) <= A064097(n).
a(n) = A064097(n) - A334202(n) = A064097(A052126(n)). - Antti Karttunen, May 13 2020
a(A334198(n)) = n for all n >= 0. - Antti Karttunen, May 19 2020

A325135 Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 02 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
or largest hook
  o o o o o o
  o
  o
both of which have complement
  o o o o
  o o
which has size 6, so a(715) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325134.

Table[If[n==1,0,Total[Most[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]-1]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325135(n) = if(1==n,0,(1+A056239(n)-bigomega(n)-A061395(n))); \\ Antti Karttunen, Apr 14 2019

a(n) = A056239(A325133(n)).
For n > 1:
a(n) = A056239(n) - A001222(n) - A061395(n) + 1.
a(n) = A056239(n) - A252464(n).
a(n) = A056239(n) - A325134(n) + 1.

More terms from Antti Karttunen, Apr 14 2019

A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 3, 3, 6, 3, 7, 2, 4, 4, 8, 3, 4, 5, 4, 3, 9, 3, 10, 4, 5, 6, 5, 4, 11, 7, 6, 3, 12, 4, 13, 4, 4, 8, 14, 4, 6, 4, 7, 5, 15, 5, 6, 3, 8, 9, 16, 4, 17, 10, 5, 5, 7, 5, 18, 6, 9, 5, 19, 5, 20, 11, 5, 7, 7, 6, 21, 4, 6, 12

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n.

Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is smaller, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - min(4,5) = 4.

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

The number of times k appears in the sequence is A325232(k).

Cf. A001222, A052126, A056239, A061395, A064989, A065770, A112798, A174090, A257990, A263297, A325134, A325169, A325223, A325225, A325227.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Total[primeMS[n]]-Min[Length[primeMS[n]],Max[primeMS[n]]]],{n,100}]

a(n) = A056239(n) - min(A001222(n), A061395(n)) = A056239(n) - A325225(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));

A087039 If n is prime then 1 else 2nd largest prime factor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 5, 2, 3, 2, 1, 3, 1, 2, 3, 2, 5, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 7, 5, 3, 2, 1, 3, 5, 2, 3, 2, 1, 3, 1, 2, 3, 2, 5, 3, 1, 2, 3, 5, 1, 3, 1, 2, 5, 2, 7, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 3, 7, 2, 3, 2, 5, 2, 1, 7, 3, 5, 1, 3

Offset: 1

Reinhard Zumkeller, Aug 01 2003

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A002808, A006530, A085392, A087040.

Cf. A027746, A052126.

a087039 n | null ps   = 1
          | otherwise = head ps
          where ps = tail $ reverse $ a027746_row n
-- Reinhard Zumkeller, Oct 03 2012

A087039 := proc(n)
    local pset ,t;
    if isprime(n) or n= 1 then
        1;
    else
        pset := [] ;
        for p in ifactors(n)[2] do
            pset := [op(pset),seq(op(1,p),t=1..op(2,p))] ;
        end do:
        op(-2,sort(pset)) ;
    end if;
end proc: # R. J. Mathar, Sep 14 2012

gpf[n_] := FactorInteger[n][[-1, 1]];
a[n_] := If[PrimeQ[n], 1, gpf[n/gpf[n]]];
Array[a, 105] (* Jean-François Alcover, Dec 16 2021 *)

from sympy import factorint
def a(n):
    pf = factorint(n, multiple=True)
    return 1 if len(pf) < 2 else pf[-2]
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 16 2021

a(n) = A006530(A052126(n)) = A006530(n/A006530(n));

A087040(n) = a(A002808(n)).

cp's OEIS Frontend

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

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

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A335904 Fully additive with a(2) = 0, and a(p) = 1+a(p-1)+a(p+1), for odd primes p.

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A323077 Number of iterations of map x -> (x - (largest divisor d < x)) needed to reach 1 or a prime, when starting at x = n.

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A325135 Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.

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A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

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

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

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