A057949 Numbers with more than one factorization into S-primes. See A054520 and A057948 for definition.

441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2205, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3465, 3717, 3933, 3969, 4221, 4257, 4389, 4473, 4557, 4653, 4761, 4977, 5229, 5301, 5313, 5445, 5733, 5841, 5929, 5985, 6237, 6321, 6417, 6489, 6633

Offset

1,1

Comments

Numbers with k >= 4 prime factors (with multiplicity) that are congruent to 3 mod 4, no k-1 of which are equal. - Charlie Neder, Nov 03 2018

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Example

2205 is in S = {1,5,9, ... 4i+1, ...}, 2205 = 5*9*49 = 5*21^2; 5, 9, 21 and 49 are S-primes (A057948).

Prog

(Sage)
def A057949_list(bound) :
    numterms = (bound-1)//4 + 1
    M = [1] * numterms
    for k in range(1, numterms) :
        if M[k] == 1 :
            kpower = k
            while kpower < numterms :
                step = 4*kpower+1
                for j in range(kpower, numterms, step) :
                    M[j] *= 4*k+1
                kpower = 4*kpower*k + kpower + k
    # Now M[k] contains the product of the terms p^e where p is an S-prime
    # and e is maximal such that p^e divides 4*k+1
    return [4*k+1 for k in range(numterms) if M[k] > 4*k+1]
# Eric M. Schmidt, Dec 11 2016
(PARI) ok(n)={if(n%4==1, my(f=factor(n)); my(s=[f[i, 2] | i<-[1..#f~], f[i, 1]%4==3]); vecsum(s)>=4 && vecmax(s)<vecsum(s)-1, 0)} \\ Andrew Howroyd, Nov 25 2018

Crossrefs

Cf. A054520, A057948, A057950.

Cf. A343826 (only 1 way), A343827 (exactly 2 ways), A343828 (exactly 3 ways).

Sequence in context: A207045 A252201 A252194 * A057950 A343827 A250808

Adjacent sequences: A057946 A057947 A057948 * A057950 A057951 A057952

Keyword

nonn

Author

Jud McCranie, Oct 14 2000

Extensions

Offset corrected by Eric M. Schmidt, Dec 11 2016

Status

approved